Search: a002193 -id:a002193
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A003007
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Number of n-level ladder expressions with A002193.
(Formerly M1145)
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+20
2
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OFFSET
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1,3
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A106537
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The decimal digits of sqrt(2) = A002193 split into groups as long as given by the string of digits themselves.
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+20
1
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1, 4142, 1, 3562, 37, 3095048801688, 72420, 969807, 85, 696, 7187537, 694807317667973799073247846210, 703885038, 75343276415727350138462309122970249248360558507372, 1264
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OFFSET
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1,2
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COMMENTS
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The decimal expansion is divided into chunks of lengths of 1, 4, 1, 4, 2 etc. Where a chunk would start with a leading zero, sizes are increased to have a length determined by 2 or more consecutive digits, as for a(6) which does not contain 1 but 13 digits.
The next chunk of the sequence will have 880 digits.
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LINKS
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EXAMPLE
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1 4142 1 3562 37 3095048801688 72420 969807 85 696 7187537... 1 4 1 4 2 13 5 6 2 3 7
Chunk "3095048801688" could not be divided into chunks of size 1, 3, 5, etc. because of the 0 (zero) in second position.
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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STATUS
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approved
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A167834
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Numbers with distinct digits appearing in partition of decimal expansion of square root of 2. (A002193)
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+20
1
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14, 142, 13562, 37, 3095, 48, 8016, 8, 8724, 2096, 9807, 8569, 6718, 753, 769480, 731, 76, 679, 73, 79, 907324, 7846210, 7038, 8503, 87534, 3276415, 72, 73501, 38462, 30912, 2970, 249, 2483605, 58, 5073, 721, 264, 412, 149709
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OFFSET
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1,1
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COMMENTS
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Start with decimal expansion of sqrt(2): 1.41421356237309504880168872420969807856967187537694807317667... Part the sequence to the sections with distinct digits: S={1,4},{1,4,2},{1,3,5,6,2},{3,7},{3,0,9,5},{0,4,8},{8,0,1,6},... Numbers from digits of s(n), leaving leading zeros: 14,142,13562,37,3095,48,8016,...
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LINKS
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 3, 4, 3
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OFFSET
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1,2
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COMMENTS
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sqrt(2)= 1. 41421354237...
Between 1 and 4 we place 2 and 3.
Between 4 and 1 we place 3 and 2.
Between 1 and 4 we place 2 and 3.
Between 4 and 2 we place 3 and so on.
This gives:
1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, ...
This could be called a walk (or promenade) on the digits of sqrt(2).
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LINKS
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PROG
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(Haskell)
a232244 n = a232244_list !! (n-1)
a232244_list = 1 : concat (zipWith w a002193_list $ tail a002193_list)
where w v u | v > u = [v - 1, v - 2 .. u]
| v < u = [v + 1 .. u]
| otherwise = [v]
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CROSSREFS
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KEYWORD
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nonn,easy,base
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AUTHOR
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STATUS
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approved
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A167835
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Length of sections with distinct digits in decimal expansion of square root of 2. (A002193)
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+20
0
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2, 3, 5, 2, 4, 3, 4, 1, 4, 4, 4, 4, 4, 3, 6, 3, 2, 3, 2, 2, 6, 7, 4, 4, 5, 7, 2, 5, 5, 5, 4, 3, 7, 2, 4, 3, 3, 3, 5, 1, 1, 4, 3, 3, 1, 2, 5, 2, 1, 4, 1, 3, 1, 3, 3, 5, 3, 5, 3, 3, 2, 6, 4, 4, 5, 6, 6
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OFFSET
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1,1
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LINKS
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A280546
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Index in A002193 of start of first occurrence of at least n consecutive equal digits in the decimal expansion of sqrt(2) after the decimal point.
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+20
0
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OFFSET
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1,1
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COMMENTS
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We index the digits of sqrt(2) = 1.4142135... starting with 1 (for the 1), 2 (for the 4), 3 (for the second 1), 4 (for the second 4), 5 (for the 2), and so on.
Find the index of the first digit of a run of n consecutive equal digits after the decimal point: this is a(n). For example, the "88" here 1414213562373095048801.. begins at the 19th digit, so a(2) = 19. - N. J. A. Sloane, Mar 20 2023
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LINKS
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MATHEMATICA
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Module[{nn=160000, s2}, s2=RealDigits[Sqrt[2], 10, nn][[1]]; Flatten[Table[ SequencePosition[ s2, PadRight[{}, k, x_], 1], {k, 7}], 1]][[;; , 1]] (* Harvey P. Dale, Mar 20 2023 *)
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PROG
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(PARI) string(n) = default(realprecision, n+10); my(x=sqrt(2)); floor(x*10^n)
digit(n) = string(n)-10*string(n-1)
searchstrpos(n) = my(x=1, i=1); while(1, my(y=x+1); while(digit(y)==digit(x), y++; i++); if(i >= n, return(x+1)); i=1; x++)
a(n) = searchstrpos(n)
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CROSSREFS
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KEYWORD
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nonn,base,more
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AUTHOR
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STATUS
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approved
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1, 8, 18, 16, 37, 26, 34, 52, 70, 90, 87, 116, 127, 112, 157, 212, 158, 192, 252, 252, 249, 272, 349, 276, 287, 478, 482, 334, 407, 478, 465, 488, 544, 698, 562, 504, 682, 698, 738, 736, 742, 880, 907, 826, 944, 848, 998, 1110, 976, 1106, 1217, 1112, 1060
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: (Sum_{j>=0} A002193(1-j) * x^j)^2.
Sum_{k>=0} a(k)/10^k = 2.
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EXAMPLE
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a(1) = 8 because the coefficient of x^1 in (1 + 4x + ... )^2 is 8.
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PROG
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(PARI) seq(n)={Vec(Ser(digits(sqrtint(2*100^n)))^2)} \\ Andrew Howroyd, Mar 04 2020
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A040000
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a(0)=1; a(n)=2 for n >= 1.
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+10
193
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1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
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OFFSET
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0,2
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COMMENTS
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Continued fraction expansion of sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))).
Inverse binomial transform of Mersenne numbers A000225(n+1) = 2^(n+1) - 1. - Paul Barry, Feb 28 2003
A Chebyshev transform of 2^n: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)). - Paul Barry, Oct 31 2004
An inverse Catalan transform of A068875 under the mapping g(x)->g(x(1-x)). A068875 can be retrieved using the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. A040000 and A068875 may be described as a Catalan pair. - Paul Barry, Nov 14 2004
Let m=2. We observe that a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k). Then there is a link with A113311 and A115291: it is the same formula with respectively m=3 and m=4. We can generalize this result with the sequence whose g.f. is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
With offset 1: number of permutations where |p(i) - p(i+1)| <= 1 for n=1,2,...,n-1. This is the identical permutation and (for n>1) its reversal.
Equals INVERT transform of bar(1, 1, -1, -1, ...).
With offset 1: minimum cardinality of the range of a periodic sequence with (least) period n. Of course the range's maximum cardinality for a purely periodic sequence with (least) period n is n. - Rick L. Shepherd, Dec 08 2014
With offset 1: n*a(1) + (n-1)*a(2) + ... + 2*a(n-1) + a(n) = n^2. - Warren Breslow, Dec 12 2014
With offset 1: decimal expansion of gamma(4) = 11/9 where gamma(n) = Cp(n)/Cv(n) is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Sep 11 2016
a(n) equals the number of binary sequences of length n where no two consecutive terms differ. Also equals the number of binary sequences of length n where no two consecutive terms are the same. - David Nacin, May 31 2017
a(n) is the period of the continued fractions for sqrt((n+2)/(n+1)) and sqrt((n+1)/(n+2)). - A.H.M. Smeets, Dec 05 2017
Also, number of self-avoiding walks and coordination sequence for the one-dimensional lattice Z. - Sean A. Irvine, Jul 27 2020
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REFERENCES
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A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
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LINKS
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FORMULA
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a(n) = 2 - 0^n; a(n) = Sum_{k=0..n} binomial(1, k). - Paul Barry, Oct 16 2004
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*2^(n-2*k)/(n-k). - Paul Barry, Oct 31 2004
Euler transform of length 2 sequence [2, -1]. - Michael Somos, Apr 16 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-v)*(u+v) - 2*v*(u-w). - Michael Somos, Apr 16 2007
a(n) = a(-n) for all n in Z (one possible extension to n<0). - Michael Somos, Apr 16 2007
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EXAMPLE
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sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))). - Harry J. Smith, Apr 21 2009
G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + ...
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MAPLE
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Digits := 100: convert(evalf(sqrt(2)), confrac, 90, 'cvgts'):
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MATHEMATICA
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PROG
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(PARI) allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(2)); for (n=0, 20000, write("b040000.txt", n, " ", x[n+1])); \\ Harry J. Smith, Apr 21 2009
(Haskell)
a040000 0 = 1; a040000 n = 2
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CROSSREFS
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See A003945 etc. for (1+x)/(1-k*x).
Prod_{0<=k<=n} a(k) = A000079(n). (End)
Cf. A000674 (boustrophedon transform).
Cf. Other continued fractions for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040002 (contfrac(sqrt(5)) = (2,4,4,...)), A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).
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KEYWORD
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AUTHOR
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STATUS
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approved
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A156035
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Decimal expansion of 3 + 2*sqrt(2).
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+10
134
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5, 8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7, 7, 0, 0, 7, 7, 5, 0, 6, 8, 6, 5, 5, 2, 8, 3, 1, 4, 5, 4, 7, 0, 0, 2
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OFFSET
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1,1
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COMMENTS
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Ratios b(n+1)/b(n) for all sequences of the form b(n) = 6*b(n-1) - b(n-2), for any initial values of b(0) and b(1), converge to this ratio.
Ratios b(n+1)/b(n) for all sequences of the form b(n) = 5*b(n-1) + 5*b(n-2) + b(n-3), for all b(0), b(1) and b(2) also converge to 3 + 2*sqrt(2). For example see A084158 (Pell Triangles).
Ratios of alternating values, b(n+2)/b(n), for all sequences of the form b(n) = 2*b(n-1) + b(n-2), also converge to 3 + 2*sqrt(2). These include A000129 (Pell Numbers). Also see A014176. (End)
Let ABCD be a square inscribed in a circle. When P is the midpoint of the arc AB, then the ratio (PC*PD)/(PA*PB) is equal to 3+2*sqrt(2). See the Mathematical Reflections link. - Michel Marcus, Jan 10 2017
Ratio between radii of the large circumscribed circle R and the small internal circle r drawn on the Sangaku tablet at Isaniwa Jinjya shrine in Ehime Prefecture (pictures in links). - Bernard Schott, Feb 25 2022
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REFERENCES
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Diogo Queiros-Condé and Michel Feidt, Fractal and Trans-scale Nature of Entropy, Iste Press and Elsevier, 2018, page 45.
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LINKS
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Bernard Ycart, Les Sangakus, Sangaku du Temple Isaniwa Jinya (in French).
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FORMULA
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Equals exp(arccosh(3)), since arccosh(x) = log(x+sqrt(x^2-1)). - Stanislav Sykora, Nov 01 2013
The periodic continued fraction is [5; [1, 4]]. - Stefano Spezia, Mar 17 2024
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EXAMPLE
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3 + 2*sqrt(2) = 5.828427124746190097603377448...
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MATHEMATICA
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PROG
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(Magma) SetDefaultRealField(RealField(100)); 3 + 2*Sqrt(2); // G. C. Greubel, Aug 21 2018
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CROSSREFS
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Cf. A104178 (decimal expansion of log_10(3+2*sqrt(2))).
Cf. A000129, A001109, A001541, A001542, A001652, A001653, A002315, A005319, A075870, A038723, A038725, A038761, A054488, A054489, A075848, A077413, A084158, A106328, A156156, A156157, A156158.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A014176
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Decimal expansion of the silver mean, 1+sqrt(2).
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+10
54
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2, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8, 8, 0, 1, 6, 8, 8, 7, 2, 4, 2, 0, 9, 6, 9, 8, 0, 7, 8, 5, 6, 9, 6, 7, 1, 8, 7, 5, 3, 7, 6, 9, 4, 8, 0, 7, 3, 1, 7, 6, 6, 7, 9, 7, 3, 7, 9, 9, 0, 7, 3, 2, 4, 7, 8, 4, 6, 2, 1, 0, 7, 0, 3, 8, 8, 5, 0, 3, 8, 7, 5, 3, 4, 3, 2, 7, 6, 4, 1, 5, 7
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OFFSET
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1,1
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COMMENTS
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Set c:=1+sqrt(2). Then the fractional part of c^n equals 1/c^n, if n odd. For even n, the fractional part of c^n is equal to 1-(1/c^n).
c:=1+sqrt(2) satisfies c-c^(-1)=floor(c)=2, hence c^n + (-c)^(-n) = round(c^n) for n>0, which follows from the general formula of A001622.
1/c = sqrt(2)-1.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).
Other examples of constants x satisfying the relation x-x^(-1)=floor(x) include A001622 (the golden ratio: where floor(x)=1) and A098316 (the "bronze" ratio: where floor(x)=3). (End)
In terms of continued fractions the constant c can be described by c=[2;2,2,2,...]. - Hieronymus Fischer, Oct 20 2010
An analog of Fermat theorem: for prime p, round(c^p) == 2 (mod p). - Vladimir Shevelev, Mar 02 2013
n*(1+sqrt(2)) is the perimeter of a 45-45-90 triangle with hypotenuse n. - Wesley Ivan Hurt, Apr 09 2016
This algebraic integer of degree 2, with minimal polynomial x^2 - 2*x - 1, is also the length ratio diagonal/side of the second largest diagonal in the regular octagon (not counting the side). The other two diagonal/side ratios are A179260 and A121601. - Wolfdieter Lang, Oct 28 2020
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
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LINKS
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FORMULA
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Silver mean = 2 + Sum_{n>=0} (-1)^n/(P(n-1)*P(n)), where P(n) is the n-th Pell number (A000129). - Vladimir Shevelev, Feb 22 2013
Limit_{n->oo} exp(asinh(cos(Pi/n))) = sqrt(2) + 1. - Geoffrey Caveney, Apr 23 2014
exp(asinh(cos(Pi/2 - log(sqrt(2)+1)*i))) = exp(asinh(sin(log(sqrt(2)+1)*i))) = i. - Geoffrey Caveney, Apr 23 2014
Equals lim_{n->oo} S(n+1, 2*sqrt(2))/S(n, 2*sqrt(2)), with the Chebyshev S(n,x) polynomial (see A049310). (End)
An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 8*k + 6 for k >= 0.
For example, taking k = 0 and k = 1 yields
sqrt(2) + 1 = 15/(6 + (1*3)/(12 + (5*7)/(12 + (9*11)/(12 + (13*15)/(12 + ... + (4*n + 1)*(4*n + 3)/(12 + ... )))))) and
sqrt(2) + 1 = (715/21) * 1/(14 + (1*3)/(28 + (5*7)/(28 + (9*11)/(28 + (13*15)/(28 + ... + (4*n + 1)*(4*n + 3)/(28 + ... )))))). (End)
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EXAMPLE
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2.414213562373095...
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MAPLE
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MATHEMATICA
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RealDigits[1 + Sqrt@ 2, 10, 111] (* Or *)
Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[
Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}], {Blue, Circle[{0, 0}, 1]}]]] Circs[4] (* Charles R Greathouse IV, Jan 14 2013 *)
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PROG
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CROSSREFS
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Apart from initial digit the same as A002193.
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KEYWORD
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AUTHOR
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STATUS
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approved
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