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Number of n-level ladder expressions with A002193.
(Formerly M1145)
+20
2
1, 1, 2, 4, 8, 17, 38, 89, 208
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
The decimal digits of sqrt(2) = A002193 split into groups as long as given by the string of digits themselves.
+20
1
1, 4142, 1, 3562, 37, 3095048801688, 72420, 969807, 85, 696, 7187537, 694807317667973799073247846210, 703885038, 75343276415727350138462309122970249248360558507372, 1264
COMMENTS
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.
EXAMPLE
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.
Numbers with distinct digits appearing in partition of decimal expansion of square root of 2. ( A002193)
+20
1
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
COMMENTS
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,...
A walk based on the digits of sqrt(2) ( A002193).
+20
1
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
COMMENTS
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).
PROG
(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]
Length of sections with distinct digits in decimal expansion of square root of 2. ( A002193)
+20
0
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
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.
+20
0
2, 19, 150, 953, 2708, 32414, 158810, 4602784, 472173970, 472173970
COMMENTS
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
MATHEMATICA
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 *)
PROG
(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)
a(n) = [x^n] (Sum_{j>=0} A002193(1-j) * x^j)^2.
+20
0
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
FORMULA
G.f.: (Sum_{j>=0} A002193(1-j) * x^j)^2.
Sum_{k>=0} a(k)/10^k = 2.
EXAMPLE
a(1) = 8 because the coefficient of x^1 in (1 + 4x + ... )^2 is 8.
PROG
(PARI) seq(n)={Vec(Ser(digits(sqrtint(2*100^n)))^2)} \\ Andrew Howroyd, Mar 04 2020
a(0)=1; a(n)=2 for n >= 1.
+10
193
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
COMMENTS
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
REFERENCES
A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
FORMULA
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
EXAMPLE
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 + ...
MAPLE
Digits := 100: convert(evalf(sqrt(2)), confrac, 90, 'cvgts'):
PROG
(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
CROSSREFS
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,...)).
Decimal expansion of 3 + 2*sqrt(2).
+10
135
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
COMMENTS
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
REFERENCES
Diogo Queiros-Condé and Michel Feidt, Fractal and Trans-scale Nature of Entropy, Iste Press and Elsevier, 2018, page 45.
LINKS
Bernard Ycart, Les Sangakus, Sangaku du Temple Isaniwa Jinya (in French).
FORMULA
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
EXAMPLE
3 + 2*sqrt(2) = 5.828427124746190097603377448...
PROG
(Magma) SetDefaultRealField(RealField(100)); 3 + 2*Sqrt(2); // G. C. Greubel, Aug 21 2018
CROSSREFS
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.
Decimal expansion of the silver mean, 1+sqrt(2).
+10
54
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
COMMENTS
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
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
FORMULA
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)
MATHEMATICA
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 *)
CROSSREFS
Apart from initial digit the same as A002193.
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