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Search: a107584 -id:a107584
Displaying 1-9 of 9 results found. page 1
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A107583 a(n) = 3^n - 3*n. +10
11
1, 0, 3, 18, 69, 228, 711, 2166, 6537, 19656, 59019, 177114, 531405, 1594284, 4782927, 14348862, 43046673, 129140112, 387420435, 1162261410, 3486784341 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n) is the number k such that the number m with n 3's and k 1's has digit product = digit sum = 3^n.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
EXAMPLE
Corresponding numbers m are 1, 3, 11133, 111111111111111111333, ...
MATHEMATICA
Table[3^m-3*m, {m, 0, 20}]
PROG
(Magma) [3^n-3*n: n in [0..30]]; // Vincenzo Librandi, Oct 22 2011
(PARI) a(n)=3^n-3*n \\ Charles R Greathouse IV, Sep 08 2012
CROSSREFS
Cf. A107584, A107585.
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, May 16 2005
EXTENSIONS
Corrected by Charles R Greathouse IV, Sep 08 2012
STATUS
approved
A107585 a(n) = 5^n - 5*n. +10
10
1, 0, 15, 110, 605, 3100, 15595, 78090, 390585, 1953080, 9765575, 48828070, 244140565, 1220703060, 6103515555, 30517578050, 152587890545, 762939453040, 3814697265535, 19073486328030, 95367431640525, 476837158203020, 2384185791015515, 11920928955078010, 59604644775390505, 298023223876953000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Numbers a(n)=k such that the number m with n 5's and k 1's has digit product = digit sum = 5^n.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-11,5)
FORMULA
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3), n >= 3. - Vincenzo Librandi, Oct 26 2011
G.f.: ( -1-26*x^2+7*x ) / ( (5*x-1)*(x-1)^2 ). - R. J. Mathar, Oct 26 2011
EXAMPLE
Corresponding numbers m are 1, 5, 11111111111111155, ...
MATHEMATICA
Table[5^m-5*m, {m, 0, 10}]
LinearRecurrence[{7, -11, 5}, {1, 0, 15}, 30] (* Harvey P. Dale, Oct 21 2015 *)
PROG
(Magma) [(5^n - 5*n): n in [0..25]]; // Vincenzo Librandi, Dec 16 2010
(PARI) a(n)=5^n-5*n \\ Charles R Greathouse IV, Oct 26 2011
CROSSREFS
Cf. A107583, A107584.
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, May 16 2005
EXTENSIONS
Corrected by Charles R Greathouse IV, Sep 08 2012
STATUS
approved
A198396 a(n) = 6^n-6*n. +10
2
1, 0, 24, 198, 1272, 7746, 46620, 279894, 1679568, 10077642, 60466116, 362796990, 2176782264, 13060693938, 78364164012, 470184984486, 2821109907360, 16926659444634, 101559956668308, 609359740010382, 3656158440062856, 21936950640377730 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).
FORMULA
a(n) = 8*a(n-1)-13*a(n-2)+6*a(n-3) for n>2.
G.f.: (1-8*x+37*x^2)/((1-6*x)*(1-x)^2). - Vincenzo Librandi, Jan 04 2013
MATHEMATICA
CoefficientList[Series[(1 - 8*x + 37*x^2)/((1 - 6*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
Table[6^n-6n, {n, 0, 30}] (* or *) LinearRecurrence[{8, -13, 6}, {1, 0, 24}, 30] (* Harvey P. Dale, Jul 25 2019 *)
PROG
(Magma) [6^n-6*n: n in [0..25]]
(PARI) a(n)=6^n-6*n \\ Charles R Greathouse IV, Jul 06 2017
CROSSREFS
Cf. A005803, A107583, A107584, A107585.
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Oct 26 2011
STATUS
approved
A198397 7^n - 7*n. +10
2
1, 0, 35, 322, 2373, 16772, 117607, 823494, 5764745, 40353544, 282475179, 1977326666, 13841287117, 96889010316, 678223072751, 4747561509838, 33232930569489, 232630513987088, 1628413597910323, 11398895185373010, 79792266297611861 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).
FORMULA
a(n) = 9*a(n-1)-15*a(n-2)+7*a(n-3) for n>2.
G.f.: (1 - 9*x + 50*x^2)/((1 - 7*x)*(1 -x)^2). - Vincenzo Librandi, Jan 04 2013
MATHEMATICA
CoefficientList[Series[(1 - 9*x + 50*x^2)/((1 - 7*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
PROG
(Magma) [7^n-7*n: n in [0..25]];
(PARI) a(n)=7^n-7*n \\ Charles R Greathouse IV, Jul 06 2017
CROSSREFS
Cf. A005803, A107583, A107584, A107585.
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Oct 26 2011
STATUS
approved
A198398 8^n - 8*n. +10
2
1, 0, 48, 488, 4064, 32728, 262096, 2097096, 16777152, 134217656, 1073741744, 8589934504, 68719476640, 549755813784, 4398046510992, 35184372088712, 281474976710528, 2251799813685112, 18014398509481840, 144115188075855720 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).
FORMULA
a(n) = 10*a(n-1)-17*a(n-2)+8*a(n-3) for n>2.
G.f.: (1 - 10*x + 65*x^2)/((1 - 8*x)*(1 -x)^2). - Vincenzo Librandi, Jan 04 2013
MATHEMATICA
CoefficientList[Series[(1 - 10*x + 65*x^2)/((1 - 8*x)*(1 -x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
LinearRecurrence[{10, -17, 8}, {1, 0, 48}, 30] (* Harvey P. Dale, Mar 13 2023 *)
PROG
(Magma) [8^n-8*n: n in [0..25]];
(PARI) a(n)=8^n-8*n \\ Charles R Greathouse IV, Jul 06 2017
CROSSREFS
Cf. A005803, A107583, A107584, A107585.
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Oct 26 2011
STATUS
approved
A198399 9^n - 9*n. +10
2
1, 0, 63, 702, 6525, 59004, 531387, 4782906, 43046649, 387420408, 3486784311, 31381059510, 282429536373, 2541865828212, 22876792454835, 205891132094514, 1853020188851697, 16677181699666416, 150094635296998959 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).
FORMULA
a(0)=1, a(1)=0, a(2)=63, a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
G.f.: (1 - 11*x + 82*x^2)/((1 - 9*x)*(1 - x)^2). - Vincenzo Librandi, Jan 04 2013
MATHEMATICA
CoefficientList[Series[(1 - 11*x + 82*x^2)/((1 - 9*x)*(1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
PROG
(Magma) [9^n-9*n: n in [0..25]];
(PARI) a(n)=9^n-9*n \\ Charles R Greathouse IV, Jul 06 2017
CROSSREFS
Cf. A005803, A107583, A107584, A107585.
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Oct 26 2011
STATUS
approved
A198400 10^n-10*n. +10
2
1, 0, 80, 970, 9960, 99950, 999940, 9999930, 99999920, 999999910, 9999999900, 99999999890, 999999999880, 9999999999870, 99999999999860, 999999999999850, 9999999999999840, 99999999999999830, 999999999999999820 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).
FORMULA
a(n) = 12*a(n-1)-21*a(n-2)+10*a(n-3) for n>2.
G.f.: (1-12*x+101*x^2)/((1-10*x)*(1-x)^2). - Vincenzo Librandi, Jul 06 2012
MATHEMATICA
CoefficientList[Series[(1-12*x+101*x^2)/((1-10*x)*(1-x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 06 2012 *)
PROG
(Magma) [10^n-10*n: n in [0..25]];
(PARI) a(n)=10^n-10*n \\ Charles R Greathouse IV, Jul 06 2017
CROSSREFS
Cf. A005803, A107583, A107584, A107585.
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Oct 26 2011
STATUS
approved
A221906 a(n) = 4^n + 4*n. +10
2
1, 8, 24, 76, 272, 1044, 4120, 16412, 65568, 262180, 1048616, 4194348, 16777264, 67108916, 268435512, 1073741884, 4294967360, 17179869252, 68719476808, 274877907020, 1099511627856, 4398046511188 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).
FORMULA
G.f.: (1 + 2*x - 15*x^2)/((1-x)^2*(1-4*x)).
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3).
MATHEMATICA
Table[(4^n + 4 n), {n, 0, 30}] (* or *) CoefficientList[Series[(1 + 2 x - 15 x^2)/((1 - x)^2 (1 - 4 x)), {x, 0, 30}], x]
PROG
(Magma) [4^n + 4*n: n in [0..30]]; /* or */ I:=[1, 8, 24]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+4*Self(n-3): n in [1..30]];
CROSSREFS
Cf. A107584, A370659.
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 02 2013
STATUS
approved
A367629 a(n) = 2^2^(n + 1) - 2^(n + 2). +10
1
8, 240, 65504, 4294967232, 18446744073709551488, 340282366920938463463374607431768211200, 115792089237316195423570985008687907853269984665640564039457584007913129639424 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) is the absolute value totals of the characteristic polynomial coefficients of n-qubit normalized Hadamard matrices, excluding the 2nd and next-to-last nonzero entries.
a(n) is also the total of the entries in row 2^(n+1) of Pascal's triangle (A007318), excluding the 2nd and next-to-last entries.
LINKS
J Gregory Moxness, Table of n, a(n) for n = 1..10
FORMULA
a(n) = A107584(2^n).
MATHEMATICA
a[n_]:= 2^2^(n + 1) - 2^(n + 2);
PROG
(Python)
def A367629(n): return (1<<(m:=1<<n+1))-(m<<1) # Chai Wah Wu, Nov 29 2023
CROSSREFS
Cf. A007318, A107584.
KEYWORD
nonn,easy
AUTHOR
J Gregory Moxness, Nov 24 2023
STATUS
approved
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Last modified August 22 02:48 EDT 2024. Contains 375354 sequences. (Running on oeis4.)