cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A305708 Expansion of e.g.f. exp(cos(x)/exp(x) - 1).

Original entry on oeis.org

1, -1, 1, 1, -11, 43, -83, -275, 3833, -21561, 51369, 375593, -5860147, 40452371, -101676235, -1409619211, 23912208945, -189650997937, 454996127889, 11250036170129, -204691511497499, 1799897065507003, -3741969787709699, -164548323889940675, 3183842522596250537, -30356999697044585833
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2018

Keywords

Examples

			exp(cos(x)/exp(x) - 1) = 1 - x + x^2/2! + x^3/3! - 11*x^4/4! + 43*x^5/5! - 83*x^6/6! - 275*x^7/7! + ...

Crossrefs

Cf. A004211, A009116, A009216, A009235, A009255, A009276, A146559, A155585.

Programs

  • Maple
    a:=series(exp(cos(x)/exp(x)-1),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Cos[x]/Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Re[(-1 - I)^k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 25}]

A362365 The sum of the coefficients of x^k in the expansion of (x + x^2 + x^3 + x^4 + x^5 + x^6)^n with k divisible by 4.

Original entry on oeis.org

1, 9, 55, 322, 1946, 11664, 69980, 419912, 2519416, 15116544, 90699280, 544195552, 3265173536, 19591041024, 117546246080, 705277476992, 4231664861056, 25389989167104, 152339935002880, 914039610015232, 5484237660094976, 32905425960566784, 197432555763399680
Offset: 1

Views

Author

Yui Chit Chan, Apr 17 2023

Keywords

Comments

a(n) is the number of ways that the sum of n labeled 6-sided dice is divisible by 4. This is important for the game called Mahjong, where the remainder of the sum of n randomly rolled dice when divided by 4 determines the starting player. Usually n=3.

Examples

			For n=2, (x + x^2 + x^3 + x^4 + x^5 + x^6)^2 = x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 5*x^8 + 4*x^9 + 3*x^10 + 2*x^11 + x^12. So a(2) = 3 + 5 + 1 = 9.

Links

Crossrefs

Cf. A000400, A009116.

Programs

  • MATLAB
    an=(6^n+(-1+1i)^n+(-1-1i)^n)/4
  • Mathematica
    LinearRecurrence[{4, 10, 12}, {1, 9, 55}, 30] (* Paolo Xausa, Aug 30 2024 *)
  • PARI
    a(n)=polcoef(lift(Mod(x+x^2+x^3+x^4+x^5+x^6, 1-x^4)^n), 0) \\ Andrew Howroyd, Apr 17 2023

Formula

a(n) = (1/4)*6^n + (2^(n/2-1))*cos(3*Pi*n/4).
a(n) = (1/4)*(6^n + (-1+i)^n + (-1-i)^n), where i is the imaginary unit.
a(n) = (1/4)*(A000400(n) + 2*A009116(n)).
a(n) = 4*a(n-1) + 10*a(n-2) + 12*a(n-3).
G.f.: x*(1 + 5*x + 9*x^2)/((1 - 6*x)*(1 + 2*x + 2*x^2)).
E.g.f.: (1/4)*exp(6*x) + cos(x)/(2*exp(x)) - 3/4.
Limit_{n->oo} a(n)/6^n = 1/4.

A138569 First differences of A137776.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 0, 4, -2, 10, -4, 20, -4, 36, 0, 64, 8, 120, 16, 240, 16, 496, 0, 1024, -32, 2080, -64, 4160, -64, 8256, 0, 16384, 128, 32640, 256, 65280, 256, 130816, 0, 262144, -512, 524800, -1024, 1049600, -1024, 2098176, 0, 4194304, 2048, 8386560, 4096, 16773120
Offset: 0

Views

Author

Paul Curtz, May 12 2008

Keywords

Links

Crossrefs

Cf. A135356.

Formula

a(n)=4a(n-2)-6a(n-4)+4a(n-6), n>7.
a(2n)=A000749(n). a(2n+1)=(-1)^(n+1)*A009116(n-1), n>0.
O.g.f.: x(1-x)(x^4-2x^2+1+x-2x^3+2x^5)/((1-2x^2)(1-2x^2+2x^4)). - R. J. Mathar, Aug 02 2008

Extensions

Edited and extended by R. J. Mathar, Aug 02 2008

A173559 a(n)= +2*a(n-2) +4*a(n-3), n>3.

Original entry on oeis.org

1, -6, -13, -27, -50, -106, -208, -412, -840, -1656, -3328, -6672, -13280, -26656, -53248, -106432, -213120, -425856, -851968, -1704192, -3407360, -6816256, -13631488, -27261952, -54528000, -109049856, -218103808, -436211712, -872407040, -1744838656
Offset: 0

Views

Author

Paul Curtz, Feb 21 2010

Keywords

Comments

Generated by scanning the diagonal of the table generated by A143025 in the top row followed by higher order differences in the other rows:
1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8,...
7, -6, 6, -7, 7, -6, 6, -7, 7, -6, 6, -7, 7,...
-13, 12, -13, 14, -13, 12, -13, 14, -13, 12,..
25, -25, 27, -27, 25, -25, 27, -27, 25, -25,..
-50, 52, -54, 52, -50, 52, -54, 52, -50, 52, ...
102, -106, 106, -102, 102, -106, 106, -102,...

Links

Programs

  • Mathematica
    LinearRecurrence[{0,2,4},{1,-6,-13,-27},30] (* Harvey P. Dale, Jan 27 2019 *)

Formula

a(n) = ( -13*2^n-2*A009116(n))/4, n>0.
a(n+1)-2*a(n) = -A137429(n-2), n>1.
G.f.: (6*x+15*x^2+19*x^3-1)/( (2*x-1) *(2*x^2+2*x+1)).
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