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.

Previous Showing 11-13 of 13 results.

A174443 Generating function x/(1+4*x-8*x^2).

Original entry on oeis.org

0, 1, -4, 24, -128, 704, -3840, 20992, -114688, 626688, -3424256, 18710528, -102236160, 558628864, -3052404736, 16678649856, -91133837312, 497964548096, -2720928890880, 14867431948288, -81237158920192, 443888091267072, -2425449636429824, 13252903275855872
Offset: 0

Views

Author

Roger L. Bagula, Nov 28 2010

Keywords

Comments

Binet form: with a = 4/(1 + sqrt(3)); b = 4/(1 - sqrt(3)); we have a(n)=(a^n - b^n)/(a - b).
Alternative expressions for the roots are -2 +- 2*sqrt(3).
Limiting ratio a(n+1)/a(n) is:-5.464101615137755== -2-2*sqrt(3).

Links

Crossrefs

Cf. A180222 (unsigned version of this sequence).

Programs

  • Mathematica
    a = 4/(1 + Sqrt[3]); b = 4/(1 - Sqrt[3]);
f[n_] = (a^n - b^n)/(a - b);
Table[FullSimplify[f[n]], {n, 0, 30}]

Formula

a(0)=0, a(1)=1, a(n) = -4a(n-1) + 8a(n-2).
a(0)=0, a(n) = (-2)^(n-1)*A002605(n).

Extensions

Correction of the Mathematica code and better name by Joerg Arndt.
Formulae and edited by Ralf Stephan, Nov 27 2010

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1
Offset: 0

Views

Author

Philippe Deléham, Dec 06 2011

Keywords

Comments

Row-reversed variant of A123585. Row sums: 2^n.

Examples

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Crossrefs

Cf. Columns: A000045, A001629, A129707.
Diagonals: A010892, A099254, Antidiagonal sums: A158943.

Formula

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if n
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

A140184 a(n) = 2*a(n-1) + 16*a(n-2) + 16*a(n-3) for n>3 with a(1)=1, a(2)=14, a(3)=60.

Original entry on oeis.org

1, 14, 60, 360, 1904, 10528, 57280, 313472, 1711872, 9355776, 51117056, 279316480, 1526198272, 8339333120, 45566902272, 248982306816, 1360464379904, 7433716105216, 40618579197952, 221944046157824, 1212724817166336, 6626451640025088, 36207605093236736
Offset: 1

Views

Author

Gary W. Adamson, May 11 2008

Keywords

Examples

			a(5) = 1904 = 2*a(4) + 16*a(3) + 16*a(2) = 2*360 + 16*60 + 16*14.
a(4) = 360 since term (1,1) of X^4 = 360.

Links

Crossrefs

Cf. A180222.

Programs

  • Mathematica
    LinearRecurrence[{2,16,16},{1,14,60},40] (* or *) CoefficientList[Series[(-1-12 x-16 x^2)/(-1+2 x+16 x^2+16 x^3),{x,0,40}],x] (* Harvey P. Dale, May 03 2011 *)

Formula

G.f.: -x*(1+12*x+16*x^2) / ( (2*x+1)*(8*x^2+4*x-1) ). - Harvey P. Dale, May 03 2011
a(n) = (A180222(n+2) +(-2)^n)/2. - R. J. Mathar, Oct 08 2016
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.