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.

Showing 1-5 of 5 results.

A100132 a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k) * 2^(n-3k).

Original entry on oeis.org

1, 2, 4, 8, 18, 44, 112, 288, 740, 1896, 4848, 12384, 31624, 80752, 206208, 526592, 1344784, 3434272, 8770368, 22397568, 57198368, 146071744, 373034240, 952645120, 2432840256, 6212924032, 15866403584, 40519208448, 103476899968
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Comments

Binomial transform of 1,1,1,1,3,3,7,7,41,... (g.f. (1-x)(1+x)^2/(1-2x^2-x^4)).

Links

Crossrefs

Cf. A100131, A100133.

Programs

  • Mathematica
    LinearRecurrence[{4,-4,0,2},{1,2,4,8},30] (* Harvey P. Dale, Jun 07 2016 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-2*k, 2*k)*2^(n-3*k)); \\ Michel Marcus, Oct 09 2021

Formula

G.f.: (1-2x)/((1-2x)^2-2x^4).
a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)2^(n-3k/2)(1+(-1)^k)/2. - Paul Barry, Jan 22 2005

A100133 a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k) * 3^k * 2^(n-4k).

Original entry on oeis.org

1, 2, 4, 8, 19, 50, 136, 368, 985, 2618, 6940, 18392, 48763, 129338, 343120, 910304, 2415025, 6406898, 16996852, 45090728, 119620579, 317340098, 841868632, 2233386320, 5924932489, 15718204970, 41698695820, 110622122360
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Comments

Binomial transform of 1,1,1,1,4,4,10,10,28,28,76,... (g.f. (1-x)(1+x)^2/(1-2x^2-2x^4)).

Links

Crossrefs

Cf. A100131, A100132.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-2*k,2*k) * 3^k * 2^(n-4*k)); \\ Michel Marcus, Oct 09 2021
  • PARI
    my(p=Mod('x, 'x^4-4*'x^3+4*'x^2-3)); a(n) = subst(lift(p^n),'x,2); \\ Kevin Ryde, Feb 02 2023

Formula

G.f.: (1-2x)/((1-2x)^2-3x^4).
a(n) = 4*a(n-1) - 4*a(n-2) + 3*a(n-4). [corrected by Kevin Ryde, Feb 02 2023]

A100137 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-6k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 65, 136, 296, 672, 1584, 3840, 9473, 23566, 58736, 146080, 361760, 891328, 2184961, 5331476, 12958684, 31400160, 75910320, 183220800, 441787201, 1064687642, 2565404524, 6181873208, 14899796416, 35922756992, 86635757825
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Comments

Binomial transform of 1,1,1,1,1,1,2,2,2,5,5,11,11,... with g.f. (1-x)^2(1+x)^2/(1-3x^2+3x^4-2x^6)=(1+x)(1-x^2)^2/((1-x^2)^3-x^6).

Links

Crossrefs

Cf. A024493, A100131, A100134, A100137, A100138.

Programs

  • Mathematica
    Table[Sum[Binomial[n-3k,3k]2^(n-6k),{k,0,Floor[n/6]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,0,0,1},{1,2,4,8,16,32},31] (* Harvey P. Dale, Mar 19 2015 *)

Formula

G.f.: (1-2x)^2/((1-2x)^3 - x^6).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + a(n-6).

A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 32, 21, 6, 1, 1, 7, 31, 65, 65, 31, 7, 1, 1, 8, 43, 116, 161, 116, 43, 8, 1, 1, 9, 57, 189, 341, 341, 189, 57, 9, 1, 1, 10, 73, 288, 645, 842, 645, 288, 73, 10, 1, 1, 11, 91, 417, 1121, 1827, 1827, 1121, 417, 91
Offset: 0

Views

Author

Paul Barry, May 31 2005

Keywords

Comments

Or as a square array read by antidiagonals, T(n,k) = Sum_{j=0..n} binomial(k,j)*binomial(n+k-j,k)*((j+1) mod 2).
A symmetric number triangle based on 1/(1-x^2).
The construction of a symmetric triangle in this example is general. Let f(n) be a sequence, preferably with f(0)=1. Then T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*f(j) yields a symmetric triangle. When f(n)=1^n, we get Pascal's triangle. When f(n)=2^n, we get the Delannoy triangle (see A008288). In general, f(n)=k^n yields a (1,k,1)-Pascal triangle (see A081577, A081578). Row sums of triangle are A100131. Diagonal sums of the triangle are A108351. Triangle mod 2 is A106465.

Examples

			Triangle rows begin
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 13, 13,  5,  1;
  1,  6, 21, 32, 21,  6,  1;
As a square array read by antidiagonals, rows begin
  1, 1,  1,   1,   1,    1,    1, ...
  1, 2,  3,   4,   5,    6,    7, ...
  1, 3,  7,  13,  21,   31,   43, ...
  1, 4, 13,  32,  65,  116,  189, ...
  1, 5, 21,  65, 161,  341,  645, ...
  1, 6, 31, 116, 341,  842, 1827, ...
  1, 7, 43, 189, 645, 1827, 4495, ...

Programs

  • PARI
    trgn(nn) = {for (n= 0, nn, for (k = 0, n, print1(sum(j=0, n-k, binomial(k,j)*binomial(n-j,k)*((j+1) % 2)), ", ");); print(););} \\ Michel Marcus, Sep 11 2013

Formula

Row k (and column k) has g.f. (1+C(k,2)x^2)/(1-x)^(k+1).

A209971 a(n) = A000129(n) + n.

Original entry on oeis.org

0, 2, 4, 8, 16, 34, 76, 176, 416, 994, 2388, 5752, 13872, 33474, 80796, 195040, 470848, 1136706, 2744228, 6625128, 15994448, 38613986, 93222380, 225058704, 543339744, 1311738146, 3166815988, 7645370072, 18457556080, 44560482178, 107578520380, 259717522880
Offset: 0

Views

Author

N. J. A. Sloane, Mar 25 2012

Keywords

References

  • Paul Brickman, Problem in August 2011 issue of Fibonacci Quarterly. [Brickman has several problems in this issue, and I am not sure now which one I was referring to. - N. J. A. Sloane, Jan 22 2019]

Links

Crossrefs

Cf. A000129.

Programs

  • PARI
    concat(0, Vec( 2*x*(1 - 2*x) / ((1 - x)^2*(1 - 2*x - x^2)) + O(x^50))) \\ Colin Barker, Nov 06 2017

Formula

G.f.: 2*x*(-1+2*x) / ( (x^2+2*x-1)*(x-1)^2 ). a(n) = 2*A100131(n-1). - R. J. Mathar, Mar 27 2012
From Colin Barker, Nov 06 2017: (Start)
a(n) = (-(1-sqrt(2))^n + (1+sqrt(2))^n) / (2*sqrt(2)) + n.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4) for n>3.
(End)
Showing 1-5 of 5 results.

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

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