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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A131300 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.

Original entry on oeis.org

1, 2, 3, 7, 14, 27, 49, 86, 147, 247, 410, 675, 1105, 1802, 2931, 4759, 7718, 12507, 20257, 32798, 53091, 85927, 139058, 225027, 364129, 589202, 953379, 1542631, 2496062, 4038747, 6534865, 10573670, 17108595, 27682327, 44790986, 72473379, 117264433, 189737882
Offset: 0

Views

Author

Gary W. Adamson, Jun 27 2007

Keywords

Comments

Row sums of A131299 and A131301.
a(n)/a(n-1) tends to phi.

Examples

			a(4) = 14 = (1 + 1 + 7 + 4 + 1).

Links

Crossrefs

Cf. A131269, A131299, A131301.

Programs

  • Magma
    /* By first comment: */ [&+[3*Binomial(n-Floor((k+1)/2), Floor(k/2))-2: k in [0..n]]: n in [0..37]]; // Bruno Berselli, May 03 2012
  • Maple
    seq(add(3*binomial(floor((n+k)/2),k)-2,k=0..n),n=0..50); # Nathaniel Johnston, Jun 29 2011
  • Mathematica
    LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 7}, 38] (* Bruno Berselli, May 03 2012 *)
  • Maxima
    makelist(expand(3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1)), n, 0, 37); /* Bruno Berselli, May 03 2012 */
  • PARI
    Vec((1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2)+O(x^38)) \\ Bruno Berselli, May 03 2012

Formula

From Bruno Berselli, May 03 2012: (Start)
G.f.: (1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2).
a(n) = (3*A131269(n)-n-1)/2.
a(n) = 3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1). (End)
a(n) = 3*A000045(n+2)-2*(n+1). - R. J. Mathar, Mar 24 2018

Extensions

Terms after a(9) from Nathaniel Johnston, Jun 29 2011
New definition from Bruno Berselli, May 03 2012

A131299 Triangle T(n,k) = 3*binomial(n-floor((k+1)/2), floor(k/2))-2 with k=0..n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 7, 4, 1, 1, 1, 10, 7, 7, 1, 1, 1, 13, 10, 16, 7, 1, 1, 1, 16, 13, 28, 16, 10, 1, 1, 1, 19, 16, 43, 28, 28, 10, 1, 1, 1, 22, 19, 61, 43, 58, 28, 13, 1, 1, 1, 25, 22, 82, 61, 103, 58, 43, 13, 1, 1, 1, 28, 25, 106, 82, 166
Offset: 0

Views

Author

Gary W. Adamson, Jun 27 2007

Keywords

Comments

Row sums are in A131300. Reversed row triangle = A131301.
From R. J. Mathar, Apr 08 2013: (Start)
The matrix inverse starts
1;
-1, 1;
0, -1, 1;
0, 3, -4, 1;
0, -6, 9, -4, 1;
0, 30, -45, 21, -7, 1;
0, -132, 198, -93, 33, -7, 1;
0, 984, -1476, 693, -246, 54, -10, 1;
0, -6756, 10134, -4758, 1689, -372, 72, -10, 1;
0, 66972, -100458, 47166, -16743, 3687, -714, 102, -13, 1;
(End)

Examples

			Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  1,  4,  1;
  1,  1,  7,  4,  1;
  1,  1, 10,  7,  7,  1;
  1,  1, 13, 10, 16,  7,  1;
  ...

Links

Crossrefs

Cf. A065941, A000012, A131300, A131301.

Programs

Formula

3*A065941 - 2*A000012 as infinite lower triangular matrices.

Extensions

Better definition from Bruno Berselli, May 03 2012
Showing 1-2 of 2 results.

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