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 1-3 of 3 results.

A131328 Row sums of triangle A131327.

Original entry on oeis.org

1, 4, 5, 12, 17, 32, 49, 84, 133, 220, 353, 576, 929, 1508, 2437, 3948, 6385, 10336, 16721, 27060, 43781, 70844, 114625, 185472, 300097, 485572, 785669, 1271244, 2056913, 3328160, 5385073, 8713236, 14098309, 22811548, 36909857, 59721408, 96631265, 156352676
Offset: 0

Views

Author

Gary W. Adamson, Jun 28 2007

Keywords

Comments

a(n)/a(n-1) tends to phi. (Cf. A062114).

Examples

			a(3) = 12 = sum of row 3 terms of A131327: (3 + 5 + 3 + 1).
a(3) = (9 + 3) since we add terms of A131326: (1, 3, 4, 9, 13,...) to A052952: (0, 1, 1, 3, 4,...), getting (9 + 3 ) = 12.

Links

Crossrefs

Cf. A062114, A052952, A131324, A131325, A131326, A131327.

Programs

  • PARI
    Vec((1 + 3*x - x^2) / ((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

Formula

a(n+1) = A131326(n) + A052952(n+1).
a(n) = -3*(1+(-1)^n)/2 +4*A000045(n+1). - R. J. Mathar, Aug 13 2012
G.f.: ( 1+3*x-x^2 ) / ( (x-1)*(1+x)*(x^2+x-1) ). - R. J. Mathar, Aug 13 2012
From Colin Barker, Jul 12 2017: (Start)
a(n) = (2^(1-n)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) - 3 for n even.
a(n) = (2^(1-n)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
(End)

Extensions

More terms from Colin Barker, Jul 12 2017

A131325 Triangle |3*|A049310(n,k)| - 2| read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 4, 2, 1, 1, 2, 7, 2, 1, 2, 7, 2, 10, 2, 1, 1, 2, 16, 2, 13, 2, 1, 2, 10, 2, 28, 2, 16, 2, 1, 1, 2, 28, 2, 43, 2, 19, 2, 1, 2, 13, 2, 58, 2, 61, 2, 22, 2, 1, 1, 2, 43, 2, 103, 2, 82, 2, 25, 2, 1, 2, 16, 2, 103, 2, 166, 2, 106, 2, 28, 2, 1, 1, 2, 61, 2, 208, 2, 250, 2
Offset: 0

Views

Author

Gary W. Adamson, Jun 28 2007

Keywords

Examples

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  4,  2,  1;
  1,  2,  7,  2,  1;
  2,  7,  2, 10,  2,  1;
  1,  2, 16,  2, 13,  2,  1;
  ...

Crossrefs

Cf. A049310, A131324, A131326 (row sums), A131327.

Programs

Extensions

Definition corrected by David Scambler, Aug 12 2012

A131324 2*A049310 - A000012(signed).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 7, 1, 19, 1, 11, 1, 1, 1, 1, 19, 1, 29, 1, 13, 1, 1, 1, 9, 1, 39, 1, 41, 1, 15, 1, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 28 2007

Keywords

Comments

Row sums = A062114: (1, 2, 3, 6, 9, 16, 25, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  1,  5,  1,  1;
  1,  5,  1,  7,  1,  1,
  1,  1, 11,  1,  9,  1,  1;
  1,  7,  1, 19,  1, 11,  1,  1;
  ...

Crossrefs

Cf. A049310, A062114, A131325, A131326, A131327.

Formula

2*A049310 - A000012(signed + - + - ... by columns).
Showing 1-3 of 3 results.

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