A133218 Indices of triangular numbers (A000217) that are also decagonal (A001107).

0, 1, 4, 55, 154, 1885, 5248, 64051, 178294, 2175865, 6056764, 73915375, 205751698, 2510946901, 6989500984, 85298279275, 237437281774, 2897630548465, 8065878079348, 98434140368551, 274002417416074, 3343863141982285, 9308016314067184, 113592912687029155

Offset

1,3

Links

Table of n, a(n) for n=1..24.

Index entries for linear recurrences with constant coefficients, signature (1, 34, -34, -1, 1).

Formula

For n>5, a(n) = 34*a(n-2) - a(n-4) + 16.

For n>6, a(n) = a(n-1) + 34*a(n-2) - 34*a(n-3) - a(n-4) + a(n-5).

For n>1, a(n) = 1/8 * ((4 + sqrt(2)*(-1)^n)*(1+sqrt(2))^(2*n - 3) + (4 - sqrt(2)*(-1)^n)*(1-sqrt(2))^(2*n-3) - 4).

a(n) = floor(1/8 * (4 + sqrt(2)*(-1)^n)* (1+sqrt(2))^(2*n-3)).

G.f.: x^2*(2*x^4+3*x^3-17*x^2-3*x-1)/((x-1)*(x^2+6*x+1)*(x^2-6*x+1)).

lim (n -> Infinity, a(2n+1)/a(2n)) = 1/7*(43 + 30*sqrt(2)).

lim (n -> Infinity, a(2n)/a(2n-1)) = 1/7*(11 + 6*sqrt(2)).

Example

The third number which is both triangular (A000217) and decagonal (A001107) is A133216(3)=10. Since this is the fourth triangular number, we have a(3) = 4.

Mathematica

LinearRecurrence[{1, 34, -34, -1, 1 }, {0, 1, 4, 55, 154, 1885}, 24 ]

Crossrefs

Cf. A000217, A001107, A133216, A133217.

Sequence in context: A217124 A064439 A352510 * A190441 A151576 A204107

Adjacent sequences: A133215 A133216 A133217 * A133219 A133220 A133221

Keyword

nonn

Author

Richard Choulet, Oct 11 2007; Ant King, Nov 04 2011

Extensions

Entry revised by Max Alekseyev, Nov 06 2011

Status

approved