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

Author

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

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A000217, A001107, A133216, A133217.

Programs

Mathematica

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

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)).

Extensions

Entry revised by Max Alekseyev, Nov 06 2011