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A133218 Indices of triangular numbers (A000217) that are also decagonal (A001107). 2
0, 1, 4, 55, 154, 1885, 5248, 64051, 178294, 2175865, 6056764, 73915375, 205751698, 2510946901, 6989500984, 85298279275, 237437281774, 2897630548465, 8065878079348, 98434140368551, 274002417416074, 3343863141982285, 9308016314067184, 113592912687029155 (list; graph; refs; listen; history; text; internal format)
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)).

GF: (1 - 31*x^2 + 51*x^3 - 2*x^4 - 3*x^5) / ((1 - x ) * (1 - 6*x + x^2 ) * (1 + 6*x + x^2)).

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 }, {1, 4, 55, 154, 1885 }, 23 ]

CROSSREFS

Cf. A000217, A001107, A133216, A133217.

Sequence in context: A077658 A217124 A064439 * 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

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Last modified June 21 16:12 EDT 2018. Contains 305624 sequences. (Running on oeis4.)