A253826 Indices of centered octagonal numbers (A016754) which are also triangular numbers (A000217).

1, 18, 595, 20196, 686053, 23305590, 791703991, 26894630088, 913625718985, 31036379815386, 1054323288004123, 35815955412324780, 1216688160731038381, 41331581509442980158, 1404057083160330286975, 47696609245941786776976, 1620280657278860420130193

Offset: 1

Author

Colin Barker, Jan 16 2015

Keywords

• nonn
• easy

Comments

Also positive integers y in the solutions to x^2 - 8*y^2 + x + 8*y - 2 = 0, the corresponding values of x being A008843.

Also the indices of centered octagonal numbers (A016754) which are also hexagonal numbers (A000384). Also positive numbers y in the solutions to 4x^2-8y^2-2x+8y-2=0. - Colin Barker, Jan 25 2015

Examples

			18 is in the sequence because the 18th centered octagonal number is 1225, which is also the 49th triangular number.
18 is in the sequence because the 18th centered octagonal number 1225 is also the 25th hexagonal number. - _Colin Barker_, Jan 25 2015
		

Links

• Colin Barker, Table of n, a(n) for n = 1..654
• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Crossrefs

Cf. A000217, A008843, A016754, A046177.

Programs

PARI

Vec(x*(17*x-1)/((x-1)*(x^2-34*x+1)) + O(x^100))

Formula

a(n) = 35*a(n-1)-35*a(n-2)+a(n-3).

G.f.: x*(17*x-1) / ((x-1)*(x^2-34*x+1)).

a(n) = sqrt((-2-(17-12*sqrt(2))^n-(17+12*sqrt(2))^n)*(2-(17-12*sqrt(2))^(1+n)-(17+12*sqrt(2))^(1+n)))/(8*sqrt(2)). - Gerry Martens, Jun 04 2015