A251730 Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to the sum of two pentagonal numbers P(m) and P(m+1) for some m.

10, 152, 2130, 29680, 413402, 5757960, 80198050, 1117014752, 15558008490, 216695104120, 3018173449202, 42037733184720, 585510091136890, 8155103542731752, 113585939507107650, 1582048049556775360, 22035086754287747402, 306909166510471688280

Offset: 1

Author

Colin Barker, Dec 07 2014

Keywords

• nonn
• easy

Comments

Also nonnegative integers y in the solutions to 3*x^2-y^2+2*x-2*y = 0, the corresponding values of x being A122770.

Examples

			10 is in the sequence because T(10)+T(11) = 55+56 = 121 = 51+70 = P(6)+P(7).
		

Links

• Colin Barker, Table of n, a(n) for n = 1..874
• Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Crossrefs

Cf. A000217, A000326, A122770.

Programs

Mathematica

LinearRecurrence[{15,-15,1},{10,152,2130},30] (* Harvey P. Dale, Jun 08 2016 *)

PARI

Vec(-2*x*(x+5)/((x-1)*(x^2-14*x+1)) + O(x^100))

Formula

a(n) = 15*a(n-1)-15*a(n-2)+a(n-3).

G.f.: -2*x*(x+5) / ((x-1)*(x^2-14*x+1)).

a(n) = (-6-(7-4*sqrt(3))^n*(-3+sqrt(3))+(3+sqrt(3))*(7+4*sqrt(3))^n)/6. - Colin Barker, Mar 05 2016