A122513 Numbers n such that 1+2n+3n^2 is a triangular number.

0, 1, 46, 135, 4540, 13261, 444906, 1299475, 43596280, 127335321, 4271990566, 12477562015, 418611479220, 1222673742181, 41019652973026, 119809549171755, 4019507379877360, 11740113145089841, 393870703575008286, 1150411278669632695, 38595309442970934700

Offset: 1

Author

Zak Seidov, Oct 20 2006

Keywords

• nonn
• easy

Comments

The y solution to the generalized Pell equation x^2 + x = 2 + 4*y + 6*y^2. - T. D. Noe, Apr 28 2011

Also numbers n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to a hexagonal number. - Colin Barker, Dec 15 2014

Also numbers n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to a triangular number. - Colin Barker, Dec 15 2014

Examples

			Corresponding values of triangular numbers tri = m(m+1)/2 and m's are
tri = 1, 6, 6441, 54946, 61843881, 527588886, 593824936321
m = 1, 3, 113, 331, 11121, 32483, 1089793.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).

Crossrefs

Cf. A000217 (triangular numbers), A086285 (numbers n such that 1+2n+3n^2 is prime).

Cf. A000326, A000384, A245783, A249164.

Programs

Maple

ivs:=[0,1,46,135,4540]:
rec:= a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5):
f:= gfun:-rectoproc({rec, seq(a(i)=ivs[i],i=1..5)},a(n),remember):
seq(f(n),n=1..100); # Robert Israel, Dec 15 2014

Mathematica

triQ[n_] := IntegerQ[ Sqrt[8n + 1]]; lst = {}; Do[ If[ triQ[1 + 2n + 3n^2], AppendTo[lst, n]; Print@n], {n, 0, 65000000}] (* Robert G. Wilson v, Jan 08 2007 *)
LinearRecurrence[{1, 98, -98, -1, 1}, {1, 46, 135, 4540, 13261}, 30] (* T. D. Noe, Apr 28 2011 *)

PARI

concat(0, Vec(x^2*(5*x^3+9*x^2-45*x-1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))) \\ Colin Barker, Dec 15 2014

Formula

a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5). - Colin Barker, Dec 15 2014

G.f.: x^2*(5*x^3+9*x^2-45*x-1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)). - Colin Barker, Dec 15 2014

a(n) = (-(5/8)*sqrt(6)-3/2)*(5-2*sqrt(6))^n+(-3/2+(5/8)*sqrt(6))*(5+2*sqrt(6))^n-1/3+(-(1/3)*sqrt(6)-5/6)*(-5+2*sqrt(6))^n+((1/3)*sqrt(6)-5/6)*(-5-2*sqrt(6))^n. - Robert Israel, Dec 15 2014

Extensions

a(8) and a(9) from Robert G. Wilson v, Jan 08 2007

a(10) and a(11) from Donovan Johnson, Apr 28 2011

Extended by T. D. Noe, Apr 28 2011