A081065 Numbers n such that n^2 = (1/3)*(n+floor(sqrt(3)*n*floor(sqrt(3)*n))).

2, 24, 330, 4592, 63954, 890760, 12406682, 172802784, 2406832290, 33522849272, 466913057514, 6503259955920, 90578726325362, 1261598908599144, 17571805994062650, 244743685008277952, 3408839784121828674, 47479013292697323480, 661297346313640700042

Offset: 1

Author

Benoit Cloitre, Apr 15 2003

Keywords

• nonn
• easy

Comments

a(n)/2 gives indices of pentagonal numbers which are also triangular.

a(n) itself gives x-values solving the Diophantine equation 2*x^2 + (x-1)^2 = y^2.

Links

• Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Crossrefs

Cf. A046090, A046174, A046175.

Programs

Mathematica

LinearRecurrence[{15,-15,1},{2,24,330},20] (* Harvey P. Dale, Mar 14 2016 *)

PARI

Vec(2*(1-3*x)/((1-x)*(1-14*x+x^2)) + O(x^40)) \\ Michel Marcus, Nov 17 2014

Formula

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

a(n) = 14*a(n-1) - a(n-2) - 4. [Sture Sjöstedt, May 02 2011]

G.f.: 2*(1-3*x)/((1-x)*(1-14*x+x^2)). - Bruno Berselli, Nov 11 2011