A273368 Numbers k such that 10*k+9 is a perfect square.

0, 4, 16, 28, 52, 72, 108, 136, 184, 220, 280, 324, 396, 448, 532, 592, 688, 756, 864, 940, 1060, 1144, 1276, 1368, 1512, 1612, 1768, 1876, 2044, 2160, 2340, 2464, 2656, 2788, 2992, 3132, 3348, 3496, 3724, 3880, 4120, 4284, 4536

Offset: 0

Author

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, 2, -2, -1, 1).

Crossrefs

Cf. A132356, A273365, A273366, A273367.

Cf. A033583 (perfect squares ending in 0 in base 10 with final 0 removed).

Programs

Mathematica

CoefficientList[Series[4*x*(x^2+3x+1)/((1-x)^3*(1+x)^2), {x,0,50}], x] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {0, 4, 16, 28, 52}, 50] (* G. C. Greubel, May 20 2016 *)

PARI

is(n)=issquare(10*n+9) \\ Charles R Greathouse IV, Jan 31 2017

Formula

a(2n) = 10*n^2 + 6*n, n>=0.

a(2n-1) = 10*n^2 - 6*n, n>=1.

G.f.: 4*x*(x^2+3x+1)/((1-x)^3*(1+x)^2).

From G. C. Greubel, May 21 2016: (Start)

E.g.f.: (1/2)*((5*x^2 + 9*x)*cosh(x) + (5*x^2 + 11*x -1)*sinh(x)).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)

a(n) = 4*A085787(n). - R. J. Mathar, Jun 03 2016