A129557 Numbers k > 0 such that k^2 is a centered pentagonal number (A005891).

1, 4, 34, 151, 1291, 5734, 49024, 217741, 1861621, 8268424, 70692574, 313982371, 2684456191, 11923061674, 101938642684, 452762361241, 3870983965801, 17193046665484, 146995452057754, 652883010927151, 5581956194228851, 24792361368566254

Offset: 1

Author

Alexander Adamchuk, Apr 20 2007

Keywords

• nonn
• easy

Comments

Corresponding numbers m such that centered pentagonal number A005891(m) = (5*m^2 + 5*m + 2)/2 is a perfect square are listed in A129556 = {0, 2, 21, 95, 816, 3626, 31005, ...}.

Also positive integers x in the solutions to 2*x^2 - 5*y^2 + 5*y - 2 = 0, the corresponding values of y being A254332. - Colin Barker, Jan 28 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Centered Pentagonal Number.
• Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Crossrefs

Cf. A005891 (centered pentagonal numbers).

Cf. A129556 (k such that A005891(k) is a perfect square).

Cf. A000290, A005667, A254332, A254333.

Programs

Mathematica

Do[ f=(5n^2+5n+2)/2; If[ IntegerQ[ Sqrt[f] ], Print[ Sqrt[f] ] ], {n,1,40000} ]
CoefficientList[Series[(1-x)*(1+5*x+x^2)/((1+6*x-x^2)*(1-6*x-x^2)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)

PARI

A129557()={ for(n=1,1000000000, f=(5*n^2+5*n+2)/2 ; if(issquare(f), print1(sqrtint(f), ", ") ; ); ) ; } \\ R. J. Mathar, Oct 11 2007

PARI

Vec(x*(1-x)*(1+5*x+x^2)/((1+6*x-x^2)*(1-6*x-x^2)) + O(x^100)) \\ Colin Barker, Jan 28 2015

Formula

a(n) = sqrt( (5*A129556(n)^2 + 5*A129556(n) + 2)/2 ).

For n >= 5, a(n) = 38*a(n-2) - a(n-4). - Max Alekseyev, May 08 2009

G.f.: x*(1-x)*(1 + 5*x + x^2)/((1 + 6*x - x^2)*(1 - 6*x - x^2)). - Colin Barker, Apr 11 2012

From Andrea Pinos, Oct 07 2022: (Start)

The ratios of successive terms converge to two different limits:

lower: D = lim_{n->oo} a(2n)/a(2n-1) = (7 + 2*sqrt(10))/3;

upper: E = lim_{n->oo} a(2n+1)/a(2n) = (13 + 4*sqrt(10))/3.

So lim_{n->oo} a(n+2)/a(n) = D*E = 19 + 6*sqrt(10).

a(n) = (A005667(n) - (-1)^n*A005667(n-1))/4. (End)

Extensions

More terms from R. J. Mathar, Oct 11 2007

More terms from Max Alekseyev, May 08 2009