A133273 Indices of centered decagonal numbers which are also decagonal numbers.

1, 10, 171, 3060, 54901, 985150, 17677791, 317215080, 5692193641, 102142270450, 1832868674451, 32889493869660, 590178020979421, 10590314883759910, 190035489886698951, 3410048503076821200, 61190837565496082641, 1098025027675852666330, 19703259660599851911291

Offset: 1

Author

Richard Choulet, Oct 16 2007

Keywords

• nonn
• easy

Comments

Numbers k such that 80*k^2 - 80*k + 25 is a square.

Also the indices of centered square numbers which are also centered pentagonal numbers. - Colin Barker, Jan 01 2015

Links

• Colin Barker, Table of n, a(n) for n = 1..798
• Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Crossrefs

Cf. A001107, A001844, A005891, A010532, A062786, A128922.

Programs

Mathematica

LinearRecurrence[{19,-19,1},{1,10,171},20] (* Harvey P. Dale, Oct 09 2020 *)

PARI

Vec(x*(-1+9*x)/((-1+x)*(1-18*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 01 2015

Formula

a(n+2) = 18*a(n+1) - a(n) - 8.

a(n+1) = 9*a(n) - 4 + sqrt(80*a(n)^2 - 80*a(n) + 25).

G.f.: x*(-1+9*x)/(-1+x)/(1 - 18*x + x^2). - R. J. Mathar, Nov 14 2007

a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). - Colin Barker, Jan 01 2015

Product_{n>=2} (1 - 1/a(n)) = 2/sqrt(5) (= A010532 / 10). - Amiram Eldar, Dec 02 2024

Extensions

More terms from Paolo P. Lava, Nov 25 2008