A202563 Numbers which are both decagonal and pentagonal.

1, 12376, 118837251, 1141075274626, 10956604668124501, 105205316882256186876, 1010181441746819238261751, 9699762098447641443533149126, 93137114659112811393986059649001, 894302565257039116557412701216561376, 8587093138460974938071465363095362686251

Offset

1,2

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (sqrt(3)+sqrt(2))^8 = 4801+1960*sqrt(6).

Intersection of A000326 and A001107. - Michel Marcus, Jun 20 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Formula

G.f.: x*(1+2773*x+126*x^2) / ((1-x)*(1-9602*x+x^2)).

a(n) = 9602*a(n-1)-a(n-2)+2900.

a(n) = 9603*a(n-1)-9603*a(n-2)+a(n-3).

a(n) = 1/192*(25*((sqrt(3)+sqrt(2))^(8*n-6)+(sqrt(3)-sqrt(2))^(8*n-6))-58).

a(n) = floor(25/192*(sqrt(3)+sqrt(2))^(8*n-6)).

Example

The second natural number which is both pentagonal and decagonal is 12376. Hence a(2) = 12376.

Mathematica

LinearRecurrence[{9603, -9603, 1}, {1, 12376, 118837251}, 11]

Prog

(Maxima) makelist(expand((25*((sqrt(3)+sqrt(2))^(8*n-6)+(sqrt(3)-sqrt(2))^(8*n-6))-58)/192), n, 1, 11); /* Bruno Berselli, Dec 22 2011 */
(Magma) I:=[1, 12376, 118837251]; [n le 3 select I[n] else 9603*Self(n-1)-9603*Self(n-2)+1*Self(n-3): n in [1..15]]; // Vincenzo Librandi, Jan 24 2012

Crossrefs

Cf. A202564, A202565, A000326, A001107.

Sequence in context: A236426 A235841 A140938 * A061731 A278892 A248084

Adjacent sequences: A202560 A202561 A202562 * A202564 A202565 A202566

Keyword

nonn,easy

Author

Ant King, Dec 21 2011

Status

approved