A161729 a(n) = ((4+sqrt(3))*(8+2*sqrt(3))^n-(4-sqrt(3))*(8-2*sqrt(3))^n)/(2*sqrt(3)).

1, 16, 204, 2432, 28304, 326400, 3750592, 43036672, 493555968, 5658988544, 64878906368, 743795097600, 8527018430464, 97754949812224, 1120674238611456, 12847530427547648, 147285426432966656, 1688495240694988800, 19357081676605554688, 221911554309549457408, 2544016621769302474752

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

Keywords

• nonn
• easy

Comments

Eighth binomial transform of A162466.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..943
• Index entries for linear recurrences with constant coefficients, signature (16,-52).

Crossrefs

Cf. A162466, A161728, A153594.

Programs

Mathematica

Join[{a=1,b=16},Table[c=16*b-52*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
LinearRecurrence[{16,-52},{1,16},20] (* Harvey P. Dale, Dec 23 2020 *)

PARI

F=nfinit(x^2-3); for(n=0, 17, print1(nfeltdiv(F, ((4+x)*(8+2*x)^n-(4-x)*(8-2*x)^n), (2*x))[1], ",")) \\ Klaus Brockhaus, Jun 19 2009

PARI

Vec(1/(1-16*x+52*x^2)+O(x^25)) \\ M. F. Hasler, Dec 03 2014

Formula

a(n) = 16*a(n-1) - 52(n-2) for n > 1; a(0) = 1, a(1) = 16.

G.f.: 1/(1 - 16*x + 52*x^2). - Klaus Brockhaus, Jun 19 2009

a(n) = 2^n*A153594(n). - M. F. Hasler, Dec 03 2014

E.g.f.: exp(8*x)*(3*cosh(2*sqrt(3)*x) + 4*sqrt(3)*sinh(2*sqrt(3)*x))/3. - Stefano Spezia, Dec 31 2022

Extensions

Extended beyond a(5) by Klaus Brockhaus, Jun 19 2009

Edited by Klaus Brockhaus, Jul 05 2009, and M. F. Hasler, Dec 03 2014