A152056 a(n) = ((9+sqrt(3))^n + (9-sqrt(3))^n)/2.

1, 9, 84, 810, 8028, 81324, 837648, 8734392, 91882512, 972602640, 10340011584, 110257202592, 1178108743104, 12605895573696, 135013638364416, 1446985635811200, 15514677652177152, 166399318145915136, 1785042869756654592, 19151624840238402048, 205495903283272178688

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..970

Index entries for linear recurrences with constant coefficients, signature (18,-78).

Formula

From Philippe Deléham, Nov 26 2008: (Start)

a(n) = 18*a(n-1) - 78*a(n-2), n > 1; a(0)=1, a(1)=9.

G.f.: (1-9*x)/(1-18*x+78*x^2).

a(n) = (Sum_{k=0..n} A098158(n,k)*3^(3*k))/3^n. (End)

Mathematica

LinearRecurrence[{18, -78}, {1, 9}, 25] (* Paolo Xausa, Jun 04 2026 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r3>:=NumberField(x^2-3); S:=[ ((9+r3)^n+(9-r3)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008
(PARI) a(n)=([0, 1; -78, 18]^n*[1; 9])[1, 1] \\ Charles R Greathouse IV, Jun 04 2026

Crossrefs

Cf. A098158.

Sequence in context: A181353 A385251 A370029 * A086627 A024118 A166902

Adjacent sequences: A152053 A152054 A152055 * A152057 A152058 A152059

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008

Extensions

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008

More terms from Paolo Xausa, Jun 04 2026

Status

approved