A120743 a(n) = (1/2)*(1 + 3*i)^n + (1/2)*(1 - 3*i)^n where i = sqrt(-1).

1, -8, -26, 28, 316, 352, -2456, -8432, 7696, 99712, 122464, -752192, -2729024, 2063872, 31417984, 42197248, -229785344, -881543168, 534767104, 9884965888, 14422260736, -70005137408, -284232882176, 131585609728, 3105500041216

Offset: 1

Author

Creighton Dement, Jun 11 2007

Keywords

• easy
• sign

Comments

From R. J. Mathar, Jun 15 2007: (Start)

These are the row sums of the triangle A013610 after every 2nd column is deleted, then every 2nd column reversed in sign, creating an intermediate irregular triangle with entries C(n,2*k)*(-9)^k, k = 0..floor(n/2):

1;

1, -9;

1, -27;

1, -54, 81;

1, -90, 405;

1, -135, 1215, -729;

1, -189, 2835, -5103;

1, -252, 5670, -20412, 6561;

1, -324, 10206, -61236, 59049;

1, -405, 17010, -153090, 295245, -59049; (End)

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[A*B] with A = + 1.5i' + .5j' + .5k' + .5e and B = 'ji' + e

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• Index entries for linear recurrences with constant coefficients, signature (2, -10).

Crossrefs

Cf. A006495

Programs

Magma

[ n eq 1 select 1 else n eq 2 select -8 else 2*Self(n-1) -10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 24 2011

Mathematica

LinearRecurrence[{2,-10}, {1,-8}, 30] (* G. C. Greubel, Nov 09 2018 *)

PARI

x='x+O('x^30); Vec((1-10*x)/(1-2*x+10*x^2)) \\ G. C. Greubel, Nov 09 2018

Formula

a(n) = 2*a(n-1) - 10*a(n-2).

G.f.: x*(1-10*x)/(10*x^2 - 2*x + 1).

a(n) mod 9 = 1. - Paul Curtz, Apr 20 2011

G.f.: G(0)/(2*x) - 1/x, where G(k) = 1 + 1/(1 - x*(9*k+1)/(x*(9*k+10) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013

E.g.f.: exp(x)*cos(3*x). - Sergei N. Gladkovskii, May 29 2013

a(n) = A190958(n)-10*A190958(n-1). - R. J. Mathar, Dec 13 2022

Extensions

a(0)=1: a(n) is main diagonal of A009116(n). - Paul Curtz, Jul 22 2011

Edited by Jon E. Schoenfield, Nov 09 2018