A100828 Expansion of (1+2*x-2*x^3-3*x^2)/((x-1)*(x+1)*(x^2+2*x-1)).

1, 4, 7, 18, 41, 100, 239, 578, 1393, 3364, 8119, 19602, 47321, 114244, 275807, 665858, 1607521, 3880900, 9369319, 22619538, 54608393, 131836324, 318281039, 768398402, 1855077841, 4478554084, 10812186007, 26102926098, 63018038201

Offset: 0

Author

Creighton Dement, Jan 06 2005; revised Aug 22 2005

Keywords

• easy
• nonn

Comments

A floretion-generated sequence relating NSW and Pell numbers.

Elements of odd index in the sequence gives A002315. a(n+2) - a(n) = A002203(n+2).

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[B*C} with B = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and C = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Robert Munafo, Sequences Related to Floretions
• Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Crossrefs

Cf. A002315, A002203.

Programs

PARI

Vec((1 + 2*x - 3*x^2 - 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Apr 29 2019

Formula

a(n) = (u^(n+1)+1)*(v^(n+1)+1)/2 with u = 1+sqrt(2), v = 1-sqrt(2). - Vladeta Jovovic, May 30 2007

From Colin Barker, Apr 29 2019: (Start)

G.f.: (1 + 2*x - 3*x^2 - 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x - x^2)).

a(n) = (1 + (-1)^(1+n) + (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n)) / 2.

a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3.

(End)