A159582 Expansion of (1+6*x+x^2-2*x^3)/((x^2+2*x-1)*(x^2-2*x-1)), bisection is NSW numbers.

1, 6, 7, 34, 41, 198, 239, 1154, 1393, 6726, 8119, 39202, 47321, 228486, 275807, 1331714, 1607521, 7761798, 9369319, 45239074, 54608393, 263672646, 318281039, 1536796802, 1855077841, 8957108166, 10812186007, 52205852194, 63018038201, 304278004998

Offset: 0

Author

Creighton Dement, Apr 16 2009

Keywords

• easy
• nonn

Comments

Define c = [0, 7, 0, 41, 0, 239, 0, 1393, 0, 8119, 0, 47321, ...] where (c(2n+1)) = A002315(n+1) (NSW numbers). Then (a(n)) has the property c(2n) - a(2n) = -a(2n) = -A002315(n) and c(2n+1) - a(2n+1) = A002315(n) (NSW numbers).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A002315.

Programs

PARI

Vec((1+6*x+x^2-2*x^3) / ((x^2+2*x-1)*(x^2-2*x-1)) + O(x^50)) \\ Colin Barker, Jun 29 2017

Formula

a(n) = 3*A078057(n)/2 - (-1)^n*A078057(n)/2. - R. J. Mathar, Nov 10 2009

From Colin Barker, Jun 29 2017: (Start)

a(n) = 6*a(n-2) - a(n-4) for n>3.

a(n) = ((-(-2+sqrt(2))*(-1+sqrt(2))^n - (-1-sqrt(2))^n*(2+sqrt(2)) - 3*(-(1-sqrt(2))^n*(-2+sqrt(2)) - (1+sqrt(2))^n*(2+sqrt(2))))) / (4*sqrt(2)).

(End)