A159582 - OEIS

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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

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OFFSET

0,2

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).

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)

PROG

(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

CROSSREFS