A178159 Modified variant of A006645, the self-convolution of the Pell series.

1, 2, 8, 22, 68, 188, 532, 1444, 3921, 10446, 27704, 72714, 189912, 492760, 1273064, 3273896, 8389489, 21423994, 54550728, 138520286, 350899964, 886925652, 2237284668, 5633150988, 14159465505, 35535456518, 89053087224, 222870328210, 557074041840, 1390807477040

Offset: 1

Author

Gary W. Adamson, Dec 18 2010

Keywords

• nonn
• easy

Comments

Analogous series using the Fibonacci numbers as a generator = A089098.

Examples

			(1/2) * (1, 4, 14, 44, 131,...) + (1, 0, 2, 0, 5,...) = (1/2) * (2, 4, 16, 44, 136, 376,...) = (1, 2, 8, 22, 68, 188,...).
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,0,-12,4,4,4,4,1).

Crossrefs

Cf. A089098, A006645.

Programs

Maple

A178159 := proc(n)
    if type (n,'even') then
        (A006645(n+2) + A000129(n/2+1))/2 ;
    else
        A006645(n+2)/2 ;
    fi;
end proc: # R. J. Mathar, Jul 21 2015

PARI

Vec(-x*(2*x^3-2*x+1)/((x^2+2*x-1)^2*(x^4+2*x^2-1)) + O(x^40)) \\ Colin Barker, Jul 21 2015

Formula

(1/2) * [ (1, 4, 14, 44, 131,...) + (1, 0, 2, 0, 5,...)]; where (1, 4, 14, 44,...) = A006645, the self-convolution of the Pell series, and (1, 0, 2, 0, 5,...) = the aerated Pell series.

G.f.: -x*(2*x^3-2*x+1) / ((x^2+2*x-1)^2*(x^4+2*x^2-1)). - Colin Barker, Jul 21 2015

Extensions

Corrected by R. J. Mathar, Jul 21 2015