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

1,2

Comments

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

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).

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

Example

(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,...).

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

Prog

(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

Crossrefs

Cf. A089098, A006645.

Sequence in context: A102880 A137104 A265951 * A262720 A321573 A137103

Adjacent sequences: A178156 A178157 A178158 * A178160 A178161 A178162

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 18 2010

Extensions

Corrected by R. J. Mathar, Jul 21 2015

Status

approved