OFFSET
1,1
COMMENTS
All elements of this sequence are multiples of 3. Any proof?
This follows from the recurrence relation. - Charles R Greathouse IV, Dec 13 2013
LINKS
Colin Barker, Table of n, a(n) for n = 1..450
Index entries for linear recurrences with constant coefficients, signature (-135,120,16).
FORMULA
a(n) = (-1)^n/2 * ((2+sqrt(2))^(4*n) + (2-sqrt(2))^(4*n)) - 1. - Vaclav Kotesovec, Dec 06 2013
G.f.: 3*x*(28*x+23) / ((x-1)*(16*x^2+136*x+1)). - Colin Barker, Dec 06 2013
MAPLE
A232719:=n->add((-1)^i*binomial(8*n, 4*i), i=1..2*n); seq(A232719(n), n=1..20); # Wesley Ivan Hurt, Dec 06 2013
MATHEMATICA
A[n_] := Sum[(-1)^k Binomial[8 n, 4 k], {k, 1, 2n}]; Array[A, 33]
Table[FullSimplify[(-1)^n/2*((2+Sqrt[2])^(4*n)+(2-Sqrt[2])^(4*n))-1], {n, 1, 15}] (* Vaclav Kotesovec, Dec 06 2013 *)
CoefficientList[Series[3 (28 x + 23) / ((x - 1) (16 x^2 + 136 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)
LinearRecurrence[{-135, 120, 16}, {-69, 9231, -1254465}, 20] (* Harvey P. Dale, Jan 31 2023 *)
PROG
(PARI) a(n)=sum(k=1, 2*n, (-1)^k*binomial(8*n, 4*k)) \\ Charles R Greathouse IV, Dec 13 2013
(PARI) Vec(3*x*(28*x+23)/((x-1)*(16*x^2+136*x+1)) + O(x^100)) \\ Colin Barker, Nov 09 2014
CROSSREFS
KEYWORD
sign,easy
AUTHOR
José María Grau Ribas, Nov 28 2013
STATUS
approved