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a(n) = (2^n - (-1+i)^n - (-1-i)^n)/4 - 1 where i is the imaginary unit.
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%I #18 Nov 14 2016 06:52:24

%S 0,0,0,5,5,15,35,55,135,255,495,1055,2015,4095,8255,16255,32895,65535,

%T 130815,262655,523775,1048575,2098175,4192255,8390655,16777215,

%U 33550335,67117055,134209535,268435455,536887295,1073709055,2147516415,4294967295,8589869055,17180000255

%N a(n) = (2^n - (-1+i)^n - (-1-i)^n)/4 - 1 where i is the imaginary unit.

%H Colin Barker, <a href="/A275016/b275016.txt">Table of n, a(n) for n = 1..1000</a>

%H Omran Ahmadi, Robert Granger, <a href="https://arxiv.org/abs/1104.3882">An efficient deterministic test for Kloosterman sum zeros</a>, arXiv:1104.3882 [math.NT], 2011-2012. See 4th column of Table 1 p. 9.

%H Robert Granger, <a href="https://arxiv.org/abs/1610.06878">On the Enumeration of Irreducible Polynomials over GF(q) with Prescribed Coefficients</a>, arXiv:1610.06878 [math.AG], 2016. See 4th column of Table 1 p. 13.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,2,-4).

%F G.f.: 5*x^4/((1 - x)*(1 - 2*x)*(1 + 2*x + 2*x^2)). - _Ilya Gutkovskiy_, Nov 12 2016

%F a(n) = a(n-1) + 2*a(n-2) + 2*a(n-3) - 4*a(n-4) for n>4. - _Colin Barker_, Nov 12 2016

%F a(n) = - 2^(n-2)*a(4-n) for all n in Z. - _Michael Somos_, Nov 13 2016

%e G.f. = 5*x^4 + 5*x^5 + 15*x^6 + 35*x^7 + 55*x^8 + 135*x^9 + 255*x^10 + ...

%o (PARI) a(n) = (2^n - (-1+I)^n - (-1-I)^n)/4 -1;

%o (PARI) concat(vector(3), Vec(5*x^4/((1-x)*(1-2*x)*(1+2*x+2*x^2)) + O(x^50))) \\ _Colin Barker_, Nov 14 2016

%K nonn,easy

%O 1,4

%A _Michel Marcus_, Nov 12 2016