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A373358 a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3) +2*a(n-4) for a(0) = a(1) = 0, a(2) = 1, a(3) = 4 for n >= 4.

%I A373358 #42 Jan 11 2025 11:08:36
%S A373358 0,0,1,4,11,26,59,136,323,782,1903,4620,11175,26970,65051,156944,
%T A373358 378811,914566,2208199,5331476,12871663,31074802,75020243,181113240,
%U A373358 437244675,1055602590,2548453951,6152518684,14853499511,35859517706,86572518539,209004522016,504581529803,1218167581622
%N A373358 a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3) +2*a(n-4) for a(0) = a(1) = 0, a(2) = 1, a(3) = 4 for n >= 4.
%H A373358 Harvey P. Dale, <a href="/A373358/b373358.txt">Table of n, a(n) for n = 0..1000</a>
%H A373358 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2,2).
%F A373358 G.f.: x^2 / ( (1 - 2*x - x^2) * (1 - 2*x + 2*x^2) ).
%F A373358 E.g.f.: exp(x)*(2*cosh(sqrt(2)*x) - 2*(cos(x)+sin(x)) + sqrt(2)*sinh(sqrt(2)*x))/6.
%F A373358 a(n) = A373245(n+1) - A114203(n+1).
%F A373358 a(0) = 0, a(n) = A373245(n-1) + A146559(n-1).
%F A373358 Binomial transform of 0, 0, followed by A077893 = abs(A077953) = abs(A077980).
%F A373358 a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3) +2*a(n-4) for n >= 4.
%F A373358 From _Thomas Scheuerle_, Jun 03 2024: (Start)
%F A373358 a(n) = (A000129(n+1) - A009545(n+1))/3.
%F A373358 a(n) = (-i*sqrt(2)*(1-i)^(n+1) + i*sqrt(2)*(1+i)^(n+1) - (1-sqrt(2))^(n+1) + (1+sqrt(2))^(n+1))/(6*sqrt(2)).
%F A373358 a(n) = 2^n*(hypergeom([1/2 - n/2, -n/2], [-n], -1) - hypergeom([1/2 - n/2, -n/2], [-n], 2))/3. (End)
%t A373358 LinearRecurrence[{4, -5, 2, 2}, {0, 0, 1, 4}, 50] (* _Paolo Xausa_, Jun 19 2024 *)
%t A373358 nxt[{a_,b_,c_,d_}]:={b,c,d,4d-5c+2b+2a}; NestList[nxt,{0,0,1,4},40][[;;,1]] (* _Harvey P. Dale_, Jan 11 2025 *)
%o A373358 (PARI) a(n) = ((([2, 1; 1, 0]^(n+1))[2, 1]) - (1+I)^(n-1) - (1-I)^(n-1))/3 \\ _Thomas Scheuerle_, Jun 03 2024
%Y A373358 Cf. A000129, A009545, A077893, A077953, A077980, A114203, A146559, A373245.
%K A373358 nonn,easy
%O A373358 0,4
%A A373358 _Paul Curtz_, Jun 02 2024