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.

0, 0, 1, 4, 11, 26, 59, 136, 323, 782, 1903, 4620, 11175, 26970, 65051, 156944, 378811, 914566, 2208199, 5331476, 12871663, 31074802, 75020243, 181113240, 437244675, 1055602590, 2548453951, 6152518684, 14853499511, 35859517706, 86572518539, 209004522016, 504581529803, 1218167581622

Offset: 0

Author

Paul Curtz, Jun 02 2024

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2,2).

Crossrefs

Cf. A000129, A009545, A077893, A077953, A077980, A114203, A146559, A373245.

Programs

Mathematica

LinearRecurrence[{4, -5, 2, 2}, {0, 0, 1, 4}, 50] (* Paolo Xausa, Jun 19 2024 *)
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 *)

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

Formula

G.f.: x^2 / ( (1 - 2*x - x^2) * (1 - 2*x + 2*x^2) ).

E.g.f.: exp(x)*(2*cosh(sqrt(2)*x) - 2*(cos(x)+sin(x)) + sqrt(2)*sinh(sqrt(2)*x))/6.

a(n) = A373245(n+1) - A114203(n+1).

a(0) = 0, a(n) = A373245(n-1) + A146559(n-1).

Binomial transform of 0, 0, followed by A077893 = abs(A077953) = abs(A077980).

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

From Thomas Scheuerle, Jun 03 2024: (Start)

a(n) = (A000129(n+1) - A009545(n+1))/3.

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)).

a(n) = 2^n*(hypergeom([1/2 - n/2, -n/2], [-n], -1) - hypergeom([1/2 - n/2, -n/2], [-n], 2))/3. (End)