A395902 Triple convolution of the modified Fibonacci sequence F(3n)/2 (A001076).

0, 0, 0, 1, 12, 99, 688, 4326, 25464, 143018, 775536, 4092207, 21126564, 107136597, 535272864, 2640788828, 12888228336, 62312915172, 298810304672, 1422537514989, 6728698090428, 31643981809111, 148044975160272, 689370668876130, 3196344091219560, 14762394097629870

Offset

0,5

Links

Table of n, a(n) for n=0..25.

D. Dmytryshyn, D. Gray, V. Khamitov, and Alex Stokolos, Convolved numbers of k-section of the Fibonacci sequence: properties, consequences, arXiv:2603.08636 [math.CA], 2026; see also alternate link, Herald of Advanced Information Technology. 2026. Vol. 9, No. 2. P. 129-139.

Index entries for linear recurrences with constant coefficients, signature (12,-45,40,45,12,1).

Formula

a(n) = Sum_{j=1..n} y(j)*y1(n-j), where y(n) = F(3*n)/2 and y1(n) is the first-order convolution of y.

G.f.: x^3/(1 - 4*x - x^2)^3.

a(n) = (-1)^(n-3) * GegenbauerC(n-3, 3, 2*I).

a(j) = (4*(j-1)*a(j-1) + (j+1)*a(j-2)) / (j-3) for j>=4, with a(0..2)=0, a(3)=1.

a(n) = 12*a(n-1) - 45*a(n-2) + 40*a(n-3) + 45*a(n-4) + 12*a(n-5) + a(n-6) for n>=6, with a(0..5) = (0, 0, 0, 1, 12, 99).

a(n) = (1/800) * ( 1/2*(n-2)*(n-1)*F(3*n+6) + (n-2)*(n+2)*F(3*n) + 1/2*(n+1)*(n+2)*F(3*n-6) ).

a(n) = (q^(-3*n-6) / (q^3+q^(-3))^5) * ( 1/2*(n-2)*(n-1)*(q^(6*n+12) - (-1)^n) + (n-2)*(n+2)*(q^(6*n+6) - (-1)^n*q^6) + 1/2*(n+1)*(n+2)*(q^(6*n) - (-1)^n*q^12) ), where q = (1+sqrt(5))/2.

E.g.f.: exp(2*x) * ( (1/10 - 7/200*x)*cosh(sqrt(5)*x) + sqrt(5)*(9/200*x^2 - 1/50*x + 7/1000)*sinh(sqrt(5)*x) ).

Maple

N := 25; a[0]:= 0; a[1]:= 0; a[2]:= 0; a[3]:= 1;
for j from 4 to N do
a[j]:=1/(j-3)*(4*(j-1)*a[j-1]+(j+1)*a[j-2]):
end:
seq(a[k], k = 0 .. N);

Mathematica

LinearRecurrence[{12, -45, 40, 45, 12, 1}, {0, 0, 0, 1, 12, 99}, 26] (* Amiram Eldar, May 10 2026 *)

Crossrefs

Cf. A395431, A001076, A000045, A001622.

Sequence in context: A090230 A133652 A004058 * A389191 A052178 A322721

Adjacent sequences: A395899 A395900 A395901 * A395903 A395904 A395905

Keyword

nonn,easy

Author

Alex Stokolos, May 08 2026

Status

approved