A394613 a(n) = a(n-1) + a(n-3) + a(n-4), a(0) = 1, a(1) = 4, a(2) = 16, a(3) = 20.

1, 4, 16, 20, 25, 45, 81, 126, 196, 322, 529, 851, 1369, 2220, 3600, 5820, 9409, 15229, 24649, 39878, 64516, 104394, 168921, 273315, 442225, 715540, 1157776, 1873316, 3031081, 4904397, 7935489, 12839886, 20775364, 33615250, 54390625, 88005875, 142396489, 230402364, 372798864

Offset

0,2

Links

Table of n, a(n) for n=0..38.

Tomás Guardia, Douglas Jiménez, and Alexander McCurdy, Fiboquadratic numbers and Rithmomachia, Recreational Mathematics Magazine, Vol. 11, No. 18 (2024), pp. 17-29.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Formula

a(n) = (4*F(n/2) + F((n/2)-1))^2 if n is even and (4*F((n-1)/2) + F(((n-1)/2)-1))*(4*F(((n-1)/2)+1) + F((n-1)/2)) if n is odd where F(n) = Fibonacci(n).

G.f.: (1 + 3*x + 12*x^2 + 3*x^3)/((1 + x^2)*(1 - x - x^2)). - Stefano Spezia, Apr 09 2026

Mathematica

LinearRecurrence[{1, 0, 1, 1}, {1, 4, 16, 20}, 40] (* Amiram Eldar, Apr 07 2026 *)
a[n_]:=If[EvenQ[n], (4*Fibonacci[n/2] + Fibonacci[(n/2)-1])^2, (4*Fibonacci[(n-1)/2] + Fibonacci[((n-1)/2)-1])*(4*Fibonacci[((n-1)/2)+1] + Fibonacci[(n-1)/2])]; Array[a, 39, 0] (* Stefano Spezia, Apr 09 2026 *)

Crossrefs

Cf. A000045, A006498, A006499.

Sequence in context: A277887 A216033 A071966 * A349521 A326781 A326788

Adjacent sequences: A394610 A394611 A394612 * A394614 A394619 A394620

Keyword

nonn,easy

Author

Alexander McCurdy, Mar 26 2026

Status

approved