A364145 a(n) is the sum of the first 2*n nonzero n-bonacci numbers.

0, 2, 7, 28, 116, 480, 1968, 8000, 32320, 130048, 521984, 2092032, 8377344, 33529856, 134164480, 536756224, 2147237888, 8589410304, 34358624256, 137436594176, 549750833152, 2199012769792, 8796071002112, 35184325951488, 140737391886336, 562949752094720

Offset: 0

Author

Muhammad Adam Dombrowski and Greg Dresden, Jul 10 2023

Keywords

• nonn
• easy

Comments

For our purposes, for n > 0 fixed we define the k-th n-bonacci number T(n,k) as equal to 0 for k <= 0, equal to 1 for k=1, and then equal to the sum of the previous n numbers for k > 1. For n=2, then, we get T(2,k) equal to F(n) = A000045(n), the Fibonacci numbers. For n=3, then, T(3,k) is the tribonacci numbers, and so on.

a(n) is thus defined as Sum_{k=1..2*n} T(n,k).

Examples

			For n=3, a(3) is the sum of the first 6 nonzero tribonacci numbers, found at A000073. This gives a(3) = 1 + 1 + 2 + 4 + 7 + 13 = 28.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-20,16).

Crossrefs

Cf. A000045, A000073, A000078, A092921, A144406.

Programs

Mathematica

T[n_, k_] := SeriesCoefficient[Series[x/(1 - Sum[x^i, {i, 1, n}]), {x, 0, k + 1}], k]; Table[Sum[T[n, k], {k, 1, 2n}], {n, 1, 30}]

Formula

a(n) = (2*4^n - (n-1)*2^n)/4 for n>=1.

a(n) = Sum_{i=1..2*n} A092921(n,i).

G.f.: -x*(12*x^2-9*x+2)/((4*x-1)*(2*x-1)^2). - Alois P. Heinz, Jul 11 2023

E.g.f.: exp(2*x)*(1 - 2*x - cosh(2*x) + 5*sinh(2*x))/4. - Stefano Spezia, Jul 12 2023