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

0,2

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).

Links

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

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

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A012855 A224066 A150646 * A128611 A061539 A232970

Adjacent sequences: A364142 A364143 A364144 * A364146 A364147 A364148

Keyword

nonn,easy

Author

Muhammad Adam Dombrowski and Greg Dresden, Jul 10 2023

Status

approved