A101220 a(n) = Sum_{k=0..n} Fibonacci(n-k)*n^k.

0, 1, 3, 14, 91, 820, 9650, 140601, 2440317, 49109632, 1123595495, 28792920872, 816742025772, 25402428294801, 859492240650847, 31427791175659690, 1234928473553777403, 51893300561135516404, 2322083099525697299278

Offset

0,3

Comments

In what follows a(i,j,k) denotes a three-dimensional array, the terms a(n) are defined as a(n,n,n) in that array. - Joerg Arndt, Jan 03 2021

Previous name was: Three-dimensional array: a(i,j,k) = expansion of (x(1+i*x-j*x))/((-1+j*x)(-1+x+x^2)), read by a(n,n,n).

a(i,j,k) = the k-th value of the convolution of the Fibonacci numbers [A000045] with the powers of i = Sum[a(i-1,j,m], {m=0...k}], both for i = j and i > 0; a(i,j,k) = a(i-1,j,k) + a(j,j,k-1), for i,k > 0; a(i,1,k) = Sum[a(i-1,0,m), {m=0...k}], for i > 0. a(1,1,k) = Fib(k+2) - 1; a(2,1,k) = Fib(k+3) - 2; a(3,1,k) = Luc(k+2) - 3; a(4,1,k) = 4Fib(k+1) + Fib(k) - 4; a(1,2,k) = 2^k - Fib(k+1); a(2,2,k) = 2^(k+1) - Fib(k+3); a(3,2,k) = 3(2^k - Fib(k+2)) + Fib(k); a(4,2,k) = 2^(k+2) - Fib(k+4) - Fib(k+2); a(1,3,k) = (3^k + Luc(k-1)) / 5, for k > 0; a(2,3,k) = (6(3^(k-1)) - Luc(k)) / 5, for k > 0; a(3,3,k) = (3^(k+1) - Luc(k+2)) / 5; a(4,3,k) = (4(3^k) - Luc(k+2) - Luc(k+1)) / 5...all for which Fib(k) denotes the k-th Fibonacci number and Luc(k) denotes the k-th Lucas number [A000032].

Links

Table of n, a(n) for n=0..18.

Eric Weisstein's World of Mathematics, Fibonacci Number

Eric Weisstein's World of Mathematics, Lucas Number

Formula

a(i, j, 0) = 0, a(i, j, 1) = 1, a(i, j, 2) = i+1; a(i, j, k) = ((j+1)*a(i, j, k-1)) - ((j-1)*a(i, j, k-2)) - (j*a(i, j, k-3)), for k > 2.

a(i, j, k) = Fib(k) + i*a(j, j, k-1), for i, k > 0, where Fib(k) denotes the k-th Fibonacci number.

a(i, j, k) = (Phi^k - (-Phi)^-k + i(((j^k - Phi^k) / (j - Phi)) - ((j^k - (-Phi)^-k) / (j - (-Phi)^-1)))) / sqrt(5), where Phi denotes the golden mean/ratio [A001622].

i^k = a(i-1, i, k) + a(i-2, i, k+1). A104161(k) = Sum[a(k-m, 0, m), {m=0...k}].

a(i, j, 0) = 0, a(i, j, 1) = 1, a(i, j, 2) = i+1, a(i, j, 3) = i(j+1) + 2; a(i, j, k) = ((j+2)a(i, j, k-1)) - ((2j)a(i, j, k-2)) - a(i, j, k-3) + (ja(i, j, k-4)), for k > 3. a(i, j, 0) = 0, a(i, j, 1) = 1; a(i, j, k) = a(i, j, k-1) + a(i, j, k-2) + (ij^(k-2)), for k > 1.

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

a(n, n, n) = Sum[Fibonacci(n-k)n^k, {k, 0, n}]. - Ross La Haye, Jan 14 2006

Sum[C(k,m)(i-1)^m,{m,0,k}] = a(i-1,i,k) + a(i-2,i,k+1), for i > 1. - Ross La Haye, May 29 2006

a(3, 3, k+1) - a(3, 3, k) = A106517(k). a(1, 1, k) = A001924(k) - A001924(k-1), for k > 0; a(2, 1, k) = A001891(k) - A001891(k-1), for k > 0; a(3, 1, k) = A023537(k) - A023537(k-1), for k > 0; Sum[a(i-j+1, 0, j), {j, 0, i+1}] - Sum[a(i-j, 0, j), {j, 0, i}] = A001595(i). - Ross La Haye, Jun 03 2006

a(i,j,k) = a(j,j,k) + (i-j)a(j,j,k-1), for k > 0.

a(n) ~ n^(n-1). - Vaclav Kotesovec, Jan 03 2021

Example

a(1,3,3) = 6 because a(1,3,0) = 0, a(1,3,1) = 1, a(1,3,2) = 2 and 4*2 - 2*1 - 3*0 = 6.

Mathematica

Join[{0}, Table[Sum[Fibonacci[n-k]*n^k, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 03 2021 *)

Prog

(PARI) a(n)=sum(k=0, n, fibonacci(n-k)*n^k) \\ Joerg Arndt, Jan 03 2021

Crossrefs

a(0, j, k) = A000045(k); a(1, 0, k) = A000045(k+1), for k > 0; a(1, 1, k) = A000071(k+2); a(1, 2, k) = A027934(k-1), for k > 0; a(1, 3, k) = A098703(k); a(2, 0, k) = A000032(k), for k > 0; a(2, 1, k) = A001911(k); a(2, 2, k) = A008466(k+1); a(3, 0, k) = A000285(k-1), for k > 0; a(3, 1, k) = A027961(k); a(3, 3, k) = A094688(k); a(4, 0, k) = A022095(k-1), for k > 0; a(4, 1, k) = A053311(k-1), for k > 0; a(4, 2, k) = A027974(k-1), for k > 0; a(5, 0, k) = A022096(k-1), for k > 0; a(6, 0, k) = A022097(k-1), for k > 0; a(i, 0, i) = A094588(i).

a(2, 3, k) = A106517(k-1), for k > 0; a(1, 2, k+1) - a(1, 2, k) = A099036(k); a(3, 2, k+1) - a(3, 2, k) = A104004(k); a(4, 2, k+1) - a(4, 2, k) = A027973(k); a(1, 3, k+1) - a(1, 3, k) = A099159(k).

a(i, 0, i+1) = A007502(i+1); a(i, 0, i+2) = A088209(i).

a(i, 0, k) = A109754(i, k).

a(14, 0, k) = A022105(k-1), for k > 0.

a(7, 0, k) = A022098(k-1), for k > 0; a(8, 0, k) = A022099(k-1), for k > 0; a(9, 0, k) = A022100(k-1), for k > 0; a(10, 0, k) = A022101(k-1), for k > 0; a(11, 0, k) = A022102(k-1), for k > 0; a(12, 0, k) = A022103(k-1), for k > 0; a(13, 0, k) = A022104(k-1), for k > 0; a(15, 0, k) = A022106(k-1), for k > 0; a(16, 0, k) = A022107(k-1), for k > 0; a(17, 0, k) = A022108(k-1), for k > 0; a(18, 0, k) = A022109(k-1), for k > 0; a(19, 0, k) = A022110(k-1), for k > 0.

a(i, i+1, 3) = A002522(i+1); a(i, i+1, 4) = A071568(i+1).

a(2^i-2, 0, k+1) = A118654(i, k), for i > 0.

Sequence in context: A125788 A183611 A259903 * A078456 A195134 A089462

Adjacent sequences: A101217 A101218 A101219 * A101221 A101222 A101223

Keyword

nonn,easy,changed

Author

Ross La Haye, Dec 14 2004

Extensions

New name from Joerg Arndt, Jan 03 2021

Status

approved