A094588 -id:A094588 - cp's OEIS frontend

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

Ross La Haye, Dec 14 2004

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-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_{m=0..k} a(i-1,j,m), 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_{m=0..k} a(i-1,0,m), for i > 0. With F = Fibonacci and L = Lucas, then a(1,1,k) = F(k+2) - 1; a(2,1,k) = F(k+3) - 2; a(3,1,k) = L(k+2) - 3; a(4,1,k) = 4*F(k+1) + F(k) - 4; a(1,2,k) = 2^k - F(k+1); a(2,2,k) = 2^(k+1) - F(k+3); a(3,2,k) = 3(2^k - F(k+2)) + F(k); a(4,2,k) = 2^(k+2) - F(k+4) - F(k+2); a(1,3,k) = (3^k + L(k-1))/5, for k > 0; a(2,3,k) = (2 * 3^k - L(k)) /5, for k > 0; a(3,3,k) = (3^(k+1) - L(k+2))/5; a(4,3,k) = (4 * 3^k - L(k+2) - L(k+1))/5, etc..

			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.

G. C. Greubel, Table of n, a(n) for n = 0..385
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Lucas Number

a(0, j, k) = A000045(k).
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, k) = A109754(i, k).
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.
Sequences of the form a(n, 0, k): A000045(k+1) (n=1), A000032(k) (n=2), A000285(k-1) (n=3), A022095(k-1) (n=4), A022096(k-1) (n=5), A022097(k-1) (n=6), A022098(k-1) (n=7), A022099(k-1) (n=8), A022100(k-1) (n=9), A022101(k-1) (n=10), A022102(k-1) (n=11), A022103(k-1) (n=12), A022104(k-1) (n=13), A022105(k-1) (n=14), A022106(k-1) (n=15), A022107(k-1) (n=16), A022108(k-1) (n=17), A022109(k-1) (n=18), A022110(k-1) (n=19), A088209(k-2) (n=k-2), A007502(k) (n=k-1), A094588(k) (n=k).
Sequences of the form a(1, n, k): A000071(k+2) (n=1), A027934(k-1) (n=2), A098703(k) (n=3).
Sequences of the form a(2, n, k): A001911(k) (n=1), A008466(k+1) (n=2), A106517(k-1) (n=3).
Sequences of the form a(3, n, k): A027961(k) (n=1), A094688(k) (n=3).
Sequences of the form a(4, n, k): A053311(k-1) (n=1), A027974(k-1) (n=2).
Cf. A001622, A001891, A001924, A023537, A104161, A106517.

A101220:= func< n | (&+[n^k*Fibonacci(n-k): k in [0..n]]) >;
[A101220(n): n in [0..30]]; // G. C. Greubel, Jun 01 2025

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

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

def A101220(n): return sum(n^k*fibonacci(n-k) for k in range(n+1))
print([A101220(n) for n in range(31)]) # G. C. Greubel, Jun 01 2025

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) = Fibonacci(k) + i*a(j, j, k-1), for i, k > 0.
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_{m=0..k} a(k-m, 0, m).
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) - 2*j*a(i, j, k-2) - a(i, j, k-3) + j*a(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) + i * j^(k-2), for k > 1.
G.f.: x*(1 + (i-j)*x)/((1-j*x)*(1-x-x^2)).
a(n, n, n) = Sum_{k=0..n} Fibonacci(n-k) * n^k. - Ross La Haye, Jan 14 2006
Sum_{m=0..k} binomial(k,m)*(i-1)^m = a(i-1,i,k) + a(i-2,i,k+1), for i > 1. - Ross La Haye, May 29 2006
From Ross La Haye, Jun 03 2006: (Start)
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_{j=0..i+1} a(i-j+1, 0, j) - Sum_{j=0..i} a(i-j, 0, j) = A001595(i). (End)
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

New name from Joerg Arndt, Jan 03 2021

A109754 Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 3, 0, 1, 4, 4, 5, 5, 0, 1, 5, 5, 7, 8, 8, 0, 1, 6, 6, 9, 11, 13, 13, 0, 1, 7, 7, 11, 14, 18, 21, 21, 0, 1, 8, 8, 13, 17, 23, 29, 34, 34, 0, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55, 0, 1, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89

Offset: 0

Ross La Haye, Aug 11 2005; corrected Apr 14 2006

Lower triangular version is at A117501. - Ross La Haye, Apr 12 2006

			Table starts:
[0] 0, 1,  1,  2,  3,  5,  8, 13,  21,  34, ...
[1] 0, 1,  2,  3,  5,  8, 13, 21,  34,  55, ...
[2] 0, 1,  3,  4,  7, 11, 18, 29,  47,  76, ...
[3] 0, 1,  4,  5,  9, 14, 23, 37,  60,  97, ...
[4] 0, 1,  5,  6, 11, 17, 28, 45,  73, 118, ...
[5] 0, 1,  6,  7, 13, 20, 33, 53,  86, 139, ...
[6] 0, 1,  7,  8, 15, 23, 38, 61,  99, 160, ...
[7] 0, 1,  8,  9, 17, 26, 43, 69, 112, 181, ...
[8] 0, 1,  9, 10, 19, 29, 48, 77, 125, 202, ...
[9] 0, 1, 10, 11, 21, 32, 53, 85, 138, 223, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows: A000045(j); A000045(j+1), for j > 0; A000032(j), for j > 0; A000285(j-1), for j > 0; A022095(j-1), for j > 0; A022096(j-1), for j > 0; A022097(j-1), for j > 0. Diagonals: a(i, i) = A094588(i); a(i, i+1) = A007502(i+1); a(i, i+2) = A088209(i); Sum[a(i-j, j), {j=0...i}] = A104161(i). a(i, j) = A101220(i, 0, j).
Rows 7 - 19: A022098(j-1), for j > 0; A022099(j-1), for j > 0; A022100(j-1), for j > 0; A022101(j-1), for j > 0; A022102(j-1), for j > 0; A022103(j-1), for j > 0; A022104(j-1), for j > 0; A022106(j-1), for j > 0; A022107(j-1), for j > 0; A022108(j-1), for j > 0; A022109(j-1), for j > 0; A022110(j-1), for j > 0.
a(2^i-2, j+1) = A118654(i, j), for i > 0.
Cf. A117501.

A := (n, k) -> ifelse(k = 0, 0,
      n*combinat:-fibonacci(k-1) + combinat:-fibonacci(k)):
seq(seq(A(n - k, k), k = 0..n), n = 0..6); # Peter Luschny, May 28 2022

T[n_, 0]:= 0; T[n_, 1]:= 1; T[n_, 2]:= n - 1; T[n_, 3]:= n - 1; T[n_, n_]:= Fibonacci[n]; T[n_, k_]:= T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Jan 07 2017 *)

a(i, 0) = 0, a(i, j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0.
a(i, 0) = 0, a(i, 1) = 1, a(i, 2) = i+1, a(i, j) = a(i, j-1) + a(i, j-2), for j > 2.
G.f.: (x*(1 + ix))/(1 - x - x^2).
Sum_{j=0..i+1} a(i-j+1, j) - Sum_{j=0..i} a(i-j, j) = A001595(i). - Ross La Haye, Jun 03 2006

More terms from G. C. Greubel, Jan 07 2017

A354044 a(n) = 2(-i)^n(nsin(c(n+1)) - isin(-cn))/sqrt(5) where c = arccos(i/2).

Original entry on oeis.org

0, 2, 5, 11, 23, 45, 86, 160, 293, 529, 945, 1673, 2940, 5134, 8917, 15415, 26539, 45525, 77842, 132716, 225685, 382877, 648165, 1095121, 1846968, 3109850, 5228261, 8777315, 14716223, 24643389, 41220110, 68873848, 114964805, 191719849, 319436697, 531789785

Offset: 0

Peter Luschny, May 16 2022

Jianing Song, Table of n, a(n) for n = 0..1000
Peter Luschny, Illustration of A354044.
Peter Luschny, The Fibonacci Function.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045 (the Fibonacci numbers), A007502, A088209, A094588, A136391, A178521, A264147, A353595.

function fibrec(n::Int)
    n == 0 && return (BigInt(0), BigInt(1))
    a, b = fibrec(div(n, 2))
    c = a * (b * 2 - a)
    d = a * a + b * b
    iseven(n) ? (c, d) : (d, c + d)
end
function A354044(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n + 1)
    a*(n - 1) + b
end
println([A354044(n) for n in 0:35])

c := arccos(I/2): a := n -> 2*(-I)^n*(n*sin(c*(n+1)) - I*sin(-c*n))/sqrt(5):
seq(simplify(a(n)), n = 0..35);

a(n) = fibonacci(n) + n*fibonacci(n+1) \\ Jianing Song, May 16 2022

a(n) = [x^n] ((2 - x)*x*(x + 1))/(x^2 + x - 1)^2.
a(n) = (((-1 - sqrt(5))^(-n)*(sqrt(5)*n - n - 2) + (-1 + sqrt(5))^(-n)*(sqrt(5)*n + n + 2)))/(2^(1 - n)*sqrt(5)).
a(n) = (-1)^(n - 1)*(Fibonacci(-n) - n*Fibonacci(-n - 1)).
a(n) = (-1)^(n - 1)*A353595(-n, -n). (A353595 is defined for all n in Z.)
a(n) = ((-42*n^2 + 259*n - 350)*a(n - 3) + (123*n^2 - 76*n - 446)*a(n - 2) + (207*n^2 - 885*n + 412)*a(n - 1)) / ((165*n - 542)*(n - 1)) for n >= 4.
a(n) = Fibonacci(n) + n*Fibonacci(n+1). - Jianing Song, May 16 2022

A264147 a(n) = nF(n+1) - (n+1)F(n), where F = A000045.

Original entry on oeis.org

0, -1, 1, 1, 5, 10, 22, 43, 83, 155, 285, 516, 924, 1639, 2885, 5045, 8773, 15182, 26162, 44915, 76855, 131119, 223101, 378696, 641400, 1084175, 1829257, 3081193, 5181893, 8702290, 14594830, 24446971, 40902299, 68359619, 114132765, 190373580, 317258388, 528265207

Offset: 0

Bruno Berselli, Nov 04 2015

a(n) is prime for n = 4, 7, 8, 26, 28, 52, 86, 87, 93, 97, 158, 196, 303, 2908, 3412, 4111, 4208, 6183, 6337, 9878, ...

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A001629, A007502, A354044.
Cf. A178521: n*F(n+1) + (n+1)*F(n).
Cf. A094588: n*F(n-1) + F(n).
Cf. A099920: Sum_{i=0..n} F(i)*L(n-i).
Cf. A023607: Sum_{i=0..n} F(i)*L(n+1-i).

# The function 'fibrec' is defined in A354044.
function A264147(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n)
    n*b - a*(n + 1)
end # Peter Luschny, May 16 2022

[n*Fibonacci(n+1)-(n+1)*Fibonacci(n): n in [0..40]];

A264147 := proc(n)
    n*combinat[fibonacci](n+1)-(n+1)*combinat[fibonacci](n) ;
end proc:
seq(A264147(n),n=0..10) ; # R. J. Mathar, Jun 02 2022

Table[n Fibonacci[n + 1] - (n + 1) Fibonacci[n], {n, 0, 40}]

makelist(n*fib(n+1)-(n+1)*fib(n), n, 0, 40);

for(n=0, 40, print1(n*fibonacci(n+1)-(n+1)*fibonacci(n)", "));

concat(0, Vec(-x*(1 - 3*x) / (1 - x - x^2)^2 + O(x^50))) \\ Colin Barker, Jul 27 2017

[n*fibonacci(n+1)-(n+1)*fibonacci(n) for n in (0..40)]

G.f.:  x*(-1 + 3*x)/(1 - x - x^2)^2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4).
a(n) = n*F(n-1) - F(n).
a(n) = Sum_{i=0..n} F(i)*L(n-1-i), where L() is a Lucas number (A000032).
a(n) = 3*A001629(n) - A001629(n+1).
a(n) = -(-1)^n*A178521(-n).
a(n+2) - a(n) = A007502(n+1).
Sum_{i>0} 1/a(i) = 1.39516607051636028893879220294180374...
a(n) = (-((1+sqrt(5))/2)^n*(2*sqrt(5) + (-5+sqrt(5))*n) + ((1-sqrt(5))/2)^n*(2*sqrt(5) + (5+sqrt(5))*n)) / 10. - Colin Barker, Jul 27 2017
a(n) = (-i)^n*(n*sin(c*(n+1)) - (n+1)*sin(c*n)*i)/sqrt(5/4) where c = arccos(i/2). - Peter Luschny, May 16 2022

A353595 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^(k - 1)(phi + n) - phi^(k - 1)(psi + n)) / (psi - phi) where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 1, 3, 3, 3, 5, 1, 4, 4, 5, 5, 6, 1, 5, 5, 7, 8, 8, 7, 1, 6, 6, 9, 11, 13, 13, 8, 1, 7, 7, 11, 14, 18, 21, 21, 9, 1, 8, 8, 13, 17, 23, 29, 34, 34, 10, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55, 11, 1, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89

Offset: 0

Peter Luschny, May 09 2022

The definition declares the Fibonacci numbers for all integers n and k. It gives the classical Fibonacci numbers as F(0, n) = A000045(n). A different enumeration is given in A352744.

			Array starts:
n\k 0, 1,  2,  3,  4,  5,  6,   7,   8,   9, ...
--------------------------------------------------------
[0]  0, 1,  1,  2,  3,  5,  8, 13,  21,  34, ... A000045
[1]  1, 1,  2,  3,  5,  8, 13, 21,  34,  55, ... A000045 (shifted once)
[2]  2, 1,  3,  4,  7, 11, 18, 29,  47,  76, ... A000032
[3]  3, 1,  4,  5,  9, 14, 23, 37,  60,  97, ... A104449
[4]  4, 1,  5,  6, 11, 17, 28, 45,  73, 118, ... [4] + A022095
[5]  5, 1,  6,  7, 13, 20, 33, 53,  86, 139, ... [5] + A022096
[6]  6, 1,  7,  8, 15, 23, 38, 61,  99, 160, ... [6] + A022097
[7]  7, 1,  8,  9, 17, 26, 43, 69, 112, 181, ... [7] + A022098
[8]  8, 1,  9, 10, 19, 29, 48, 77, 125, 202, ... [8] + A022099
[9]  9, 1, 10, 11, 21, 32, 53, 85, 138, 223, ... [9] + A022100

Peter Luschny, The Fibonacci Function.
Wikipedia, Cassini and Catalan identities.

Cf. A000045, A000032, A104449, A094588 (main diagonal).

Cf. A352744, A354265 (generalized Lucas numbers).

function fibrec(n::Int)
    n == 0 && return (BigInt(0), BigInt(1))
    a, b = fibrec(div(n, 2))
    c = a * (b * 2 - a)
    d = a * a + b * b
    iseven(n) ? (c, d) : (d, c + d)
end
function Fibonacci(n::Int, k::Int)
    k == 0 && return BigInt(n)
    k == 1 && return BigInt(1)
    k  < 0 && return (-1)^(k-1)*Fibonacci(-n - 1, 2 - k)
    a, b = fibrec(k - 1)
    a*n + b
end
for n in -6:6
    println([n], [Fibonacci(n, k) for k in -6:6])
end

f := n -> combinat:-fibonacci(n): F := (n, k) -> n*f(k - 1) + f(k):
seq(seq(F(n - k, k), k = 0..n), n = 0..11);
# The next implementation is for illustration only but is not recommended
# as it relies on floating point arithmetic. Illustrates the case n,k < 0.
phi := (1 + sqrt(5))/2: psi := (1 - sqrt(5))/2:
F := (n, k) -> (psi^(k-1)*(psi + n) - phi^(k-1)*(phi + n)) / (psi - phi):
for n from -6 to 6 do lprint(seq(simplify(F(n, k)), k = -6..6)) od;

(* Works also for n < 0 and k < 0. Uses a remark from Bill Gosper. *)
c := I*ArcSinh[1/2] - Pi/2;
F[n_, k_] := (n Sin[(k - 1) c] - I Sin[k c]) / (I^k Sqrt[5/4]);
Table[Simplify[F[n, k]], {n, 0, 6}, {k, 0, 6}] // TableForm

Functional equation extends Cassini's theorem:
F(n, k) = (-1)^(k - 1)*F(-n - 1, 2 - k).
F(n, k) = ((1 - phi)^(k - 1)*(1 - phi + n) - phi^(k - 1)*(phi + n))/(1 - 2*phi).
F(n, k) = n*fib(k - 1) + fib(k), where fib(n) are the classical Fibonacci numbers A000045 extended in the usual way for negative n.
F(n, k) - F(n-1, k) = fib(k-1).
F(n, k) = F(n, k-1) + F(n, k-2).
F(n, k) = (n*sin((k - 1)*c) - i*sin(k*c))/(i^k*sqrt(5/4)) where c = i*arcsinh(1/2) - Pi/2, for all n, k in Z. Based on a remark of Bill Gosper.

cp's OEIS Frontend

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

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A109754 Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.

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A354044 a(n) = 2*(-i)^n*(n*sin(c*(n+1)) - i*sin(-c*n))/sqrt(5) where c = arccos(i/2).

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A264147 a(n) = n*F(n+1) - (n+1)*F(n), where F = A000045.

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A353595 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^(k - 1)*(phi + n) - phi^(k - 1)*(psi + n)) / (psi - phi) where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.

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Formula

A354044 a(n) = 2(-i)^n(nsin(c(n+1)) - isin(-cn))/sqrt(5) where c = arccos(i/2).

A264147 a(n) = nF(n+1) - (n+1)F(n), where F = A000045.

A353595 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^(k - 1)(phi + n) - phi^(k - 1)(psi + n)) / (psi - phi) where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.