A007502 -id:A007502 - 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

A088209 Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,...(n 1's)...,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], ...

Original entry on oeis.org

1, 3, 7, 14, 28, 53, 99, 181, 327, 584, 1034, 1817, 3173, 5511, 9527, 16402, 28136, 48109, 82023, 139481, 236631, 400588, 676822, 1141489, 1921993, 3231243, 5424679, 9095126, 15230452, 25475429, 42566379, 71052157, 118489383

Offset: 0

Paul D. Hanna, Sep 23 2003

Denominators form the Les Marvin sequence: A007502(n+1).

			a(3)/A007502(4) = [1;1,1,4] = 14/9.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

a(n) = A109754(n, n+2) = A101220(n, 0, n+2).

Cf. A007502 (the denominators), A000045, A045925.

a088209 n = a088209_list !! n
a088209_list = zipWith (+) a000045_list $ tail a045925_list
-- Reinhard Zumkeller, Oct 01 2012, Mar 04 2012

# The function 'fibrec' is defined in A354044.
function A088209(n)
    n == 0 && return BigInt(1)
    a, b = fibrec(n)
    a + (n + 1)*b
end
println([A088209(n) for n in 0:32]) # Peter Luschny, May 18 2022

f[n_] := Numerator@  FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* Robert G. Wilson v, Mar 04 2012 *)
CoefficientList[Series[(1+x-x^3)/(-1+x+x^2)^2,{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2,-1},{1,3,7,14},40] (* Harvey P. Dale, Jul 13 2021 *)

G.f.: (1+x-x^3)/(1-x-x^2)^2. [Corrected by Georg Fischer, Aug 16 2021]
a(n) = Fibonacci(n) + (n+1)*Fibonacci(n+1). - Paul Barry, Apr 20 2004
a(n) = a(n-1) + a(n-2) + Lucas(n). - Yuchun Ji, Apr 23 2023

A352744 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^k(phi - n) - phi^k(psi - n)) / (phi - psi) 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

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

Offset: 0

Peter Luschny, Apr 01 2022

The definition declares the Fibonacci numbers for all integers n and k. An alternative version is A353595.
The identity F(n, k) = (-1)^k*F(1 - n, -k) holds for all integers n, k. Proof:
     F(n, k)*(2+phi) = (phi^k*(n*phi + 1) - (-phi)^(-k)*((n-1)*phi - 1))
                     = (-1)^k*(phi^(-k)*((1-n)*phi+1) - (-phi)^k*(-n*phi-1))
                     = (-1)^k*F(1-n, -k)*(2+phi).
This identity can be seen as an extension of Cassini's theorem of 1680 and of an identity given by Graham, Knuth and Patashnik in 'Concrete Mathematics' (6.106 and 6.107). The beginning of the full array with arguments in Z x Z can be found in the linked note.
The enumeration is the result of the simple form of the chosen definition. The classical positive Fibonacci numbers starting with 1, 1, 2, 3,... are in row n = 1 with offset 0. The nonnegative Fibonacci numbers starting 0, 1, 1, 2, 3,... are in row 0 with offset 1. They prolong towards -infinity with an index shifted by 1 compared to the enumeration used by Knuth. A characteristic of our enumeration is F(n, 0) = 1 for all integer n.
Fibonacci numbers vanish only for (n,k) in {(-1,2), (0,1), (1,-1), (2,-2)}. The zeros correspond to the identities (phi + 1)*psi^2 = (psi + 1)*phi^2, psi*phi = phi*psi, (phi - 1)*phi = (psi - 1)*psi and (phi - 2)*phi^2 = (psi - 2)*psi^2.
For divisibility properties see A352747.
For any fixed k, the sequence F(n, k) is a linear function of n. In other words, an arithmetic progression. This implies that F(n+1, k) = 2*F(n, k) - F(n-1, k) for all n in Z. Special case of this is Fibonacci(n+1) = 2 *Fibonacci(n) - Fibonacci(n-2). - Michael Somos, May 08 2022

			Array starts:
n\k 0, 1,  2,  3,  4,  5,  6,   7,   8,   9, ...
---------------------------------------------------------
[0] 1, 0,  1,  1,  2,  3,  5,   8,  13,  21, ... A212804
[1] 1, 1,  2,  3,  5,  8, 13,  21,  34,  55, ... A000045 (shifted once)
[2] 1, 2,  3,  5,  8, 13, 21,  34,  55,  89, ... A000045 (shifted twice)
[3] 1, 3,  4,  7, 11, 18, 29,  47,  76, 123, ... A000032 (shifted once)
[4] 1, 4,  5,  9, 14, 23, 37,  60,  97, 157, ... A000285
[5] 1, 5,  6, 11, 17, 28, 45,  73, 118, 191, ... A022095
[6] 1, 6,  7, 13, 20, 33, 53,  86, 139, 225, ... A022096
[7] 1, 7,  8, 15, 23, 38, 61,  99, 160, 259, ... A022097
[8] 1, 8,  9, 17, 26, 43, 69, 112, 181, 293, ... A022098
[9] 1, 9, 10, 19, 29, 48, 77, 125, 202, 327, ... A022099

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, sec. 6.6.
Donald Ervin Knuth, The Art of Computer Programming, Third Edition, Vol. 1, Fundamental Algorithms. Chapter 1.2.8 Fibonacci Numbers. Addison-Wesley, Reading, MA, 1997.

Alexander Bogomolny, Cassini's Identity.
Edsger W. Dijkstra, In honour of Fibonacci, in: F. L. Bauer, M. Broy, & E. W. Dijkstra (editors), Program Construction, 1979, Lecture Notes in Computer Science, Vol. 69.
Peter Luschny, The Fibonacci Function.

Rows: A212804, A000045, A000032, A000285, A022095, A022096, A022097, A022098, A022099.
Columns: A000012, A001477, A000027, A005408, A016789, A016885, A004770.
Diagonals: A088209 (main), A007502, A066982 (antidiagonal sums).
Cf. A352747, A353595 (alternative version), A354265 (generalized Lucas numbers).
Similar arrays based on the Catalan and the Bell numbers are A352680 and A352682.

# Time complexity is O(lg n).
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(1)
    k  < 0 && return (-1)^k*Fibonacci(1 - n, -k)
    a, b = fibrec(k - 1)
    a + b*n
end
for n in -6:6
    println([Fibonacci(n, k) for k in -6:6])
end

f := n -> combinat:-fibonacci(n + 1): F := (n, k) -> (n-1)*f(k-1) + f(k):
seq(seq(F(n-k, k), k = 0..n), n = 0..9);
# The next implementation is for illustration only but is not recommended
# as it relies on floating point arithmetic.
phi := (1 + sqrt(5))/2: psi := (1 - sqrt(5))/2:
F := (n, k) -> (psi^k*(phi - n) - phi^k*(psi - n)) / (phi - psi):
for n from -6 to 6 do lprint(seq(simplify(F(n, k)), k = -6..6)) od;

Table[LinearRecurrence[{1, 1}, {1, n}, 10], {n, 0, 9}] // TableForm
F[ n_, k_] := (MatrixPower[{{0, 1}, {1, 1}}, k].{{1}, {n}})[[1, 1]]; (* Michael Somos, May 08 2022 *)
c := Pi/2 - I*ArcSinh[1/2]; (* Based on a remark from Bill Gosper. *)
F[n_, k_] := 2 (I (n-1) Sin[k c] + Sin[(k+1) c]) / (I^k Sqrt[5]);
Table[Simplify[F[n, k]], {n, -6, 6}, {k, -6, 6}] // TableForm (* Peter Luschny, May 10 2022 *)

F(n, k) = ([0, 1; 1, 1]^k*[1; n])[1, 1]

{F(n, k) = n*fibonacci(k) + fibonacci(k-1)}; /* Michael Somos, May 08 2022 */

F(n, k) = F(n, k-1) + F(n, k-2) for k >= 2, otherwise 1, n for k = 0, 1.
F(n, k) = (n-1)*f(k-1) + f(k) where f(n) = A000045(n+1), the Fibonacci numbers starting with f(0) = 1.
F(n, k) = ((phi^k*(n*phi + 1) - (-phi)^(-k)*((n - 1)*phi - 1)))/(2 + phi).
F(n, k) = [x^k] (1 + (n - 1)*x)/(1 - x - x^2) for k >= 0.
F(k, n) = [x^k] (F(0, n) + F(0, n-1)*x)/(1 - x)^2 for k >= 0.
F(n, k) = (k!/sqrt(5))*[x^k] ((n-psi)*exp(phi*x) - (n-phi)*exp(psi*x)) for k >= 0.
F(n, k) - F(n-1, k) = sign(k)^(n-1)*f(k) for all n, k in Z, where A000045 is extended to negative integers by f(-n) = (-1)^(n-1)*f(n) (CMath 6.107). - Peter Luschny, May 09 2022
F(n, k) = 2*((n-1)*i*sin(k*c) + sin((k+1)*c))/(i^k*sqrt(5)) where c = Pi/2 - i*arcsinh(1/2), for all n, k in Z. Based on a remark from Bill Gosper. - Peter Luschny, May 10 2022

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

A094588 a(n) = n*F(n-1) + F(n), where F = A000045.

Original entry on oeis.org

0, 1, 3, 5, 11, 20, 38, 69, 125, 223, 395, 694, 1212, 2105, 3639, 6265, 10747, 18376, 31330, 53277, 90385, 153011, 258523, 436010, 734136, 1234225, 2072043, 3474029, 5817515, 9730748, 16258910, 27139509, 45258917, 75408775, 125538539

Offset: 0

Paul Barry, May 13 2004

This is the transform of the Fibonacci numbers under the inverse of the signed permutations matrix (see A094587).

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A007502, A045925, A088209, A354044.

a094588 n = a094588_list !! n
a094588_list = 0 : zipWith (+) (tail a000045_list)
                               (zipWith (*) [1..] a000045_list)
-- Reinhard Zumkeller, Mar 04 2012

# The function 'fibrec' is defined in A354044.
function A094588(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n - 1)
    a*n + b
end
println([A094588(n) for n in 0:34]) # Peter Luschny, May 16 2022

[n*Fibonacci(n-1)+Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

CoefficientList[Series[x (1+x-2x^2)/(1-x-x^2)^2,{x,0,40}],x]  (* Harvey P. Dale, Apr 16 2011 *)

Vec((1+x-2*x^2)/(1-x-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Mar 04 2012

G.f. : x*(1 + x - 2*x^2)/(1 - x - x^2)^2.
a(n) = A101220(n, 0, n) - Ross La Haye, Jan 28 2005
a(n) = A109754(n, n). - Ross La Haye, Aug 20 2005
a(n) = (sin(c*n)*i - n*sin(c*(n - 1)))/(i^n*sqrt(5/4)) where c = arccos(i/2). - Peter Luschny, 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

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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A088209 Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,...(n 1's)...,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], ...

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A352744 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^k*(phi - n) - phi^k*(psi - n)) / (phi - psi) 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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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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A094588 a(n) = n*F(n-1) + F(n), where F = A000045.

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A352744 Array read by ascending antidiagonals. Generalized Fibonacci numbers F(n, k) = (psi^k(phi - n) - phi^k(psi - n)) / (phi - psi) where phi = (1 + sqrt(5))/2 and psi = (1 - sqrt(5))/2. F(n, k) for n >= 0 and k >= 0.

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.