A022101 -id:A022101 - 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

A127830 a(n) = Sum_{k=0..n} (binomial(floor(k/2),n-k) mod 2).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 3, 2, 2, 3, 2, 3, 5, 5, 4, 4, 5, 4, 3, 3, 3, 4, 4, 3, 4, 5, 3, 5, 8, 8, 7, 6, 7, 7, 5, 6, 8, 7, 6, 5, 5, 5, 4, 4, 5, 6, 5, 5, 7, 6, 4, 5, 6, 7, 7, 5, 6, 8, 5, 8, 13, 13, 11, 10, 12, 11, 8, 9, 11, 11, 10, 8, 9, 10, 7, 9, 13, 12

Offset: 0

Paul Barry, Feb 01 2007

Row sums of number triangle A127829.
From Johannes W. Meijer, Jun 05 2011: (Start)
The Ze3 and Ze4 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence.
The sequences A127830(2^n-p), p>=0, are apparently all Fibonacci like sequences, i.e., the next term is the sum of the two nonzero terms that precede it; see the crossrefs. (End)

Cf.: A000045 (p=0), A000204 (p=7), A001060 (p=13), A000285 (p=14), A022095 (p=16), A022120 (p=24), A022121 (p=25), A022113 (p=28), A022096 (p=30), A022097 (p=31), A022098 (p=32), A022130 (p=44), A022137 (p=48), A022138 (p=49), A022122 (p=52), A022114 (p=53), A022123 (p=56), A022115 (p=60), A022100 (p=62), A022101 (p=63), A022103 (p=64), A022136 (p=79), A022388 (p=80), A022389 (p=88). - Johannes W. Meijer, Jun 05 2011

A127830 := proc(n) local k: option remember: add(binomial(floor(k/2), n-k) mod 2, k=0..n) end: seq(A127830(n), n=0..80); # Johannes W. Meijer, Jun 05 2011

Table[Sum[Mod[Binomial[Floor[k/2],n-k],2],{k,0,n}],{n,0,80}] (* James C. McMahon, Jan 04 2025 *)

def A127830(n): return sum(not ~(k>>1)&n-k for k in range(n+1)) # Chai Wah Wu, Jul 29 2025

a(2^n) = F(n); a(2^(n+1)+1) = L(n).

a(n) mod 2 = A000931(n+5) mod 2 = A011656(n+4).

A022315 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 10.

Original entry on oeis.org

0, 10, 11, 22, 34, 57, 92, 150, 243, 394, 638, 1033, 1672, 2706, 4379, 7086, 11466, 18553, 30020, 48574, 78595, 127170, 205766, 332937, 538704, 871642, 1410347, 2281990, 3692338, 5974329, 9666668

Offset: 0

N. J. A. Sloane

nonn

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

CoefficientList[Series[(10 x-9 x^2)/(1-2 x+x^3),{x,0,30}],x]  (* Harvey P. Dale, Mar 05 2011 *)

x='x+O('x^50); concat([0], Vec(x*(10-9*x)/( (1-x)*(1-x-x^2) ))) \\ G. C. Greubel, Aug 25 2017

From R. J. Mathar, Apr 07 2011: (Start)
G.f.: x*(10-9*x)/( (1-x)*(1-x-x^2) ).
a(n) = A022101(n) - 1. (End)
a(n) = F(n+2) + 9*F(n) - 1, where F = A000045. - G. C. Greubel, Aug 25 2017

A092778 Concatenate pairs of successive Fibonacci numbers.

Original entry on oeis.org

1, 11, 12, 23, 35, 58, 813, 1321, 2134, 3455, 5589, 89144, 144233, 233377, 377610, 610987, 9871597, 15972584, 25844181, 41816765, 676510946, 1094617711, 1771128657, 2865746368, 4636875025, 75025121393, 121393196418

Offset: 0

Jorge Coveiro, Apr 14 2004

Cf. A000045, A022101.

AA = Table[Fibonacci[n]*10^(Floor[Log[10, 10Fibonacci[n + 1]]]) + Fibonacci[n + 1], {n, 1, 12}] (* José de Jesús Camacho Medina, Dec 16 2016 *)

glue(a,b)=a*10^logint(10*b,10)+b
a(n)=glue(fibonacci(n),fibonacci(n+1)) \\ Charles R Greathouse IV, Dec 16 2016

a(n) = A000045(n)*10^(A055642(A000045(n+1))) + A000045(n+1). - José de Jesús Camacho Medina, Dec 16 2016

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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A127830 a(n) = Sum_{k=0..n} (binomial(floor(k/2),n-k) mod 2).

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A022315 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 10.

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A092778 Concatenate pairs of successive Fibonacci numbers.

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