A022103 -id:A022103 - 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).

A022375 Fibonacci sequence beginning 2, 26.

Original entry on oeis.org

2, 26, 28, 54, 82, 136, 218, 354, 572, 926, 1498, 2424, 3922, 6346, 10268, 16614, 26882, 43496, 70378, 113874, 184252, 298126, 482378, 780504, 1262882, 2043386, 3306268, 5349654, 8655922, 14005576, 22661498, 36667074, 59328572, 95995646, 155324218, 251319864, 406644082, 657963946

Offset: 0

N. J. A. Sloane

nonn

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

LinearRecurrence[{1,1}, {2,26}, 50] (* G. C. Greubel, Oct 04 2017 *)

x='x+O('x^50); Vec((2+24*x)/(1-x-x^2)) \\ G. C. Greubel, Oct 04 2017

G.f.: 2*(1+12*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

Equals 2*A022103(n). - Bruno Berselli, Oct 05 2017

Terms a(30) onward added by G. C. Greubel, Oct 04 2017

A022317 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 12.

Original entry on oeis.org

0, 12, 13, 26, 40, 67, 108, 176, 285, 462, 748, 1211, 1960, 3172, 5133, 8306, 13440, 21747, 35188, 56936, 92125, 149062, 241188, 390251, 631440, 1021692, 1653133, 2674826, 4327960, 7002787, 11330748

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).

LinearRecurrence[{2,0,-1}, {0,12,13}, 50] (* G. C. Greubel, Aug 25 2017 *)

x='x+O('x^50); concat([0], Vec(x*(12-11*x)/( (1-x)*(1-x-x^2) ))) \\ G. C. Greubel, Aug 25 2017

From R. J. Mathar, Apr 07 2011: (Start)
G.f.: x*(12-11*x)/( (1-x)*(1-x-x^2) ).
a(n) = A022103(n) - 1. (End)
a(n) = F(n+2) + 11*F(n) - 1, where F = A000045. - G. C. Greubel, Aug 25 2017

A347351 Triangle read by rows: T(n,k) is the number of links of length k in a set of all necklaces A000358 of length n, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 1, 1, 0, 1, 6, 4, 2, 1, 0, 1, 7, 3, 2, 1, 1, 0, 1, 8, 8, 3, 3, 1, 1, 0, 1, 9, 8, 7, 3, 2, 1, 1, 0, 1, 10, 18, 9, 5, 4, 2, 1, 1, 0, 1, 11, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, 12, 40, 24, 16, 8, 6, 3, 2, 1, 1, 0, 1, 13, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1

Offset: 0

Maxim Karimov and Vladislav Sulima, Aug 28 2021

Definitions:
1. A link is any 0 in any necklace from A000358 and all 1s following this 0 in this necklace to right until another 0 is encountered.
2. Length of the link is the number of elements in the link.
Sum of all elements n-row is Fibonacci(n-1)+n iff n=1 or n=p (follows from the identity for the sum of the Fibonacci numbers and the formula for the triangle T(n,k)).

			For k > 0:
   n\k |  1   2   3   4   5   6   7   8   9  10  ...
  -----+---------------------------------------
   1   |  1
   2   |  2   1
   3   |  3   0   1
   4   |  4   2   0   1
   5   |  5   1   1   0   1
   6   |  6   4   2   1   0   1
   7   |  7   3   2   1   1   0   1
   8   |  8   8   3   3   1   1   0   1
   9   |  9   8   7   3   2   1   1   0   1
  10   | 10  18   9   5   4   2   1   1   0   1
  ...
If we continue the calculation for nonpositive k, we get a table in which each row is a Fibonacci sequence, in which term(0) = A113166, term(1) = A034748.
For k <= 0:
   n\k |  0   -1   -2   -3   -4   -5   -6   -7   -8   -9 ...
  -----+------------------------------------------------
   1   |  0    1    1    2    3    5    8   13   21   34 ... A000045
   2   |  1    2    3    5    8   13   21   34   55   89 ... A000045
   3   |  1    4    5    9   14   23   37   60   97  157 ... A000285
   4   |  3    6    9   15   24   39   63  102  165  267 ... A022086
   5   |  3    9   12   21   33   54   87  141  228  369 ... A022379
   6   |  8   14   22   36   58   94  152  246  398  644 ... A022112
   7   |  8   19   27   46   73  119  192  311  503  814 ... A206420
   8   | 17   30   47   77  124  201  325  526  851 1377 ... A022132
   9   | 23   44   67  111  178  289  467  756 1223 1979 ... A294116
  10   | 41   68  109  177  286  463  749 1212 1961 3173 ... A022103
  ...

Cf. A000010, A000027, A000045, A000358, A113166, A034748.

function [res] = calcLinks(n,k)
if k==1
    res=n;
else
    d=divisors(n);
    res=0;
    for i=1:length(d)
        if d (i) >= k
            res=res+eulerPhi(n/d(i))*fiboExt(d(i)-k-1);
        end
    end
end
function [s] = fiboExt(m) % extended fibonacci function (including negative arguments)
m=sym(m); % for large fibonacci numbers
if m>=0 || mod(m,2)==1
    s=fibonacci(abs(m));
else
    s=fibonacci(abs(m))*(-1);
end

T(n, k) = if (k==1, n, sumdiv(n, d, if (d>=k, eulerphi(n/d)*fibonacci(d-k-1)))); \\ Michel Marcus, Aug 29 2021

If k=1, T(n,k)=n, otherwise T(n,k) = Sum_{d>=k, d|n} Phi(n/d)*Fibonacci(d-k-1), where Phi=A000010.

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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A022375 Fibonacci sequence beginning 2, 26.

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A022317 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 0 and a(1) = 12.

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A347351 Triangle read by rows: T(n,k) is the number of links of length k in a set of all necklaces A000358 of length n, 1 <= k <= n.

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