A001924 -id:A001924 - cp's OEIS frontend

A213579 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

1, 3, 2, 7, 5, 3, 14, 11, 7, 4, 26, 21, 15, 9, 5, 46, 38, 28, 19, 11, 6, 79, 66, 50, 35, 23, 13, 7, 133, 112, 86, 62, 42, 27, 15, 8, 221, 187, 145, 106, 74, 49, 31, 17, 9, 364, 309, 241, 178, 126, 86, 56, 35, 19, 10, 596, 507, 397, 295, 211, 146, 98, 63, 39, 21

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213580.
Antidiagonal sums: A053808.
Row 1, (1,1,2,3,5,...)**(1,2,3,4,...): A001924.
Row 2, (1,1,2,3,5,...)**(2,3,4,5,...): A023548.
Row 3, (1,1,2,3,5,...)**(3,4,5,6,...): A023552.
Row 4, (1,1,2,3,5,...)**(4,5,6,7,...): A210730.
Row 5, (1,1,2,3,5,...)**(5,6,7,8,...): A210731.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....3....7....14...26...46
2....5....11...21...38...66
3....7....15...28...50...86
4....9....19...35...62...106
5....11...23...42...74...126
6....13...27...49...86...146

Clark Kimberling, Antidiagonas n = 1..60, flattened

Cf. A000045, A213500.

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2) ))); # G. C. Greubel, Jul 08 2019

[[Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[n]; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213579 *)
r[n_]:= Table[T[n, k], {k, 40}]
d = Table[T[n, n], {n, 1, 40}] (* A213580 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A053808 *)
(* Second program *)
Table[Fibonacci[n-k+4] +k*Fibonacci[n-k+3] -(n+3), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

t(n,k) = fibonacci(n-k+4) + k*fibonacci(n-k+3) - (n+3);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[fibonacci(k+3) + n*fibonacci(k+2) -(n+k+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = n - (n-1)*x and g(x) = (1-x-x^2) *(1-x)^2.
T(n, k) = Fibonacci(k+3) + n*Fibonacci(k+2) - (n+k+2). - G. C. Greubel, Jul 08 2019

A033937 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= 3.

Original entry on oeis.org

2, 7, 17, 35, 66, 118, 204, 345, 575, 949, 1556, 2540, 4134, 6715, 10893, 17655, 28598, 46306, 74960, 121325, 196347, 317737, 514152, 831960, 1346186, 2178223, 3524489, 5702795, 9227370, 14930254, 24157716

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001924, A001891.

List([0..40], n-> Fibonacci(n+7) -3*n-11) # G. C. Greubel, Jul 05 2019

[Fibonacci(n+7) -3*n-11: n in [0..40]]; // G. C. Greubel, Jul 05 2019

Table[Fibonacci[n+7] -3*n-11, {n,0,40}] (* G. C. Greubel, Jul 05 2019 *)

vector(40, n, n--; fibonacci(n+7) -3*n-11) \\ G. C. Greubel, Jul 05 2019

[fibonacci(n+7) -3*n-11 for n in (0..40)] # G. C. Greubel, Jul 05 2019

a(n) = Fibonacci(n+7) - (11+3*n).

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

A120297 Sum of all matrix elements of n X n matrix M(i,j) = Fibonacci(i+j-1).

Original entry on oeis.org

1, 5, 20, 65, 193, 544, 1489, 4005, 10660, 28193, 74273, 195200, 512257, 1343077, 3519412, 9219105, 24144289, 63224096, 165544721, 433437125, 1134810436, 2971065025, 7778499265, 20364618240, 53315655553, 139582833989

Offset: 1

Alexander Adamchuk, Jul 11 2006

nonn

p^2 divides a(p-1) for p = 5, 11, 19, 29, 31, 41, 59, 61, 71, ... = A038872 (Primes congruent to {0, 1, 4} mod 5), also odd primes p such that where 5 is a square mod p. All squared prime divisors of a(n) also belong to A038872.

			Matrix begins:
  1  1  2  3  5
  1  2  3  5  8
  2  3  5  8 13
  3  5  8 13 21
  5  8 13 21 34

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000045, A038872, A001924, A062381.

Table[Sum[Sum[Fibonacci[i+j-1],{i,1,n}],{j,1,n}],{n,1,50}]

a(n) = Sum_{j=1..n} Sum_{i=1..n} Fibonacci(i+j-1).
a(n) = Fibonacci(2*n+3) - 2*Fibonacci(n+3) + 2. - Vladeta Jovovic, Jul 21 2006
G.f.: (1 - x^3 + 2*x^2)/((x-1)*(x^2 + x - 1)*(x^2 - 3*x + 1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009

A152687 Partial products operator applied thrice to nonzero Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 24, 8640, 746496000, 201231433728000000, 3554168771933456302080000000000, 139840535301953855934724122328694784000000000000000, 674129921807822677705190163721626103970522196466131271680000000000000000000000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

nonn

Partial products of A152686.

G. C. Greubel, Table of n, a(n) for n = 1..18

Cf. A001924, A003266, A152686.

Nest[FoldList[Times,#]&,Fibonacci[Range[10]],3] (* Harvey P. Dale, Oct 06 2017 *)

a(n) = prod(i=1, n, prod(j=1, i, prod(k=1, j, fibonacci(k)))); \\ Michel Marcus, Sep 15 2018

Edited by R. J. Mathar, Dec 12 2008

One more term (a(10)) from Harvey P. Dale, Oct 06 2017

A197649 a(n) = Sum_{k=0..n} kFibonacci(2k).

Original entry on oeis.org

0, 1, 7, 31, 115, 390, 1254, 3893, 11789, 35045, 102695, 297516, 853932, 2432041, 6881395, 19361995, 54214939, 151164018, 419910354, 1162585565, 3209268665, 8835468881, 24266461007, 66501634776, 181882282200, 496539007825, 1353272290399, 3682496714743

Offset: 0

Gary Detlefs, Oct 16 2011

There are only a small number of Fibonacci identities that can be solved for n. Some of these are
n = (-F(4*n) + 5*Sum_{k=1..n} F(2*k-1)^2)/2 (Vajda #95).
n = (F(n+3) - 2 + Sum_{k=0..n} k*F(k))/F(n+2). (A104286)
n = (a(n) + F(2*n))/F(2*n+1).
n = F(n+4) - 3 - Sum_{k=0..1} (F(k+2) - 1). (A001924)
n can also be expressed in terms of phi = (1+sqrt(5))/2:
n = floor(n*phi^3) - floor(2*n*phi).
n = (floor(2*n*phi^2) - floor(2*n*phi))/2.

Michael De Vlieger, Table of n, a(n) for n = 0..2384
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 5.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 17, 19.
E. Pérez Herrero, A small Fibonacci sum, Psychedelic Geometry Blogspot
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A023619 (inverse binomial transform).

Cf. A001871, A001924, A104286.

a:=n->sum(k*fibonacci(2*k),n= 0..n):seq(a(n), n=0..25);

Table[Sum[k*Fibonacci[2*k], {k, 0, n}], {n, 0, 50}] (* T. D. Noe, Oct 17 2011 *)

a(n) = n*F(2*n+1) - F(2*n), where F(n) = Fibonacci(n).
a(n) = ((F(2*n+1)*((n-1)*h(n-1) - (n-1)*h(n-2)) - h(n)*F(2*n))/h(n), n > 2, where h(n) is the n-th harmonic number.
From R. J. Mathar, Oct 17 2011: (Start)
G.f.: x*(1+x) / (x^2-3*x+1)^2.
a(n) = A001871(n-1) + A001871(n-2). (End)
a(n) ~ c*n*(3 + sqrt(5))^n*2^(-n), where c = (5 + sqrt(5))/10. - Stefano Spezia, Mar 29 2022
E.g.f.: 2*exp(3*x/2)*(5*x*cosh(sqrt(5)*x/2) + sqrt(5)*(2*x - 1)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Mar 04 2025

Identity 4 added by Gary Detlefs, Dec 22 2012

A202876 Symmetric matrix based on A000071, by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 10, 10, 7, 12, 18, 21, 18, 12, 20, 31, 38, 38, 31, 20, 33, 52, 66, 70, 66, 52, 33, 54, 86, 111, 122, 122, 111, 86, 54, 88, 141, 184, 206, 214, 206, 184, 141, 88, 143, 230, 302, 342, 362, 362, 342, 302, 230, 143, 232, 374, 493, 562, 602

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=A000071 (Fibonacci numbers -1), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202876 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202877 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....2....4....7....12....20
2....5....10...18...31....52
4....10...21...38...66....111
7....18...38...70...122...206
12...31...66...122..214...362

Cf. A202877, A202876.

s[k_] := -1 + Fibonacci[k + 2];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A001924 *)
Table[m[1, j], {j, 1, 12}]     (* A000071 *)
Table[m[j, j], {j, 1, 12}]     (* A202462 *)

A213576 Rectangular array: (row n) = bc, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and = convolution.

Original entry on oeis.org

1, 3, 1, 7, 4, 2, 14, 10, 7, 3, 26, 21, 17, 11, 5, 46, 40, 35, 27, 18, 8, 79, 72, 66, 56, 44, 29, 13, 133, 125, 118, 106, 91, 71, 47, 21, 221, 212, 204, 190, 172, 147, 115, 76, 34, 364, 354, 345, 329, 308, 278, 238, 186, 123, 55, 596, 585, 575, 557, 533, 498, 450, 385, 301, 199, 89

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213577.
Antidiagonal sums:  A213578.
Row 1,  (1,2,3,...)**(1,1,2,3,5,...): A001924;
Row 2,  (1,2,3,...)**(1,2,3,5,8,...): A001891;
Row 3,  (1,2,3,...)**(2,3,5,8,13,...): A033937;
Row 4,  (1,2,3,...)**(3,5,8,13,21,...): A033960;
Row 5,  (1,2,3,...)**(5,8,13,21,...): A037140;
Row 6,  (1,2,3,...)**(8,13,21,34,...): A037157.
For a guide to related arrays, see A213500.
The falling antidiagonal rows can be computed by the sum Sum_{j=0..n-k} (n-k-j+1)*Fibonacci(k+j) which can also be seen as Fibonacci(n+4) - Lucas(k+2) - (n-k)*Fibonacci(k+1). - G. C. Greubel, Jul 05 2019

			Northwest corner (the array is read by falling antidiagonals):
  1,   3,   7,  14,  26,  46,  79
  1,   4,  10,  21,  40,  72, 125
  2,   7,  17,  35,  66, 118, 204
  3,  11,  27,  56, 106, 190, 329
  5,  18,  44,  91, 172, 308, 533
  8,  29,  71, 147, 278, 498, 862

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500.

Flat(List([1..10], n-> List([1..n], k-> Fibonacci(n+4) - (n-k+1) *Fibonacci(k+1) - Fibonacci(k+3)))); # G. C. Greubel, Jul 05 2019

[[Fibonacci(n+4) -(n-k)*Fibonacci(k+1) -Lucas(k+2): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Jul 05 2019

(* First Program *)
b[n_]:= n; c[n_]:= Fibonacci[n];
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213576 *)
r[n_]:= Table[t[n, k], {k,1,40}]  (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n, 1, 40}] (* A213577 *)
s[n_]:= Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* Second Program *)
T[n_, k_]:= Fibonacci[n+4] - (n-k)*Fibonacci[k+1] - LucasL[k+2];
Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

T(n, k)= fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3);
for(n=1,10, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019

[[fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Jul 05 2019

Rows: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
Columns: T(n,k) = T(n-1,k) + T(n-2,k).
G.f. for row n: f(x)/g(x), where f(x) = F(n) - F(n-1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
T(n,k) = F(n+k+3) - k*F(n+1) - F(n+3). - Ehren Metcalfe, Jul 04 2019

A316939 Triangle read by rows formed using Pascal's rule except that n-th row begins and ends with Fibonacci(n+2).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 7, 7, 5, 8, 12, 14, 12, 8, 13, 20, 26, 26, 20, 13, 21, 33, 46, 52, 46, 33, 21, 34, 54, 79, 98, 98, 79, 54, 34, 55, 88, 133, 177, 196, 177, 133, 88, 55, 89, 143, 221, 310, 373, 373, 310, 221, 143, 89, 144, 232, 364, 531, 683, 746, 683, 531, 364, 232, 144, 233, 376, 596, 895, 1214, 1429

Offset: 0

Vincenzo Librandi, Jul 28 2018

			Triangle begins:
   1;
   2,  2;
   3,  4,   3;
   5,  7,   7,   5;
   8, 12,  14,  12,   8;
  13, 20,  26,  26,  20,  13;
  21, 33,  46,  52,  46,  33,  21;
  34, 54,  79,  98,  98,  79,  54, 34;
  55, 88, 133, 177, 196, 177, 133, 88, 55;
  ...

Cf. A316528 (row sums).
Columns k=0..2: A000045, A000071, A001924.
Other Fibonacci borders: A074829, A108617, A316938.

f:= proc(n,k) option remember;
  if k=0 or k=n then combinat:-fibonacci(n+2) else procname(n-1,k)+procname(n-1,k-1) fi
end proc:
for n from 0 to 10 do
  seq(f(n,k),k=0..n)
od; # Robert Israel, Sep 20 2018

t={}; Do[r={}; Do[If[k==0||k==n, m=Fibonacci[n + 2], m=t[[n, k]] + t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t // Flatten

Incorrect g.f. removed by Georg Fischer, Feb 18 2020

A033960 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= 4.

Original entry on oeis.org

3, 11, 27, 56, 106, 190, 329, 557, 929, 1534, 2516, 4108, 6687, 10863, 17623, 28564, 46270, 74922, 121285, 196305, 317693, 514106, 831912, 1346136, 2178171, 3524435, 5702739, 9227312, 14930194, 24157654, 39088001

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001924, A001891, A033937.

List([0..40], n-> Fibonacci(n+8) -5*n-18) # G. C. Greubel, Jul 05 2019

[Fibonacci(n+8) -5*n-18: n in [0..40]]; // G. C. Greubel, Jul 05 2019

Table[Fibonacci[n+8] -5*n-18, {n,0,40}] (* G. C. Greubel, Jul 05 2019 *)

vector(40, n, n--; fibonacci(n+8) -5*n-18) \\ G. C. Greubel, Jul 05 2019

[fibonacci(n+8) -5*n-18 for n in (0..40)] # G. C. Greubel, Jul 05 2019

a(n) = Fibonacci(n+8) - (18+5*n).

G.F.: (3+2*x)/((1-x-x^2)*(1-x)^2).

A037140 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), for k >= 5.

Original entry on oeis.org

5, 18, 44, 91, 172, 308, 533, 902, 1504, 2483, 4072, 6648, 10821, 17578, 28516, 46219, 74868, 121228, 196245, 317630, 514040, 831843, 1346064, 2178096, 3524357, 5702658, 9227228, 14930107, 24157564, 39087908, 63245717, 102333878, 165579856, 267914003

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001924, A001891, A033937, A033960.

List([0..40], n-> Fibonacci(n+9) -8*n-29) # G. C. Greubel, Jul 05 2019

[Fibonacci(n+9) -8*n-29: n in [0..40]]; // G. C. Greubel, Jul 05 2019

Table[Fibonacci[n+9] -8*n-29, {n,0,40}] (* G. C. Greubel, Jul 05 2019 *)

vector(40, n, n--; fibonacci(n+9) -8*n-29) \\ G. C. Greubel, Jul 05 2019

[fibonacci(n+9) -8*n-29 for n in (0..40)] # G. C. Greubel, Jul 05 2019

a(n) = Fibonacci(n+9) - (29+8*n).

G.f.: (5+3*x)/((1-x-x^2)*(1-x)^2).

Corrected by Franklin T. Adams-Watters, Oct 25 2006

cp's OEIS Frontend

A213579 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A033937 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= 3.

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A120297 Sum of all matrix elements of n X n matrix M(i,j) = Fibonacci(i+j-1).

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Mathematica

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A152687 Partial products operator applied thrice to nonzero Fibonacci numbers.

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A197649 a(n) = Sum_{k=0..n} k*Fibonacci(2*k).

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A202876 Symmetric matrix based on A000071, by antidiagonals.

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A213576 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and ** = convolution.

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A213579 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

A197649 a(n) = Sum_{k=0..n} kFibonacci(2k).

A213576 Rectangular array: (row n) = bc, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and = convolution.