A049667 -id:A049667 - cp's OEIS frontend

A028412 Rectangular array of numbers Fibonacci(m(n+1))/Fibonacci(m), m >= 1, n >= 0, read by downward antidiagonals.

1, 1, 1, 1, 3, 2, 1, 4, 8, 3, 1, 7, 17, 21, 5, 1, 11, 48, 72, 55, 8, 1, 18, 122, 329, 305, 144, 13, 1, 29, 323, 1353, 2255, 1292, 377, 21, 1, 47, 842, 5796, 15005, 15456, 5473, 987, 34, 1, 76, 2208, 24447, 104005, 166408, 105937, 23184, 2584, 55, 1, 123, 5777

Offset: 0

N. J. A. Sloane

Every integer-valued quotient of two Fibonacci numbers is in this array. - Clark Kimberling, Aug 28 2008

Not only does 5 divide row 5, but 50 divides (-5 + row 5), as in A214984. - Clark Kimberling, Nov 02 2012

			   1   1    1      1       1        1
   1   3    4      7      11       18
   2   8   17     48     122      323
   3  21   72    329    1353     5796
   5  55  305   2255   15005   104005
   8 144 1292  15456  166408  1866294
  13 377 5473 105937 1845493 33489287
  ...

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 142.

Clark Kimberling, Table of n, a(n) for n = 0..1829
I. Strazdins, Lucas factors and a Fibonomial generating function, in Applications of Fibonacci numbers, Vol. 7 (Graz, 1996), 401-404, Kluwer Acad. Publ., Dordrecht, 1998.

Columns include A000045, A001906, A001076, A004187, A049666, A049660, A049667, A049668, A049669, A049670.
Rows include (essentially) A000032, A047946, A083564, A103226.
Main diagonal is A051294.
Transpose is A214978.

max = 11; col[m_] := CoefficientList[ Series[ 1/(1 - LucasL[m]*x + (-1)^m*x^2), {x, 0, max}], x]; t = Transpose[ Table[ col[m], {m, 1, max}]] ; Flatten[ Table[ t[[n - m + 1, m]], {n, 1, max }, {m, n, 1, -1}]] (* Jean-François Alcover, Feb 21 2012, after Paul D. Hanna *)
f[n_] := Fibonacci[n]; t[m_, n_] := f[m*n]/f[n]
TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]] (* array *)
t = Flatten[Table[t[k, n + 1 - k], {n, 1, 120}, {k, 1, n}]] (* sequence *) (* Clark Kimberling, Nov 02 2012 *)

{T(n,m)=polcoeff(1/(1 - Lucas(m)*x + (-1)^m*x^2 +x*O(x^n)),n)}

T(n, m) = Sum_{i_1>=0} Sum_{i_2>=0} ... Sum_{i_m>=0} C(n-i_m, i_1)*C(n-i_1, i_2)*C(n-i_2, i_3)*...*C(n-i_{m-1}, i_m).

G.f. for column m >= 1: 1/(1 - Lucas(m)*x + (-1)^m*x^2), where Lucas(m) = A000204(m). - Paul D. Hanna, Jan 28 2012

More terms from Erich Friedman, Jun 03 2001
Edited by Ralf Stephan, Feb 03 2005
Better description from Clark Kimberling, Aug 28 2008

A134498 a(n) = Fibonacci(7n).

Original entry on oeis.org

0, 13, 377, 10946, 317811, 9227465, 267914296, 7778742049, 225851433717, 6557470319842, 190392490709135, 5527939700884757, 160500643816367088, 4660046610375530309, 135301852344706746049, 3928413764606871165730

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1)

Cf. A000045, A134499, A134500, A134501, A134502, A134503, A134504, A049667.

[Fibonacci(7*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[7n], {n, 0, 30}]
{a,b}={0,13};Do[Print[c={a,b}.{1,29}];a=b;b=c,{30}] (* Zak Seidov, Nov 02 2009 *)

numlib::fibonacci(7*n) $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(7*n) \\ Charles R Greathouse IV, Jun 11 2015

[fibonacci(7*n) for n in range(0, 16)] # Zerinvary Lajos, May 15 2009

G.f.: -13*x / ( -1+29*x+x^2 ). a(n) = 13*A049667(n). - R. J. Mathar, Apr 17 2011

a(n) = A000045(A008589(n)). - Michel Marcus, Nov 08 2013

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A134501 a(n) = Fibonacci(7n + 3).

Original entry on oeis.org

2, 55, 1597, 46368, 1346269, 39088169, 1134903170, 32951280099, 956722026041, 27777890035288, 806515533049393, 23416728348467685, 679891637638612258, 19740274219868223167, 573147844013817084101, 16641027750620563662096

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490-A134495, A103134, A134497 - A134504.

[Fibonacci(7*n+3): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica
```
Table[Fibonacci[7n+3], {n, 0, 30}]
```

a(n)=fibonacci(7*n+3) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-2+3*x) / (-1 + 29*x + x^2).
a(n) = 2*A049667(n+1) - 3*A049667(n). (End)
a(n) = A000045(A017017(n)). - Michel Marcus, Nov 07 2013

Offset changed to 0 by Vincenzo Librandi, Apr 16 2011

A134502 a(n) = Fibonacci(7n + 4).

Original entry on oeis.org

3, 89, 2584, 75025, 2178309, 63245986, 1836311903, 53316291173, 1548008755920, 44945570212853, 1304969544928657, 37889062373143906, 1100087778366101931, 31940434634990099905, 927372692193078999176, 26925748508234281076009

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1)

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490-A134495, A103134, A134497-A134504.

[Fibonacci(7*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica
```
Table[Fibonacci[7n+4], {n, 0, 30}]
```

a(n)=fibonacci(7*n+4) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-3-2*x) / (-1 + 29*x + x^2).
a(n) = 2*A049667(n) + 3*A049667(n+1). (End)
a(n) = A000045(A017029(n)). - Michel Marcus, Nov 07 2013

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A056565 Fibonomial coefficients.

Original entry on oeis.org

1, 21, 714, 19635, 582505, 16776144, 488605194, 14169550626, 411591708660, 11948265189630, 346934172869802, 10072785423545712, 292460526776698763, 8491396839675395415, 246543315138161480670, 7158243695757340957617, 207835653079349665473587

Offset: 0

Wolfdieter Lang, Jul 10 2000

Index entries for linear recurrences with constant coefficients, signature (21,273,-1092,-1820,1092,273,-21,-1).

Cf. A010048, A000045, A001654-8, A001076, A049666 (signed), A049667.

[ &*[Fibonacci(n+k): k in [0..6]]/3120: n in [1..16] ]; // Bruno Berselli, Apr 11 2011

(Times@@@Partition[Fibonacci[Range[30]],7,1])/3120  (* Harvey P. Dale, Apr 10 2011 *)

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 7))  \\ Joerg Arndt, May 08 2016

a(n) = A010048(n+7, 7) =: Fibonomial(n+7, 7).
G.f.: 1/p(8, n) with p(8, n) = 1 - 21*x - 273*x^2 + 1092*x^3 + 1820*x^4 - 1092*x^5 - 273*x^6 + 21*x^7 + x^8 = (1 + x - x^2) * (1 - 4*x - x^2) * (1 + 11*x - x^2) * (1 - 29*x - x^2) (n=8 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
a(n) = 29*a(n-1) + a(n-2) + ((-1)^n) * A001657(n), n >= 2, a(0)=1, a(1)=21.
G.f.: exp( Sum_{k>=1} F(8*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

Offset corrected by Seiichi Manyama, May 07 2025

A134503 a(n) = Fibonacci(7n + 5).

Original entry on oeis.org

5, 144, 4181, 121393, 3524578, 102334155, 2971215073, 86267571272, 2504730781961, 72723460248141, 2111485077978050, 61305790721611591, 1779979416004714189, 51680708854858323072, 1500520536206896083277

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1)

Cf. A000045, A134498, A134499, A134500, A134501, A134502, A134504.

[Fibonacci(7*n+5): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[7n+5], {n, 0, 30}]
LinearRecurrence[{29,1},{5,144},20] (* Harvey P. Dale, Apr 24 2017 *)

a(n)=fibonacci(7*n+5) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Apr 17 2011: (Start)
G.f.: (-5+x) / (-1 + 29*x + x^2).
a(n) = 5*A049667(n+1) - A049667(n). (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A203319 a(n) = nFibonacci(n) Sum_{d|n} 1/(d*Fibonacci(d)).

Original entry on oeis.org

1, 3, 7, 19, 26, 81, 92, 267, 358, 848, 980, 3061, 3030, 7976, 11042, 25099, 27150, 78642, 79440, 219884, 270704, 584862, 659112, 1977909, 1950651, 4735370, 6204499, 14189096, 14912642, 43168586, 41734340, 110786987, 135815060, 290854380, 339428752, 953889058, 893839230

Offset: 1

Paul D. Hanna, Jan 01 2012

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 81*x^6/6 +...
where
L(x) = x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 +...+ F(n+1)*x^n +...) +
x^2/2*(1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 +...+ F(2*n+2)*x^(2*n) +...) +
x^3/3*(1 + 4*x^3 + 17*x^6 + 72*x^9 +...+ F(3*n+3)/2*x^(3*n) +...) +
x^4/4*(1 + 7*x^4 + 48*x^8 + 329*x^12 +...+ F(4*n+4)/3*x^(4*n) +...) +
x^5/5*(1 + 11*x^5 + 122*x^10 + 1353*x^15 +...+ F(5*n+5)/5*x^(5*n) +...) +
x^6/6*(1 + 18*x^6 + 323*x^12 + 5796*x^18 +...+ F(6*n+6)/8*x^(6*n) +...) +...
here F(n) = Fibonacci(n) = A000045(n).
Equivalently,
L(x) = x/(1-x-x^2) + (x^2/2)/(1-3*x^2+x^4) + (x^3/3)/(1-4*x^3-x^6) + (x^4/4)/(1-7*x^4+x^8) +...+ (x^n/n)/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
here Lucas(n) = A000032(n).
Exponentiation of the l.g.f. equals the g.f. of A203318:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...+ A203318(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A203318, A203321; A203414 (Pell variant).

Cf. A000032 (Lucas), A000045 (Fibonacci), A001906, A001076, A004187, A049666, A049660, A049667.

a[n_] := n Fibonacci[n] DivisorSum[n, 1/(# Fibonacci[#]) &]; Array[a, 40] (* Jean-François Alcover, Dec 23 2015 *)

{a(n)=if(n<1,0, n*fibonacci(n)*sumdiv(n,d,1/(d*fibonacci(d))) )}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(sum(m=1,n+1,(x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L,n)}

{a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

Equals the logarithmic derivative of A203318.
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n satisfies:
(1) L(x) = Sum_{n>=1} x^n/n * Sum_{k>=0} F(n*k+n)/F(n) * x^(n*k) where F(n) = Fibonacci(n).
(2) L(x) = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) where Lucas(n) = A000032(n).
(3) L(x) = Sum_{n>=1} x^n/n * 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000032(n).
(4) L(x) = Sum_{n>=1} x^n/n * G_n(x^n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.

A056567 Fibonomial coefficients.

Original entry on oeis.org

1, 55, 4895, 352440, 27372840, 2063912136, 157373300370, 11948265189630, 908637119420910, 69056421075989160, 5249543573067466872, 399024295188779925720, 30331388438447118520355, 2305576054220330112077285, 175254358052498673797874685, 13321629629800423781409595728

Offset: 0

Wolfdieter Lang, Jul 10 2000

Cf. A010048, A000045, A001654-8, A056565-6, A001076 (signed), A049666, A049667 (signed), A049669.

a[n_] := (1/2227680) Times @@ Fibonacci[n + Range[9]]; Array[a, 20, 0] (* Giovanni Resta, May 08 2016 *)

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 9))  \\ Joerg Arndt, May 08 2016

a(n) = A010048(n+9, 9) = Fibonomial(n+9, 9).
G.f.: 1/p(10, n) with p(10, n)= 1 - 55*x - 1870*x^2 + 19635*x^3 + 85085*x^4 - 136136*x^5 - 85085*x^6 + 19635*x^7 + 1870*x^8 - 55*x^9 - x^10 = (1 - x - x^2)*(1 + 4*x - x^2)*(1 - 11*x - x^2)*(1 + 29*x - x^2)*(1 - 76*x - x^2) (n=10 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
Recursion: a(n) = 76*a(n-1) + a(n-2)+((-1)^n)*A056565(n), n >= 2, a(0)=1, a(1)=55.
G.f.: exp( Sum_{k>=1} F(10*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

A305413 a(n) = Fibonacci(11*n)/89.

Original entry on oeis.org

0, 1, 199, 39602, 7880997, 1568358005, 312111123992, 62111682032413, 12360536835574179, 2459808941961294034, 489514339987133086945, 97415813466381445596089, 19386236394149894806708656, 3857958458249295447980618633, 767753119428003944042949816623

Offset: 0

Vincenzo Librandi, Jun 05 2018

S. Falcon, Generalized Fibonacci Sequences Generated from a k-Fibonacci Sequence, Journal of Mathematics Research, Vol. 4, No. 2 (2012), 97-100.
Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (199,1).

Cf. A000045, A167398.

Cf. similar sequences: F(3*n)/2 (A001076), F(4*n)/3 (A004187), F(5*n)/5 (A049666), F(6*n)/8 (A049660), F(7*n)/13 (A049667), F(8*n)/21 (A049668), F(9*n)/34 (A049669), F(10*n)/55 (A049670), F(11*n)/89 (this sequence), F(12*n)/144 (A253368).

Magma
```
[Fibonacci(11*n)/89: n in [0..30]];
```

Fibonacci[11 Range[0, 20]]/89
LinearRecurrence[{199,1},{0,1},20] (* Harvey P. Dale, Aug 03 2024 *)

a(n) = fibonacci(11*n)/89 \\ Felix Fröhlich, Jul 30 2019

G.f.: x/(1 - 199*x - x^2).
a(n) = 199*a(n-1) + a(n-2) for n>1, a(0)=0, a(1)=1.
a(n) = A167398(n)/89.
For n >= 1, a(n) equals the denominator of the continued fraction [199, 199, ..., 199] (with n copies of 199). The numerator of that continued fraction is a(n+1). - Greg Dresden and Shaoxiong Yuan, Jul 29 2019

cp's OEIS Frontend

A028412 Rectangular array of numbers Fibonacci(m(n+1))/Fibonacci(m), m >= 1, n >= 0, read by downward antidiagonals.

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A134498 a(n) = Fibonacci(7n).

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A134501 a(n) = Fibonacci(7n + 3).

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A134502 a(n) = Fibonacci(7n + 4).

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A056565 Fibonomial coefficients.

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A134503 a(n) = Fibonacci(7n + 5).

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A203319 a(n) = n*Fibonacci(n) * Sum_{d|n} 1/(d*Fibonacci(d)).

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A203319 a(n) = nFibonacci(n) Sum_{d|n} 1/(d*Fibonacci(d)).