A055870 -id:A055870 - cp's OEIS frontend

A001657 Fibonomial coefficients: column 5 of A010048.

1, 8, 104, 1092, 12376, 136136, 1514513, 16776144, 186135312, 2063912136, 22890661872, 253854868176, 2815321003313, 31222272414424, 346260798314872, 3840089017377228, 42587248616222024, 472299787252290712, 5237885063192296801, 58089034826620525728

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 8*x + 104*x^2 + 1092*x^3 + 12376*x^4 + 136136*x^5 + 1514513*x^6 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 17.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (8,40,-60,-40,8,1).

Cf. A010048, A001654-A001658, A065563.

with(combinat) : a:=n-> 1/30*fibonacci(n)*fibonacci(n+1)*fibonacci(n+2)*fibonacci(n+3)*fibonacci(n+4): seq(a(n), n=1..19); # Zerinvary Lajos, Oct 07 2007
A001657:=-1/(z**2+11*z-1)/(z**2-4*z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation

f[n_] := Times @@ Fibonacci[Range[n + 1, n + 5]]/30; t = Table[f[n], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
LinearRecurrence[{8,40,-60,-40,8,1},{1,8,104,1092,12376,136136},20] (* Harvey P. Dale, Nov 30 2019 *)

a(n)=(n->(n^5-n)/30)(fibonacci(n+3)) \\ Charles R Greathouse IV, Apr 24 2012

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 5))  \\ Joerg Arndt, May 08 2016

a(n) = A010048(5+n, 5) (or fibonomial(5+n, 5)).
G.f.: 1/(1-8*x-40*x^2+60*x^3+40*x^4-8*x^5-x^6) = 1/((1-x-x^2)*(1+4*x-x^2)*(1-11*x-x^2)) (see Comments to A055870).
a(n) = 11*a(n-1) + a(n-2) + ((-1)^n)*fibonomial(n+3, 3), n >= 2; a(0)=1, a(1)=8; fibonomial(n+3, 3)= A001655(n).
a(n) = Fibonacci(n+3)*(Fibonacci(n+3)^4-1)/30. - Gary Detlefs, Apr 24 2012
a(n) = (A049666(n+3) + 2*(-1)^n*A001076(n+3) - 3*A000045(n+3))/150, n >= 0, with A049666(n) = F(5*n)/5, A001076(n) = F(3*n)/2 and A000045(n) = F(n). From the partial fraction decomposition of the o.g.f. and recurrences. - Wolfdieter Lang, Aug 23 2012
a(n) = a(-6-n) * (-1)^n for all n in Z. - Michael Somos, Sep 19 2014
0 = a(n)*(-a(n+1) - 3*a(n+2)) + a(n+1)*(-8*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 19 2014
G.f.: exp( Sum_{k>=1} F(6*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

Corrected and extended by Wolfdieter Lang, Jun 27 2000

A056573 Sixth power of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 1, 64, 729, 15625, 262144, 4826809, 85766121, 1544804416, 27680640625, 496981290961, 8916100448256, 160005726539569, 2871098559212689, 51520374361000000, 924491486192068809, 16589354847268067929

Offset: 0

Wolfdieter Lang, Jul 10 2000

A divisibility sequence; that is, if n divides m, then a(n) divides a(m).

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, p. 85, (exercise 1.2.8. Nr. 30) and p. 492 (solution).

Vincenzo Librandi, Table of n, a(n) for n = 0..124
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876. Mathematical Reviews, MR2959001. Zentralblatt MATH, Zbl 1255.05003.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059. Mathematical Reviews, MR2980853. Zentralblatt MATH, Zbl 1255.05004.
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1).

Cf. A000045, A007598, A055870, A056570, A056571, A056572, A056588.

Sixth row of array A103323.

[Fibonacci(n)^6: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011

with(combinat): A056573:=n->fibonacci(n)^6: seq(A056573(n), n=0..30); # Wesley Ivan Hurt, Jun 29 2015

f[n_]:=Fibonacci[n]^6; lst={}; Do[AppendTo[lst,f[n]],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
Fibonacci[Range[0,20]]^6 (* Harvey P. Dale, Sep 21 2024 *)

a(n)=fibonacci(n)^6 \\ Charles R Greathouse IV, Jun 29 2015

a(n) = F(n)^6, where F(n) = A000045(n).
G.f.: x*p(6, x)/q(6, x) with p(6, x) := sum_{m=0..5} A056588(5, m)*x^m = (1-x)*(1 - 11*x - 64*x^2 - 11*x^3 + x^4) and q(6, x) := sum_{m=0..7} A055870(7, m)*x^m = (1+x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2) (denominator factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum_{m=0..7} A055870(7, m)*a(n-m) = 0, n >= 7; inputs: a(n), n=0..6. a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7).
From Gary Detlefs, Jan 07 2013: (Start)
a(n) = (F(3*n)^2 - (-1)^n*6*F(n)*F(3*n) + 9*F(n)^2)/25.
a(n) = (10*F(n)^3*F(3*n) - F(3*n)^2 + 9*F(n)^2)/25. (End)
a(n+1) = 2*[2*F(n+1)^2-(-1)^n]^3+3*F(n)^2*F(n+1)^2*F(n+2)^2-[F(n)^6+F(n+2)^6] = {Sum(0 <= j <= [n/2]; binomial(n-j, j))}^6, for n (this is Theorem 2.2 (vi) of Azarian's second paper in the references for this sequence). - Mohammad K. Azarian, Jun 29 2015

A001658 Fibonomial coefficients.

Original entry on oeis.org

1, 13, 273, 4641, 85085, 1514513, 27261234, 488605194, 8771626578, 157373300370, 2824135408458, 50675778059634, 909348684070099, 16317540120588343, 292806787575013635, 5254201798026392211, 94282845030238533383, 1691836875411111866723, 30358781826262552258596

Offset: 0

N. J. A. Sloane

It appears that a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 6. - John W. Layman, Apr 14 2000
Layman's formula is correct. - Wolfdieter Lang, Jul 13 2000
Layman's formula is a consequence of formula 2.8 (p. 116) of Lind (1971). - Dale Gerdemann, May 08 2016

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 74.
D. A. Lind, A Determinant Involving Binomial Coefficients, Part 1, Part 2, Fibonacci Quarterly 9.2, 1971.
Index entries for linear recurrences with constant coefficients, signature (13, 104, -260, -260, 104, 13, -1).

Cf. A001654, A001656, A001657.

f[n_] := Times @@ Fibonacci[Range[n+1, n+6]]/240; Table[f[n], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
LinearRecurrence[{13,104,-260,-260,104,13,-1},{1,13,273,4641,85085,1514513,27261234},20] (* Harvey P. Dale, Aug 24 2014 *)

b(n,k)=prod(j=1,k,fibonacci(n+j)/fibonacci(j));
vector(20,n,b(n-1,6))  \\ Joerg Arndt, May 08 2016

From Wolfdieter Lang, Jul 13 2000: (Start)
G.f.: 1/(1-13*x-104*x^2+260*x^3+260*x^4-104*x^5-13*x^6+x^7) = 1/((1+x)*(1-3*x+x^2)*(1+7*x+x^2)*(1-18*x+x^2)) (see Comments to A055870).
a(n) = 5*a(n-1)+F(n-5)*Fibonomial(n+5, 5), n >= 1, a(0) = 1; F(n) = A000045(n) (Fibonacci). a(n) = 18*a(n-1)-a(n-2)+((-1)^n)*Fibonomial(n+4, 4), n >= 2; a(0) = 1, a(1) = 13; Fibonomial(n+4, 4) = A001656(n). (End)
From Gary Detlefs, Dec 03 2012: (Start)
a(n) = F(n+1)*F(n+2)*F(n+3)*F(n+4)*F(n+5)*F(n+6)/240.
a(n) = (F(n+5)^2 - F(n+4)^2)*(F(n+3)^4 - 1)/240, where F(n) = A000045(n). (End)
Conjecture: a(n) = F(7)^(n-6) + Sum_{i=3..n-5} F(i-2)F(6)^{i-1}F(7)^{n-i-5} + Sum_{j=3..i} F(i-2)F(j-2)F(5)^{j-1}F(6)^{i-j}F(7)^{n-i-5} + Sum_{k=3..j} F(i-2)F(j-2)F(k-2)F(4)^{k-1}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5} + Sum_{l=3..k} F(i-2)F(j-2)F(k-2)F(l-2)F(3)^{l-1}F(4)^{k-l}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5} + Sum_{m=3..l} F(i-2)F(j-2)F(k-2)F(l-2)F(m-2)F(m)F(3)^{l-m}F(4)^{k-l}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5}, where F(n)=A000045(n). - Dale Gerdemann, May 08 2016
G.f.: exp( Sum_{k>=1} F(7*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

More terms from Wolfdieter Lang, Jul 13 2000

A056574 Seventh power of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 1, 128, 2187, 78125, 2097152, 62748517, 1801088541, 52523350144, 1522435234375, 44231334895529, 1283918464548864, 37281334283719577, 1082404156823183753, 31427428360210000000, 912473096871571914483

Offset: 0

Wolfdieter Lang, Jul 10 2000

A divisibility sequence; that is, if n divides m, then a(n) divides a(m).

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, p. 85, (exercise 1.2.8. Nr. 30) and p. 492 (solution).

Vincenzo Librandi, Table of n, a(n) for n = 0..115
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876. Mathematical Reviews, MR2959001. Zentralblatt MATH, Zbl 1255.05003.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059. Mathematical Reviews, MR2980853. Zentralblatt MATH, Zbl 1255.05004.
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (21,273,-1092,-1820,1092,273,-21,-1).

Cf. A000045, A007598, A056570, A056571, A056572, A056573, A056588, A055870.

Seventh row of array A103323.

[Fibonacci(n)^7: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011

f[n_]:=Fibonacci[n]^7; lst={}; Do[AppendTo[lst,f[n]],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)

a(n)=fibonacci(n)^7 \\ Charles R Greathouse IV, Jan 30 2012

a(n) = F(n)^7, where F(n) = A000045(n).
G.f.: x*p(7, x)/q(7, x) with p(7, x) := sum_{m=0..6} A056588(6, m)*x^m = 1 - 20*x - 166*x^2 + 318*x^3 + 166*x^4 - 20*x^5 - x^6 and q(7, x) := sum_{m=0..8} A055870(8, m)*x^m = (1 + x - x^2)*(1 - 4*x - x^2)*(1 + 11*x - x^2)*(1 - 29*x - x^2) (factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum_{m=0..8} A055870(8, m)*a(n-m) = 0, n >= 8; inputs: a(n), n=0..7. a(n) = 21*a(n-1) + 273*a(n-2) - 1092*a(n-3) - 1820*a(n-4) + 1092*a(n-5) + 273*a(n-6) - 21*a(n-7) - a(n-8).
a(n+1) = F(n)^7+F(n+1)^7+7*F(n)*F(n+1)*F(n+2)*[2*F(n+1)^2-(-1)^n]^2 = {Sum(0 <= j <= [n/2]; binomial(n-j, j))}^7, for n>=0 (This is Theorem 2.3 (iv) of Azarian's second paper in the references for this sequence). - Mohammad K. Azarian, Jun 29 2015

A003268 Central Fibonomial coefficients.

Original entry on oeis.org

1, 2, 6, 15, 60, 260, 1820, 12376, 136136, 1514513, 27261234, 488605194, 14169550626, 411591708660, 19344810307020, 908637119420910, 69056421075989160, 5249543573067466872, 645693859487298425256, 79413089729752455762384, 15803204856220738696714416

Offset: 0

N. J. A. Sloane

nonn

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 74.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

Central column of A010048, |A055870|.

Cf. A062073.

Table[Product[Fibonacci[k],{k,Floor[n/2]+1,n}] / Product[Fibonacci[k],{k,1,Ceiling[n/2]}],{n,2,20}] (* Vaclav Kotesovec, Apr 10 2015 *)

a(n) = (Product_{k=floor(n/2)+1..n} Fibonacci(k)) / (Product_{k=1..ceiling(n/2)} Fibonacci(k)).

a(n) ~ ((1+sqrt(5))/2)^(n^2/4 + n + 1 - (1-(-1)^n)/8) / A062073, where A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant. - Vaclav Kotesovec, May 01 2015

More terms from Vaclav Kotesovec, May 01 2015

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

A215039 a(n) = Fibonacci(2*n)^3, n>=0.

Original entry on oeis.org

0, 1, 27, 512, 9261, 166375, 2985984, 53582633, 961504803, 17253512704, 309601747125, 5555577996431, 99690802348032, 1788878864685457, 32100128763082731, 576013438873664000, 10336141770970357629, 185474538438612378103

Offset: 0

Wolfdieter Lang, Aug 10 2012

Bisection (even part) of A056570. From this follows the o.g.f., and its partial fraction decomposition leads to the explicit formula given below. The recurrences for F(2*n) and F(6*n)/8 are used in this computation. They follow from the fact that F(2*n) = S(n-1,3), and F(6*n)/8 = S(n-1,18), with Chebyshev's S(n,x) = U(n,x/2) polynomial of the second kind (see A001906 and A049660, respectively).

G. C. Greubel, Table of n, a(n) for n = 0..790
E. Kilic, Y. T. Ulutas, N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, Table 1, k=3.
Index entries for linear recurrences with constant coefficients, signature (21,-56,21,-1).

Cf. A000045, A056570, A163198 (partial sums).

List([0..20], n-> Fibonacci(2*n)^3 ); # G. C. Greubel, Dec 22 2019

[Fibonacci(2*n)^3: n in [0..20]]; // G. C. Greubel, Dec 22 2019

with(combinat); seq( fibonacci(2*n)^3, n=0..20); # G. C. Greubel, Dec 22 2019

Fibonacci[2*(Range[21]-1)]^3 (* G. C. Greubel, Dec 22 2019 *)

vector(21, n, fibonacci(2*(n-1)) ); \\ G. C. Greubel, Dec 22 2019

[fibonacci(2*n)^3 for n in (0..20)] # G. C. Greubel, Dec 22 2019

a(n) = F(2*n)^3, n>=0, with F=A000045.
O.g.f.: x*(1+6*x+x^2)/((1-3*x+x^2)*(1-18*x+x^2)) (from the bisection (even part) of A056570).
a(n) = (F(6*n) - 3*F(2*n))/5, n>=0.
a(n+2) - 18*a(n+1) + a(n) - 9*F(2*(n+1)) = 0, n>=0. From the F_n^3 recurrence (see a comment and references on A055870, use row n=4) together with the recurrence appearing in the solution of exercise 6.58, p. 315, on p. 556 of the second edition of the Graham-Knuth-Patashnik book (reference given on A007318), both with n -> 2*n. See also Koshy's book (reference given on A065563) p. 87, 1. and p. 89, 32. (with a - sign) and 33. - Wolfdieter Lang, Aug 11 2012

A056566 Fibonomial coefficients.

Original entry on oeis.org

1, 34, 1870, 83215, 3994320, 186135312, 8771626578, 411591708660, 19344810307020, 908637119420910, 42689423937884208, 2005443612183077232, 94214069697350815795, 4426039514623184676790, 207929935924379904006970, 9768275694729434277258589, 458901121999204061365680096

Offset: 0

Wolfdieter Lang, Jul 10 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A010048, A000045, A001654-8, A056565, A001906 (signed), A004187, A049660 (signed), A049668.

a[n_] := (1/65520) Times @@ Fibonacci[n + Range[8]]; 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, 8)) \\ Joerg Arndt, May 08 2016

a(n) = A010048(n+8, 8) = Fibonomial(n+8, 8).
G.f.: 1/p(9, n) with p(9, n)= 1 - 34*x - 714*x^2 + 4641*x^3 + 12376*x^4 - 12376*x^5 - 4641*x^6 + 714*x^7 + 34*x^8 - x^9 = (1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2)* (1 + 18*x + x^2)*(1 - 47*x + x^2) (n=9 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
Recursion: a(n) = 47*a(n-1) - a(n-2) + ((-1)^n)*A001658(n), n >= 2, a(0)=1, a(1)=34.
G.f.: exp( Sum_{k>=1} F(9*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

A056585 Eighth power of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 1, 256, 6561, 390625, 16777216, 815730721, 37822859361, 1785793904896, 83733937890625, 3936588805702081, 184884258895036416, 8686550888106661441, 408066367122340274881, 19170731299728100000000

Offset: 0

Wolfdieter Lang, Jul 10 2000

A divisibility sequence; that is, if n divides m, then a(n) divides a(m).

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, p. 85, (exercise 1.2.8. Nr. 30) and p. 492 (solution).

Vincenzo Librandi, Table of n, a(n) for n = 0..107
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876. Mathematical Reviews, MR2959001. Zentralblatt MATH, Zbl 1255.05003.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059. Mathematical Reviews, MR2980853. Zentralblatt MATH, Zbl 1255.05004.
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (34,714,-4641,-12376,12376,4641,-714,-34,1).

Cf. A000045, A007598, A056570, A056571, A056572, A056573, A056574, A056588, A055870.

[Fibonacci(n)^8: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011

lst={};Do[f=Fibonacci[n];AppendTo[lst, f^8], {n, 0, 4!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
Fibonacci[Range[0,20]]^8 (* Harvey P. Dale, Jul 03 2017 *)

a(n)=fibonacci(n)^8 \\ Charles R Greathouse IV, Jun 30 2015

a(n) = F(n)^8, F(n)=A000045(n).
G.f.: x*p(8, x)/q(8, x) with p(8, x) := sum_{m=0..7} A056588(7, m)*x^m = (1+x)*(1 - 34*x - 458*x^2 + 2242*x^3 - 458*x^4 - 34*x^5 + x^6) and q(8, x) := sum_{m=0..9} A055870(9, m)*x^m = (1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2)*(1 + 18*x + x^2)*(1 - 47*x + x^2) (denominator factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum_{m=0..9} A055870(9, m)*a(n-m) = 0, n >= 9; inputs: a(n), n=0..8. a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
a(n+1) = 8*F(n)^2*F(n+1)^2*[F(n)^4+F(n+1)^4+4*F(n)^2*F(n+1)^2+3*F(n)*F(n+1)*F(n+2)]-[F(n)^8+F(n+2)^8]+2*[2*F(n+1)^2-(-1)^n]^4 = {Sum(0 <= j <= [n/2]; binomial(n-j, j))}^8, for n>=0 (This is Theorem 2.2 (vii) of Azarian's second paper in the references for this sequence). - Mohammad K. Azarian, Jun 29 2015

A217471 Partial sum of fifth power of the even-indexed Fibonacci numbers.

Original entry on oeis.org

0, 1, 244, 33012, 4117113, 507401488, 62424765712, 7678070811369, 944346243245076, 116147016764564500, 14285140634333292625, 1756956185432949082176, 216091326285380812359744, 26577476188001703626949937

Offset: 0

Wolfdieter Lang, Oct 11 2012

For the o.g.f. for general powers of Fibonacci numbers F=A000045 see A056588 (row polynomials as numerators) and A055870 (row polynomials as denominator). The even part of the bisection leads to the o.g.f. for powers of F(2*n), and the partial sums of these powers are then given by dividing this o.g.f. by (1-x). For the o.g.f.s for F(n)^5 and F(2*n)^5 see A056572 and A215044, respectively.

The tables of the coefficient of the polynomials which appear in Ozeki's formula and in Melham's conjecture are found in A217472 and A217475, respectively (see References).

			a(2) = 244 = 2*(8-3)/5 - 610/20 + (832040-6765)/55^2 - 7/22.
a(2) = 244 = (1/11)*5^5 - (15/44)*5^3 + (25/44)*5 - 7/22.
a(2) = 244 = (5-1)^2*(4*5^3 + 8*5^2 - 3*5 - 14)/44
           = (4*5^3 + 8*5^2 - 3*5 - 14)*(4/11).

G. C. Greubel, Table of n, a(n) for n = 0..475
R. S. Melham, Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers, The Fibonacci Quart. 46/47 (2008/2009), no. 4, 312-315.
K. Ozeki, On Melham's sum, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110.
H. Prodinger, On a sum of Melham and its variants, The Fibonacci Quart. 46/47 (2008/2009), no. 3, 207-215.

Cf. A163198 (third powers).

Table[Sum[Fibonacci[2*k]^5, {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Apr 12 2017 *)
Accumulate[Fibonacci[Range[0,30,2]]^5] (* Harvey P. Dale, Jun 30 2025 *)

a(n) = sum(k=1, n, fibonacci(2*k)^5); \\ Michel Marcus, Feb 29 2016

a(n) = Sum_{k=0..n} F(2*k)^5, n>=0.
O.g.f.: x*(1+99*x+416*x^2+99*x^3+x^4)/((1-3*x+x^2)*(1-18*x+x^2)*(1-123*x+x^2)*(1-x)).
a(n) = 2*(F(2*(n+1)) - F(2*n))/5 - F(3*(2*n+1))/20 +
  (F(10*(n+1)) - F(10*n))/F(10)^2 - 7/22 (from the partial fraction decomposition of the o.g.f.).
a(n) = (1/11)*F(2*n+1)^5 - (15/44)*F(2*n+1)^3 + (25/44)*F(2*n+1) - 7/22 (from Ozeki reference, Theorem 2, p. 109 --- with a misprint -- and from Prodinger reference, p. 207).
a(n) =(F(2*n+1)-1)^2*(4*F(2*n+1)^3 + 8*F(2*n+1)^2 - 3*F(2*n+1) - 14)/44 (an example for Melham's conjecture, see the reference, eq. (2.7) for m=2).

cp's OEIS Frontend

A001657 Fibonomial coefficients: column 5 of A010048.

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A056573 Sixth power of Fibonacci numbers A000045.

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A001658 Fibonomial coefficients.

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A056574 Seventh power of Fibonacci numbers A000045.

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A003268 Central Fibonomial coefficients.

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A056565 Fibonomial coefficients.

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A215039 a(n) = Fibonacci(2*n)^3, n>=0.