A056588 -id:A056588 - cp's OEIS frontend

A056591 Fifth column sequence of triangle A056588.

1, 12, 166, 1784, 17840, 163504, 1418549, 11751784, 94002810, 730859800, 5554472496, 41437244784, 304478259625, 2209596042260, 15871463933950, 113044318064744, 799558820643440, 5622796403700080, 39354459839661725

Offset: 0

Wolfdieter Lang Jul 10 2000

Robert Israel, Table of n, a(n) for n = 0..1193

Cf. A000045, A056588, A010048, A000012, A000071, A056589, A056590.

g:= (72*x^9-142*x^8+276*x^7+473*x^6-112*x^5-78*x^3+46*x^2-8*x+1)/((1-x)*(1+2*x)*(1-5*x)*(1+x-x^2)*(1+3*x+x^2)*(1-3*x-9*x^2)*(1-4*x-x^2)*(1-6*x+4*x^2) *(1-7*x+x^2)):
S:= series(g,x,21):
seq(coeff(S,x,i),i=0..20); # Robert Israel, Jul 24 2018

a(n)= A056588(n+4, 4), n >= 0.
a(n)=5^(n+5)-3^(n+5)*F(n+6)-Fibonomial(n+6, 2)*2^(n+5)+Fibonomial(n+6, 3)+Fibonomial(n+6, 4); F(n)=A000045(n) (Fibonacci); Fibonomial(n, m) := A010048(n, m).
G.f.: (72*x^9-142*x^8+276*x^7+473*x^6-112*x^5-78*x^3+46*x^2-8*x+1)/((1-x)*(1+2*x)*(1-5*x)*(1+x-x^2)*(1+3*x+x^2)*(1-3*x-9*x^2)*(1-4*x-x^2)*(1-6*x+4*x^2) *(1-7*x+x^2)).

A126770 Unsigned version of A056588.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 16, 7, 1, 1, 12, 53, 53, 12, 1, 1, 20, 166, 318, 166, 20, 1, 1, 33, 492, 1784, 1784, 492, 33, 1, 1, 54, 1413, 9288, 17840, 9288, 1413, 54, 1, 1, 88, 3960, 46233, 163504, 163504, 46233, 3960, 88, 1, 1, 143, 10912, 221859, 1418549, 2616064, 1418549, 221859, 10912, 143, 1

Offset: 0

Philippe Deléham, Feb 17 2007

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 4, 4, 1;
  1, 7, 16, 7, 1;
  1, 12, 53, 53, 12, 1;
  1, 20, 166, 318, 166, 20, 1;
  1, 33, 492, 1784, 1784, 492, 33, 1;
  1, 54, 1413, 9288, 17840, 9288, 1413, 54, 1;
  1, 88, 3960, 46233, 163504, 163504, 46233, 3960, 88, 1;
  1, 143, 10912, 221859, 1418549, 2616064, 1418549, 221859, 10912, 143, 1;

Column sequences: A000012, A000071, A056589-91; A056593 (Row sums).

T[n_, 1] := 1; T[n_, n_] := 1; T[n_, k_] := Fibonacci[(n - k + 1)]*T[ n - 1, k - 1] + Fibonacci[k ]*T[n - 1, k];
Table[T[n, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Roger L. Bagula, Sep 09 2008 *)

T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,k)=F(n-k+1)*T(n-1,k-1)+F(k+1)*T(n-1,k) where F(n) = A000045(n).

A255492 Sixth diagonal of triangle in A056588.

Original entry on oeis.org

1, 20, 492, 9288, 163504, 2616064, 39369649, 561744656, 7690052788, 101717711304, 1308089742576, 16431681189504, 202396884008657, 2452192780001276, 29299135239719708, 345967823847659992, 4044638027348853616, 46886410125951835648, 539632277597240357409

Offset: 0

N. J. A. Sloane, Mar 06 2015

nonn

S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675. See p. 675.

Cf. A056588.

A056570 Third power of Fibonacci numbers (A000045).

Original entry on oeis.org

0, 1, 1, 8, 27, 125, 512, 2197, 9261, 39304, 166375, 704969, 2985984, 12649337, 53582633, 226981000, 961504803, 4073003173, 17253512704, 73087061741, 309601747125, 1311494070536, 5555577996431, 23533806109393, 99690802348032, 422297015640625

Offset: 0

Wolfdieter Lang, Jul 10 2000

Divisibility sequence; that is, if n divides m, then a(n) divides a(m).
In general, cubing the terms of a Horadam sequence with signature (c,d) will result in a fourth-order recurrence with signature (c^3+2*c*d, c^4*d+3*(c*d)^2+2*d^3, -(c*d)^3-2*c*d^4, -d^6). - Gary Detlefs, Nov 12 2021
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/3,2/3)-fences and third-squares (1/3 X 1 pieces, always placed so that the shorter sides are horizontal). A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/6,1/3)-fences and (1/6,5/6)-fences. - Michael A. Allen, Jan 11 2022

			a(4) = 27 because the fourth Fibonacci number is 3 and 3^3 = 27.
a(5) = 125 because the fifth Fibonacci number is 5 and 5^3 = 125.

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..173
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
Andrej Dujella, A bijective proof of Riordan's theorem on powers of Fibonacci numbers, Discrete Math. 199 (1999), no. 1-3, 217--220. MR1675924 (99k:05016).
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers cubed, Fib. Q. 58:5 (2020) 128-134.
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux
Mariana Nagy, Simon R. Cowell and Valeriu Beiu, Survey of Cubic Fibonacci Identities - When Cuboids Carry Weight, arXiv:1902.05944 [math.HO], 2019.
Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop and Abiodun A. Opanuga, Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp 4623-4627.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
H. C. Williams and R. K. Guy, Odd and even linear divisibility sequences of order 4, INTEGERS, 2015, #A33.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000045, A007598, A056588, A055870, A066259, A066258.
Cf. A346513 (first differences), A005968 (partial sums).
Third row of array A103323.

[Fibonacci(n)^3: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011

A056570 := proc(n) combinat[fibonacci](n)^3 ; end proc:
seq(A056570(n),n=0..20) ;

Table[Fibonacci[n]^3, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)

a(n)=fibonacci(n)^3 \\ Charles R Greathouse IV, Sep 24 2015

concat(0, Vec(x*(1-2*x-x^2)/((1+x-x^2)*(1-4*x-x^2)) + O(x^30))) \\ Colin Barker, Jun 04 2016

[fibonacci(n)^3 for n in (0..30)] # G. C. Greubel, Feb 20 2019

a(n) = A000045(n)^3.
G.f.: x*p(3, x)/q(3, x) with p(3, x) = Sum_{m=0..2} A056588(2, m)*x^m = 1 -2*x -x^2 and q(3, x) = Sum_{m=0..4} A055870(4, m)*x^m = 1 -3*x -6*x^2 +3*x^3 +x^4 = (1+x-x^2)*(1-4*x-x^2) (factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): 1*a(n) -3*a(n-1) -6*a(n-2) +3*a(n-3) +1*a(n-4) = 0, n >= 4, a(0)=0, a(1)=a(2)=1, a(3)=2^3. See 5th row of signed Fibonomial triangle for coefficients: A055870(4, m), m=0..4
a(n) = (Fibonacci(3n) - 3(-1)^n*Fibonacci(n))/5. - Ralf Stephan, May 14 2004
a(n) and a(n+1) are found as rightmost and leftmost terms (respectively) in M^n * [1 0 0 0] where M = the 4 X 4 upper triangular Pascal's triangle matrix [1 3 3 1 / 1 2 1 0 / 1 1 0 0 / 1 0 0 0]. E.g., Ma(4) = 27, a(5) = 125. M^4 * [1 0 0 0] = [125 75 45 27]; where 75 = A066259(4) and 45 = A066258(3). The characteristic polynomial of M = x^4 - 3x^3 - 6x^2 + 3x + 1. a(n)/a(n-1) of the sequence and companions tend to 2+sqrt(5) = 4.2360679..., an eigenvalue of M and a root of the polynomial. - Gary W. Adamson, Oct 31 2004
From R. J. Mathar, Oct 16 2006: (Start)
Sum_{j=0..n} binomial(n,j)*a(j) = (2^n*A001906(n) + 3*A000045(n))/5.
Sum_{j=0..n} (-1)^j*binomial(n,j)*a(j) = ((-2)^n*A000045(n) - 3*A001906(n))/5. (End)
G.f.: x*(1-2*x-x^2)/((1+x-x^2)*(1-4*x-x^2)). - Colin Barker, Feb 28 2012
a(n) = F(n-2)*F(n+1)^2 + F(n-1)*(-1)^n. - J. M. Bergot, Mar 17 2016
a(n) = ((-3*(1/2*(-1-sqrt(5)))^n-(2-sqrt(5))^n+3*(1/2*(-1+sqrt(5)))^n+(2+sqrt(5))^n)) / (5*sqrt(5)). - Colin Barker, Jun 04 2016
a(n) = F(n-1)*F(n)*F(n+1) + F(n)*(-1)^(n-1). - Tony Foster III, Apr 11 2018
5*a(n) = L(2*n-1)*F(n+2) - L(2*n+1)*F(n-2) - 7*(-1)^n*F(n), where L(n) = A000032(n). - Peter Bala, Nov 12 2019
F(n+1)*F(n)*F(n-1) = 2*Sum_{j=1..n-1} P(j)*a(n-j) for n>0, where Pell number P(n) = A000129(n). - Michael A. Allen, Jan 11 2022

A056572 Fifth power of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 1, 32, 243, 3125, 32768, 371293, 4084101, 45435424, 503284375, 5584059449, 61917364224, 686719856393, 7615646045657, 84459630100000, 936668172433707, 10387823949447757, 115202670521319424, 1277617458486664901

Offset: 0

Wolfdieter Lang, Jul 10 2000

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..135
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop, Abiodun A. Opanuga, Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp 4623-4627.
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 (8,40,-60,-40,8,1).

Cf. A000045, A007598, A056570-1, A056588, A055870.

Fifth row of array A103323.

[Fibonacci(n)^5: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011

Table[f=Fibonacci[n];f^5,{n,0,12}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
Fibonacci[Range[0,20]]^5 (* Harvey P. Dale, Aug 07 2022 *)

a(n)=fibonacci(n)^5 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = F(n)^5, F(n)=A000045(n).
G.f.: x*p(5, x)/q(5, x) with p(5, x) := sum(A056588(4, m)*x^m, m=0..4)= 1-7*x-16*x^2+7*x^3+x^4 and q(5, x) := sum(A055870(6, m)*x^m, m=0..6)= 1-8*x-40*x^2+60*x^3+40*x^4-8*x^5-x^6 = (1-x-x^2)*(1+4*x-x^2)*(1-11*x-x^2) (factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum(A055870(6, m)*a(n-m), m=0..6) = 0, n >= 6; inputs: a(n), n=0..5. a(n) = +8*a(n-1) +40*a(n-2) -60*a(n-3) -40*a(n-4) +8*a(n-5) +a(n-6).
a(n) = (10*F(n) + 5*(-1)^(n+1)*F(3*n) + F(5*n))/25, n >= 0. See the general comment on A111418 regarding the Ozeki reference; here the row 10, 5, 1 of that triangle applies. - Wolfdieter Lang, Aug 25 2012
a(n) = (F(n)^2*(F(3*n)-(-1)^n*3*F(n)))/5. - Gary Detlefs, Jan 07 2013
a(n) = F(n-2)*F(n-1)*F(n)*F(n+1)*F(n+2) + F(n). - Tony Foster III, Apr 11 2018

A056571 Fourth power of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 1, 16, 81, 625, 4096, 28561, 194481, 1336336, 9150625, 62742241, 429981696, 2947295521, 20200652641, 138458410000, 949005240561, 6504586067281, 44583076827136, 305577005139121, 2094455819300625, 14355614096087056

Offset: 0

Wolfdieter Lang, Jul 10 2000

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

a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/4,3/4)-fences and quarter-squares (1/4 X 1 pieces, always placed so that the shorter sides are horizontal). A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/8,3/8)-fences and (1/8,7/8)-fences. - Michael A. Allen, Jan 11 2022

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 31.
Donald 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).
Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop and Abiodun A. Opanuga, Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp. 4623-4627.

Vincenzo Librandi, Table of n, a(n) for n = 0..151
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.
Alfred Brousseau, A sequence of power formulas, Fib. Quart., Vol. 6, No. 1 (1968), pp. 81-83.
Andrej Dujella, A bijective proof of Riordan's theorem on powers of Fibonacci numbers, Discrete Math., Vol. 199, No. 1-3 (1999), pp. 217-220. MR1675924 (99k:05016).
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers cubed, Fib. Q. 58:5 (2020) 128-134.
Hideyuki Ohtsuka, Problem N-1220, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 55, No. 4 (2017), p. 368; Gelin-Cesàro Identity Yields a Telescoping Product, Solution to Problem H-790 by Ramya Dutta, ibid., Vol. 56, No. 4 (2018), p. 372.
John Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J., Vol. 29, No. 1 (1962), pp. 5-12.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000045, A001622, A007598, A056570, A056588, A055870.
First differences of A005969.
Fourth row of array A103323.

[Fibonacci(n)^4: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011

A056571 := proc(n)
    combinat[fibonacci](n)^4 ;
end proc:
seq(A056571(n),n=0..10) ; # R. J. Mathar, Jan 23 2022

Fibonacci[Range[0, 25]]^4 (* Wesley Ivan Hurt, Jan 11 2022 *)

a(n)=fibonacci(n)^4 \\ Charles R Greathouse IV, Oct 07 2016

a(n) = F(n)^4 = A007598(n)^2, F(n) = A000045(n).
G.f.: x*p(4, x)/q(4, x) with p(4, x) := sum(A056588(3, m)*x^m, m=0..3) = 1 - 4*x - 4*x^2 + x^3 = (1+x)*(1 - 5*x + x^2) and q(4, x) := Sum_{m=0..5} A055870(5, m)*x^m = 1 - 5*x - 15*x^2 + 15*x^3 + 5*x^4 - x^5 = (1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2) (denominator factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): 1*a(n) - 5*a(n-1) - 15*a(n-2) + 15*a(n-3) + 5*a(n-4) - 1*a(n-5) = 0, n >= 5; inputs: a(n), n=0..4.
(1/25)*((-1)^n*(2*F(2*n-2) - 6*F(2*n+1)) + 2*F(4*n-1) + F(4*n) + 6). - Ralf Stephan, May 14 2004
a(n) = F(n-2)*F(n-1)*F(n+1)*F(n+2) + 1 = A244855(n)+1.
Sum_{j=0..n} binomial(n,j)*a(j) = (3^n*A005248(n) - 4*(-1)^n*A000032(n) + 6*2^n)/25. Sum_{j=0..n} (-1)^j*binomial(n,j)*a(j) = -5^((n+1)/2-2)*(A001906(n) + 4*A000045(n)) if n odd. - R. J. Mathar, Oct 16 2006
a(n) = (F(n)*F(3n) - 3*F(n)^2*(-1)^n)/5. - Gary Detlefs, Dec 26 2010
Product_{n>=3} (1 - 1/a(n)) = phi^5/12, where phi is the golden ratio (A001622)(Ohtsuka, 2017). - Amiram Eldar, Dec 02 2021

cp's OEIS Frontend

A056589 Third column sequence of unsigned triangle A056588.

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A056592 Row sums of signed triangle A056588.

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A056593 Row sums of unsigned triangle A056588.

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A056590 Fourth column sequence of triangle A056588.

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A056591 Fifth column sequence of triangle A056588.

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A126770 Unsigned version of A056588.

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A255492 Sixth diagonal of triangle in A056588.

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A056570 Third power of Fibonacci numbers (A000045).

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A056572 Fifth power of Fibonacci numbers A000045.

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A056571 Fourth power of Fibonacci numbers A000045.