A005969 Sum of fourth powers of Fibonacci numbers. (Formerly M2106)

1, 2, 18, 99, 724, 4820, 33381, 227862, 1564198, 10714823, 73457064, 503438760, 3450734281, 23651386922, 162109796922, 1111115037483, 7615701104764, 52198777931900, 357775783071021, 2452231602371646, 16807845698458702

Offset

1,2

References

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 19.

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Kunle Adegoke, Sums of fourth powers of Fibonacci and Lucas numbers, arXiv:1706.00407 [math.NT], 2017.

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 (6,10,-30,10,6,-1)

Formula

a(n) = Sum_{i=0..n} A056571(i).

G.f.: x*(1+x)*(x^2-5*x+1)/ ( (x^2+3*x+1)*(x^2-7*x+1)*(x-1)^2 ). - Ralf Stephan, Apr 23 2004

a(n) = (1/25)*(F(4n+2)-(-1)^n*4*F(2n+1)+6n+3) where F(n)=A000045(n). - Benoit Cloitre, Sep 13 2004. [Corrected by David Lambert (dave.lambert(AT)comcast.net), Mar 28 2008]

Maple

with(combinat): l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+fibonacci(i)^4; printf(`%d, `, l[i]) od: # James A. Sellers, May 29 2000
A005969:=(z+1)*(z**2-5*z+1)/(z**2-7*z+1)/(z**2+3*z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation, offset zero

Mathematica

CoefficientList[Series[(1+x)*(x^2-5*x+1)/((x^2+3*x+1)*(x^2-7*x+1)*(x- 1)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 02 2017 *)
LinearRecurrence[{6, 10, -30, 10, 6, -1}, {1, 2, 18, 99, 724, 4820}, 30] (* G. C. Greubel, Jan 17 2018 *)

Prog

(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; -1, 6, 10, -30, 10, 6]^n*[0; 1; 2; 18; 99; 724])[1, 1] \\ Charles R Greathouse IV, Sep 28 2015
(Magma) [(1/25)*(Fibonacci(4*n+2)-(-1)^n*4*Fibonacci(2*n+1)+6*n+3): n in [1..25]]; // Vincenzo Librandi, Jun 02 2017

Crossrefs

Cf. A001654, A098531, A098532, A098533, A119285, A000071, A005968, A128697.

Sequence in context: A087291 A267691 A219758 * A094251 A345969 A101570

Adjacent sequences: A005966 A005967 A005968 * A005970 A005971 A005972

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 29 2000

Status

approved