A005972 Partial sums of fourth powers of Lucas numbers. (Formerly M5358)

1, 82, 338, 2739, 17380, 122356, 829637, 5709318, 39071494, 267958135, 1836197336, 12586569192, 86266785673, 591288786874, 4052734152890, 27777904133691, 190392453799372, 1304969641560028, 8944394070807629

Offset

1,2

References

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 21.

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

Formula

G.f.: x*(1+76*x-164*x^2-79*x^3+16*x^4)/((1-x)^2*(1+3*x+x^2)*(1-7*x+x^2)). - Ralf Stephan, Apr 23 2004

a(n) = A000045(4*n+2) + 4*(-1)^n*(A000045(n)^2 + A000045(n+1)^2) + 6*n-5. - Vaclav Kotesovec, Nov 19 2012

Maple

lucas := proc(n) option remember: if n=1 then RETURN(1) fi: if n=2 then RETURN(3) fi: lucas(n-1)+lucas(n-2) end: l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+lucas(i)^4; printf(`%d, `, l[i]) od: # James A. Sellers, May 29 2000
A005972:=(1+76*z-164*z**2-79*z**3+16*z**4)/(z**2-7*z+1)/(z**2+3*z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation

Mathematica

Accumulate[LucasL[Range[20]]^4] (* Harvey P. Dale, Jul 17 2011 *)
Table[Fibonacci[4*n+2]+(-1)^n*(4*Fibonacci[n]^2+4*Fibonacci[n+1]^2)+6*n-5, {n, 1, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)
CoefficientList[Series[(1 + 76 x - 164 x^2 - 79 x^3 + 16 x^4) / ((1 - x)^2 (1 + 3 x + x^2) (1 - 7 x + x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 03 2012 *)

Crossrefs

Sequence in context: A116341 A102956 A031696 * A082972 A288920 A304416

Adjacent sequences: A005969 A005970 A005971 * A005973 A005974 A005975

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 29 2000

Definition clarified by Harvey P. Dale, Jul 17 2011

Status

approved