A005971 Partial sums of cubes of Lucas numbers. (Formerly M5198)

1, 28, 92, 435, 1766, 7598, 31987, 135810, 574786, 2435653, 10316252, 43702500, 185123261, 784200368, 3321916912, 14071880655, 59609419066, 252509590018, 1069647725567, 4531100578950, 19194049901126, 81307300410353

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

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.: (1+24*x-23*x^2-8*x^3)/((1-x)*(1+x-x^2)*(1-4*x-x^2)). - Ralf Stephan, Apr 23 2004

a(n) = A000032(3*n+2)/2+3*(-1)^n*A000032(n-1)+3/2. - 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)^3; printf(`%d, `, l[i]) od: # James A. Sellers, May 29 2000
A005971:=(-1-24*z+23*z**2+8*z**3)/(z-1)/(z**2+4*z-1)/(z**2-z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[LucasL[3*n+2]/2+3*(-1)^n*LucasL[n-1]+3/2, {n, 1, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)
CoefficientList[Series[(1 + 24 x - 23 x^2 - 8 x^3) / ((1-x) (1+x-x^2) (1-4*x-x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 03 2013 *)
Accumulate[LucasL[Range[30]]^3] (* Harvey P. Dale, Oct 11 2021 *)

Crossrefs

Sequence in context: A331765 A333280 A192256 * A189808 A130085 A211495

Adjacent sequences: A005968 A005969 A005970 * A005972 A005973 A005974

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 29 2000

Definition clarified by Harvey P. Dale, Oct 11 2021

Status

approved