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Partial sums of fourth powers of Lucas numbers.
(Formerly M5358)
1

%I M5358 #45 Apr 13 2022 13:25:18

%S 1,82,338,2739,17380,122356,829637,5709318,39071494,267958135,

%T 1836197336,12586569192,86266785673,591288786874,4052734152890,

%U 27777904133691,190392453799372,1304969641560028,8944394070807629

%N Partial sums of fourth powers of Lucas numbers.

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

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

%H Vincenzo Librandi, <a href="/A005972/b005972.txt">Table of n, a(n) for n = 1..1000</a>

%H Kunle Adegoke, <a href="https://arxiv.org/abs/1706.00407">Sums of fourth powers of Fibonacci and Lucas numbers</a>, arXiv:1706.00407 [math.NT], 2017.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%F 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

%F a(n) = A000045(4*n+2) + 4*(-1)^n*(A000045(n)^2 + A000045(n+1)^2) + 6*n-5. - _Vaclav Kotesovec_, Nov 19 2012

%p 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

%p 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

%t Accumulate[LucasL[Range[20]]^4] (* _Harvey P. Dale_, Jul 17 2011 *)

%t 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 *)

%t 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 *)

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, May 29 2000

%E Definition clarified by _Harvey P. Dale_, Jul 17 2011