%I #31 Jan 11 2022 05:48:33

%S 4,36,196,1444,9604,66564,454276,3118756,21362884,146458404,

%T 1003749124,6880038916,47155859716,323212716324,2215328606404,

%U 15184099435684,104073336269956,713329336067076,4889231802533764,33511293841053604,229689823620354244,1574317475335503396,10790532493690424836

%N a(n) = A127546(n)^2.

%D Giacomo Candido, A Relationship Between the Fourth Powers of the Terms of the Fibonacci Series, Scripta Mathematica, Vol. 17, No. 3-4 (1951), p. 230.

%D Shalosh B. Ekhad and Doron Zeilberger, Automatic Counting of Tilings of Skinny Plane Regions, in: Simon R. Blackburn, Stefanie Gerke and Mark Wildon, eds., Surveys in Combinatorics 2013, Cambridge University Press, 2013, pp. 363-378.

%H Claudi Alsina and Roger B. Nelsen, <a href="https://www.jstor.org/stable/27643032">On Candido's Identity</a>, Mathematics Magazine, Vol. 80, No. 3 (2007), pp. 226-228; <a href="https://www.maa.org/sites/default/files/pdf/cms_upload/alsina6-200726895.pdf">alternative link</a>.

%H Alexander Bogomolny, <a href="https://www.cut-the-knot.org/arithmetic/algebra/CandidoIdentity.shtml">Candido's Identity</a>, Cut the Knot.

%H Brother Alfred Brousseau, <a href="https://www.fq.math.ca/Scanned/10-4/brousseau2.pdf">Ye Olde Fibonacci Curiosity Shoppe</a>, The Fibonacci Quarterly, Vol. 10, No. 4 (1972), pp. 441-443.

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://arxiv.org/abs/1206.4864">Automatic Counting of Tilings of Skinny Plane Regions</a>, arXiv preprint, arXiv:1206.4864 [math.CO], 2012.

%H R. S. Melham, <a href="https://www.fq.math.ca/Papers1/42-2/quartmelham02_2004.pdf">Ye Olde Fibonacci Curiosity Shoppe Revisited</a>, The Fibonacci Quarterly, Vol. 42, No. 2 (2004), pp. 155-160.

%H Roger B. Nelsen, <a href="https://www.jstor.org/stable/30044141">Proof without Words: Candido's Identity</a>, Mathematics Magazine, Vol. 78, No. 2 (2005), p. 131.

%H Darko Veljan, <a href="https://doi.org/10.5592/CO/CCD.2016.08">A note on Candido's identity and Heron's formula</a>, Proceedings of the 1st Croatian Combinatorial Days, Zagreb, September 29-30, 2016 (2016).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Candido%27s_identity">Candido's identity</a>.

%F Empirical g.f.: -4*(x^4-4*x^3-11*x^2+4*x+1) / ((x-1)*(x^2-7*x+1)*(x^2+3*x+1)). - _Colin Barker_, Jul 22 2013

%F a(n) = 4*(4*((-1)^(n + 1)*Lucas(2*(n + 1)) + Lucas(4*(n + 1))) + 9)/25. - _Ehren Metcalfe_, Feb 18 2017

%F a(n) = 2 * (F(n)^4 + F(n+1)^4 + F(n+2)^4), where F(n) is the n-th Fibonacci number (A000045) (Candido, 1951). - _Amiram Eldar_, Jan 11 2022

%t Table[Total[Fibonacci[Range[n, n + 2]]^2]^2, {n, 0, 22}] (* or *)

%t Table[4 (4 ((-1)^(n + 1) LucasL[2 (n + 1)] + LucasL[4 (n + 1)]) + 9)/25, {n, 0, 22}] (* _Michael De Vlieger_, Feb 18 2017 *)

%Y Cf. A000045, A127546.

%K nonn

%O 0,1

%A _N. J. A. Sloane_, Dec 16 2012