A354337 - OEIS

%I #17 Aug 23 2022 09:35:58

%S 19,149,1039,7139,48949,335519,2299699,15762389,108037039,740496899,

%T 5075441269,34787591999,238437702739,1634276327189,11201496587599,

%U 76776199786019,526231901914549,3606847113615839,24721697893396339,169445038140158549,1161393569087713519

%N a(n) is the integer w such that (L(2*n)^2, -L(2*n + 1)^2, w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

%C Subsequence of A017377.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).

%F a(n) = (125 - 2*A005248(n)^6 + 2*A002878(n)^6)^(1/3).

%F a(n) = Lucas(4*n+2) + Lucas(4n-1) - 3 = 2*A056914(n)-3 = 15*A092521(n) + A288913(n-1).

%F a(n) = 2*A081017(n) - 1.

%F a(n) = 10*A089508(n) + 9.

%F a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

%F G.f.: x*(19 - 3*x - x^2)/((1 - x)*(1 - 7*x + x^2)). - _Stefano Spezia_, Jun 22 2022

%e 2*(L(4)^2)^3 + 2*(-L(5)^2)^3 + (149)^3 = 2*(49)^3 + 2*(-121)^3 + (149)^3 = 125, a(2) = 149.

%t LinearRecurrence[{7,-1},{19,149},21]-1 + LucasL[2*Range[21]-3]^2

%Y Cf. A000032, A002878, A005248, A017377, A081017, A089508, A092521, A288913.

%Y Cf. A337929, A354336.

%K nonn,easy

%O 1,1

%A _XU Pingya_, Jun 20 2022