A228105 - OEIS

%I #28 Sep 08 2022 08:46:05

%S 0,432,27648,314928,1769472,6750000,20155392,50824368,113246208,

%T 229582512,432000000,765314352,1289945088,2085181488,3252759552,

%U 4920750000,7247757312,10427429808,14693280768,20323820592,27648000000,37050964272,48980118528

%N a(n) = 432*n^6.

%C For any n > 0, the equation y^2 = x^3 - a(n) has exactly one solution in natural numbers (x = 12*n^2 and y = 36*n^3).

%H Arkadiusz Wesolowski, <a href="/A228105/b228105.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = A008585(n)*A008591(n)*A016744(n).

%F G.f.: 432*x*(1 + x)*(1 + 56*x + 246*x^2 + 56*x^3 + x^4) / (1 - x)^7.

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6. - _Colin Barker_, Dec 11 2017

%e a(2) = 432*2^6 = 27648.

%p seq(432*n^6, n=0..22);

%t Table[432*n^6, {n, 0, 22}]

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,432,27648,314928,1769472,6750000,20155392},40] (* _Harvey P. Dale_, Apr 06 2018 *)

%o (Magma) [432*n^6 : n in [0..22]];

%o (PARI) concat(0, Vec(432*x*(1 + x)*(1 + 56*x + 246*x^2 + 56*x^3 + x^4) / (1 - x)^7 + O(x^40))) \\ _Colin Barker_, Dec 11 2017

%Y Cf. A134109.

%K nonn,easy

%O 0,2

%A _Arkadiusz Wesolowski_, Aug 10 2013