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%I #21 Jan 06 2024 00:59:05
%S 5327263,684056220943393618,87836547552751547393253180439,
%T 11278691501915643258450349467913578516874,
%U 1448245468880558621537182415402996832263200922550703,185962612575832140241603356412217415201039246491645779158754978
%N Numbers x such that (x + 103)^3 - x^3 is a square.
%H G. C. Greubel, <a href="/A147528/b147528.txt">Table of n, a(n) for n = 1..50</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (128405450991,-128405450991,1).
%F a(n+2) = 128405450990*a(n+1) - a(n) + 6612880725882.
%F G.f.: 103*x*(51721 + 64202725495*x - 51722*x^2) / ((1-x)*(1 -128405450990*x +x^2)). - _Colin Barker_, Oct 21 2014
%e a(1) = 5327263 because the first relation is : (5327263 + 103)^3 - 5327263^3 = 93645643^2.
%p seq(coeff(series(103*x*(51721 +64202725495*x -51722*x^2)/((1-x)*(1 -128405450990*x +x^2)), x, n+1), x, n), n = 1..20); # _G. C. Greubel_, Jan 10 2020
%t LinearRecurrence[{128405450991, -128405450991, 1}, {5327263, 684056220943393618, 87836547552751547393253180439}, 20] (* _G. C. Greubel_, Jan 10 2020 *)
%o (PARI) Vec(103*x*(51721+64202725495*x-51722*x^2)/((1-x)*(1-128405450990*x+x^2)) + O(x^20)) \\ _Colin Barker_, Oct 21 2014
%o (Magma) I:=[5327263, 684056220943393618, 87836547552751547393253180439]; [n le 3 select I[n] else 128405450991*Self(n-1) - 128405450991*Self(n-2) + Self(n-3): n in [1..20]]; // _G. C. Greubel_, Jan 10 2020
%o (Sage)
%o def A147528_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 103*x*(51721 + 64202725495*x - 51722*x^2) / ((1-x)*(1 -128405450990*x +x^2)) ).list()
%o a=A147528_list(20); a[1:] # _G. C. Greubel_, Jan 10 2020
%o (GAP) a:=[5327263, 684056220943393618, 87836547552751547393253180439];; for n in [4..20] do a[n]:=128405450991*a[n-1] - 128405450991*a[n-2] + a[n-3]; od; a; # _G. C. Greubel_, Jan 10 2020
%Y Cf. A147527, A147529, A147530.
%K nonn,easy
%O 1,1
%A _Richard Choulet_, Nov 06 2008
%E Editing and a(6) from _Colin Barker_, Oct 21 2014
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