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Numbers x such that there exists n in N : (x+1)^3 - x^3 = 103*n^2.
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%I #26 Sep 08 2022 08:45:38

%S 51721,6641322533431006,852782015075257741682069713,

%T 109501859241899449111168441436054160358,

%U 14060635620199598267351285586436862449157290510201,1805462258017787769335954916623470050495526664967434749114126

%N Numbers x such that there exists n in N : (x+1)^3 - x^3 = 103*n^2.

%H Colin Barker, <a href="/A147530/b147530.txt">Table of n, a(n) for n = 1..90</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) + 62402725494.

%F G.f.: x*(51721 + 64202725495*x - 51722*x^2)/((1-x)*(1 - 128405450990*x + x^2)). - _Colin Barker_, Oct 21 2014, corrected Jul 13 2016

%e a(1)=51721 because the first relation is : 51722^3 - 51721^3 = 103*8827^2.

%p seq(coeff(series(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 12 2020

%t LinearRecurrence[{128405450991,-128405450991,1}, {51721, 6641322533431006, 852782015075257741682069713}, 20] (* _G. C. Greubel_, Jan 12 2020 *)

%o (PARI) Vec(x*(51721+64202725495*x-51722*x^2)/((1-x)*(1-128405450990*x+x^2)) + O(x^10)) \\ _Colin Barker_, Oct 21 2014, corrected Jul 13 2016

%o (PARI) isok(x) = issquare(((x+1)^3-x^3)/103) \\ _Colin Barker_, Jul 13 2016

%o (Magma) I:=[51721, 6641322533431006]; [n le 2 select I[n] else 128405450990*Self(n-1) - Self(n-2) + 62402725494: n in [1..20]]; // _G. C. Greubel_, Jan 12 2020

%o (Sage)

%o def A147530_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( x*(51721+64202725495*x-51722*x^2)/((1-x)*(1-128405450990*x + x^2)) ).list()

%o a=A147530_list(20); a[1:] # _G. C. Greubel_, Jan 12 2020

%o (GAP) a:=[51721, 6641322533431006];; for n in [3..20] do a[n]:=128405450990*a[n-1] -a[n-2] +62402725494; od; a; # _G. C. Greubel_, Jan 12 2020

%Y Cf. A147527, A147528, A147529.

%K nonn,easy

%O 1,1

%A _Richard Choulet_, Nov 06 2008

%E Editing and a(6) from _Colin Barker_, Oct 21 2014

%E a(3) to a(6) corrected by _Colin Barker_, Jul 13 2016