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A147527 Numbers k such that there exists x in N : (x + 103)^3 - x^3 = k^2.

%I A147527 #26 Oct 04 2024 05:44:53
%S A147527 93645643,12024611022569890927,1544025601332411913276450522087,
%T A147527 198261303679194296628699373223979621125203,
%U A147527 25457832112792289938442435570354101121237746019778883,3268924413670798537740342016261657034171968745307560952072318967
%N A147527 Numbers k such that there exists x in N : (x + 103)^3 - x^3 = k^2.
%H A147527 G. C. Greubel, <a href="/A147527/b147527.txt">Table of n, a(n) for n = 1..50</a>
%H A147527 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (128405450990,-1).
%F A147527 a(n+2) = 128405450990*a(n+1) - a(n).
%F A147527 G.f.: 93645643*x*(1-x)/(1 - 128405450990*x + x^2). - _Colin Barker_, Oct 21 2014
%F A147527 a(n) = sqrt((A147528(n) + 103)^3 - A147528(n)^3). - _Michel Marcus_, Jan 10 2020
%e A147527 a(1)=93645643 because the first relation is (5327263 + 103)^3 - 5327263^3 = 93645643^2.
%p A147527 seq(coeff(series(93645643*x*(1-x)/(1 - 128405450990*x + x^2), x, n+1), x, n), n = 0..20); # _G. C. Greubel_, Jan 10 2020
%t A147527 LinearRecurrence[{128405450990,-1}, {93645643, 12024611022569890927}, 20] (* _G. C. Greubel_, Jan 10 2020 *)
%o A147527 (PARI) Vec(93645643*x*(1-x)/(1-128405450990*x+x^2) + O(x^20)) \\ _Colin Barker_, Oct 21 2014
%o A147527 (Magma) I:=[93645643, 12024611022569890927]; [n le 2 select I[n] else 128405450990*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 10 2020
%o A147527 (Sage)
%o A147527 def A147527_list(prec):
%o A147527     P.<x> = PowerSeriesRing(ZZ, prec)
%o A147527     return P( 93645643*x*(1-x)/(1 - 128405450990*x + x^2) ).list()
%o A147527 a=A147527_list(20); a[1:] # _G. C. Greubel_, Jan 10 2020
%o A147527 (GAP) a:=[93645643, 12024611022569890927];; for n in [3..20] do a[n]:=128405450990*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 10 2020
%Y A147527 Cf. A147528, A147529, A147530.
%K A147527 easy,nonn
%O A147527 1,1
%A A147527 _Richard Choulet_, Nov 06 2008
%E A147527 Editing and a(6) from _Colin Barker_, Oct 21 2014