A147528 Numbers x such that (x + 103)^3 - x^3 is a square.

5327263, 684056220943393618, 87836547552751547393253180439, 11278691501915643258450349467913578516874, 1448245468880558621537182415402996832263200922550703, 185962612575832140241603356412217415201039246491645779158754978

Offset: 1

Author

Richard Choulet, Nov 06 2008

Keywords

• nonn
• easy

Examples

			a(1) = 5327263 because the first relation is : (5327263 + 103)^3 - 5327263^3 = 93645643^2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..50
• Index entries for linear recurrences with constant coefficients, signature (128405450991,-128405450991,1).

Crossrefs

Cf. A147527, A147529, A147530.

Programs

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

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

Maple

seq(coeff(series(103*x*(51721 [phone]*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

Mathematica

LinearRecurrence[{128405450991, -128405450991, 1}, {5327263, 684056220943393618, 87836547552751547393253180439}, 20] (* G. C. Greubel, Jan 10 2020 *)

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

Sage

def A147528_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 103*x*(51721 + 64202725495*x - 51722*x^2) / ((1-x)*(1 -128405450990*x +x^2)) ).list()
a=A147528_list(20); a[1:] # G. C. Greubel, Jan 10 2020

Formula

a(n+2) = 128405450990*a(n+1) - a(n) + 6612880725882.

G.f.: 103*x*(51721 + 64202725495*x - 51722*x^2) / ((1-x)*(1 -128405450990*x +x^2)). - Colin Barker, Oct 21 2014

Extensions

Editing and a(6) from Colin Barker, Oct 21 2014