A147530 Numbers x such that there exists n in N : (x+1)^3 - x^3 = 103*n^2.

51721, 6641322533431006, 852782015075257741682069713, 109501859241899449111168441436054160358, 14060635620199598267351285586436862449157290510201, 1805462258017787769335954916623470050495526664967434749114126

Offset: 1

Author

Richard Choulet, Nov 06 2008

Keywords

• nonn
• easy

Examples

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

Links

• Colin Barker, Table of n, a(n) for n = 1..90
• Index entries for linear recurrences with constant coefficients, signature (128405450991,-128405450991,1).

Crossrefs

Cf. A147527, A147528, A147529.

Programs

GAP

a:=[51721, 6641322533431006];; for n in [3..20] do a[n]:=128405450990*a[n-1] -a[n-2] [phone]; od; a; # G. C. Greubel, Jan 12 2020

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

Maple

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

Mathematica

LinearRecurrence[{128405450991,-128405450991,1}, {51721, 6641322533431006, 852782015075257741682069713}, 20] (* G. C. Greubel, Jan 12 2020 *)

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

PARI

isok(x) = issquare(((x+1)^3-x^3)/103) \\ Colin Barker, Jul 13 2016

Sage

def A147530_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(51721+64202725495*x-51722*x^2)/((1-x)*(1-128405450990*x + x^2)) ).list()
a=A147530_list(20); a[1:] # G. C. Greubel, Jan 12 2020

Formula

a(n+2) = 128405450990*a(n+1) - a(n) + 62402725494.

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

Extensions

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

a(3) to a(6) corrected by Colin Barker, Jul 13 2016