A147529 Numbers n such that there exists x in N : (x+1)^3 - x^3 = 103*n^2.

8827, 1133434915879903, 145539221541371657392445143, 18688029378753350610679552570834161667, 2399644840493193509137754319007833077692312755187, 308127477959355126566155341338642382333110448233345362623463

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..50

Index entries for linear recurrences with constant coefficients, signature (128405450990,-1).

Formula

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

For n >= 0, a(n) = (8827/2)*( (64202725495 + 3652365444*sqrt(309))^n + (64202725495 - 3652365444*sqrt(309))^n ) - (103443*sqrt(309)/412)*( (64202725495 - 3652365444 *sqrt(309))^n - (64202725495 + 3652365444*sqrt(309))^n ). - Paolo P. Lava, Nov 25 2008

G.f.: 8827*x*(1-x) / (1 - 128405450990*x + x^2). - Colin Barker, Oct 21 2014

Example

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

Maple

seq(coeff(series(8827*x*(1-x)/(1-128405450990*x+x^2), x, n+1), x, n), n = 1..20); # G. C. Greubel, Jan 12 2020

Mathematica

LinearRecurrence[{128405450990, -1}, {8827, 1133434915879903}, 20] (* G. C. Greubel, Jan 12 2020 *)

Prog

(PARI) Vec(8827*x*(1-x)/(1-128405450990*x+x^2) + O(x^20)) \\ Colin Barker, Oct 21 2014
(Magma) I:=[8827, 1133434915879903]; [n le 2 select I[n] else 128405450990*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 12 2020
(Sage)
def A147529_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 8827*x*(1-x)/(1-128405450990*x+x^2) ).list()
a=A147529_list(20); a[1:] # G. C. Greubel, Jan 12 2020
(GAP) a:=[8827, 1133434915879903];; for n in [3..20] do a[n]:=128405450990*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 12 2020

Crossrefs

Cf. A147527, A147528, A147530.

Sequence in context: A109027 A237761 A186582 * A255089 A255078 A031592

Adjacent sequences: A147526 A147527 A147528 * A147530 A147531 A147532

Keyword

easy,nonn

Author

Richard Choulet, Nov 06 2008

Extensions

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

Status

approved