cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

93645643, 12024611022569890927, 1544025601332411913276450522087, 198261303679194296628699373223979621125203, 25457832112792289938442435570354101121237746019778883, 3268924413670798537740342016261657034171968745307560952072318967
Offset: 1

Views

Author

Richard Choulet, Nov 06 2008

Keywords

Examples

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

Links

Crossrefs

Cf. A147528, A147529, A147530.

Programs

  • 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
  • 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
  • Maple
    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
  • Mathematica
    LinearRecurrence[{128405450990,-1}, {93645643, 12024611022569890927}, 20] (* G. C. Greubel, Jan 10 2020 *)
  • PARI
    Vec(93645643*x*(1-x)/(1-128405450990*x+x^2) + O(x^20)) \\ Colin Barker, Oct 21 2014
  • Sage
    def A147527_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 93645643*x*(1-x)/(1 - 128405450990*x + x^2) ).list()
a=A147527_list(20); a[1:] # G. C. Greubel, Jan 10 2020

Formula

a(n+2) = 128405450990*a(n+1) - a(n).
G.f.: 93645643*x*(1-x)/(1 - 128405450990*x + x^2). - Colin Barker, Oct 21 2014
a(n) = sqrt((A147528(n) + 103)^3 - A147528(n)^3). - Michel Marcus, Jan 10 2020

Extensions

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

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

Original entry on oeis.org

5327263, 684056220943393618, 87836547552751547393253180439, 11278691501915643258450349467913578516874, 1448245468880558621537182415402996832263200922550703, 185962612575832140241603356412217415201039246491645779158754978
Offset: 1

Views

Author

Richard Choulet, Nov 06 2008

Keywords

Examples

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

Links

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 +64202725495*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

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

Original entry on oeis.org

8827, 1133434915879903, 145539221541371657392445143, 18688029378753350610679552570834161667, 2399644840493193509137754319007833077692312755187, 308127477959355126566155341338642382333110448233345362623463
Offset: 1

Views

Author

Richard Choulet, Nov 06 2008

Keywords

Examples

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

Links

Crossrefs

Cf. A147527, A147528, A147530.

Programs

  • 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
  • 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
  • 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 *)
  • PARI
    Vec(8827*x*(1-x)/(1-128405450990*x+x^2) + O(x^20)) \\ Colin Barker, Oct 21 2014
  • Sage
    def A147529_list(prec):
    P. = 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

Formula

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

Extensions

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

A274971 Numbers k such that (x+1)^3 - x^3 = k*y^2 has integer solutions.

Original entry on oeis.org

1, 7, 19, 31, 37, 43, 61, 67, 79, 91, 103, 127, 139, 151, 157, 163, 169, 199, 211, 217, 223, 247, 271, 283, 307, 313, 331, 343, 349, 367, 373, 379, 397, 403, 427, 439, 463, 469, 487, 499, 511, 523, 547, 553, 571, 577, 607, 613, 619, 631, 643, 661, 679, 691
Offset: 1

Views

Author

Colin Barker, Jul 13 2016

Keywords

Examples

			7 is in the sequence because, for instance, (167^3-166^3)/7 = 11881 = 109^2.

Links

Crossrefs

Cf. A001921 (k=1), A144929 (k=7), A145124 (k=19), A145323 (k=31), A145700 (k=37), A145336 (k=43), A274972 (k=61), A145212 (k=67), A145309 (k=79), A145530 (k=91), A147530 (k=103), A145720 (k=127).
Cf. A003215 is a subsequence; A004611 contains this sequence.

Programs

  • Mathematica
    A004611=Select[Range[500],And@@(Mod[#,3]==1&)/@(First/@FactorInteger[#])&]; Select[A004611,Reduce[x^2+3== 12*#*y^2,{x,y},Integers]=!=False &] (* Ray Chandler, Jul 24 2016 *)

Extensions

More terms using solver at Alpern link by Ray Chandler, Jul 23 2016
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.