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 1-3 of 3 results.

A104240 Nonnegative integers n such that 13*n^2 + 13*n + 1 is a square.

Original entry on oeis.org

0, 7, 144, 504, 9727, 187560, 654840, 12626287, 243453384, 849982464, 16388911447, 316002305520, 1103276584080, 21272794432567, 410170749112224, 1432052156154024, 27612070784561167, 532401316345361880, 1858802595411339720, 35840446605565962847
Offset: 0

Views

Author

Gerald McGarvey, Apr 02 2005

Keywords

Comments

The next terms appear to be 243453384, 849982464, 16388911447 (confirmed by Pierre CAMI).

Links

Crossrefs

Cf. similar sequences indexed in A222390. [Bruno Berselli, Feb 19 2013]

Programs

  • Magma
    m:=19; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((7+137*x+360*x^2+137*x^3+7*x^4)/((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)))); // Bruno Berselli, Feb 19 2013
  • Mathematica
    LinearRecurrence[{1, 0, 1298, -1298, 0, -1, 1}, {0, 7, 144, 504, 9727, 187560, 654840}, 20] (* Bruno Berselli, Feb 19 2013 *)
  • PARI
    for(n=0,12626287,if(issquare(13*n*(n+1)+1),print1(n,",")))

Formula

a(0)=0, a(1)=7, a(2)=144, a(3)=504, a(4)=9727, a(6)=187560 and then a(n) = 1298*a(n-3)+648-a(n-6). - Pierre CAMI, Apr 05 2005
G.f.: x*(7+137*x+360*x^2+137*x^3+7*x^4)/((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)). - Bruno Berselli, Feb 19 2013
a(n) = a(n-1)+1298*a(n-3)-1298*a(n-4)-a(n-6)+a(n-7). - Bruno Berselli, Feb 19 2013

Extensions

More terms from Pierre CAMI, Apr 05 2005

A105838 Nonnegative integers n such that 11*n^2 + 11*n + 1 is a square.

Original entry on oeis.org

0, 39, 159, 15720, 63480, 6256719, 25265079, 2490158640, 10055438160, 991076882199, 4002039122799, 394446108956760, 1592801515436040, 156988560287908479, 633931001104421319, 62481052548478618080, 252302945638044249120, 24867301925734202087559
Offset: 1

Views

Author

Pierre CAMI, Apr 22 2005

Keywords

Links

Crossrefs

Cf. A105837 (square roots of 11*a(n)^2+11*a(n)+1).
Cf. similar sequences indexed in A222390. [Bruno Berselli, Feb 20 2013]

Programs

  • Magma
    m:=17; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(3*(13+40*x+13*x^2)/((1-x)*(1-20*x+x^2)*(1+20*x+x^2)))); // Bruno Berselli, Feb 20 2013
  • Mathematica
    LinearRecurrence[{1, 398, -398, -1, 1}, {0, 39, 159, 15720, 63480}, 18] (* Bruno Berselli, Feb 20 2013 *)
  • Maxima
    makelist(expand(-1/2+((11+2*sqrt(11)*(-1)^n)*(10-3*sqrt(11))^(2*floor(n/2))+(11-2*sqrt(11)*(-1)^n)*(10+3*sqrt(11))^(2*floor(n/2)))/44), n, 1, 18); /* Bruno Berselli, Feb 20 2013 */

Formula

a(1)=0, a(2)=39, a(3)=398*a(1)+198-a(2), a(4)=398*a(2)+198-a(1), a(n) = 398*a(n-2) + 198 - a(n-4) for n>4.
From Bruno Berselli, Feb 20 2013: (Start)
G.f.: 3*x*(13+40*x+13*x^2)/((1-x)*(1-20*x+x^2)*(1+20*x+x^2)).
a(n) = a(-n+1) = -1/2+((11+2*t*(-1)^n)*(10-3*t)^(2*floor(n/2))+(11-2*t*(-1)^n)*(10+3*t)^(2*floor(n/2)))/44, where t=sqrt(11). (End)

Extensions

More terms from Bruno Berselli, Feb 20 2013

A222393 Nonnegative integers m such that 18*m*(m+1)+1 is a square.

Original entry on oeis.org

0, 4, 12, 152, 424, 5180, 14420, 175984, 489872, 5978292, 16641244, 203085960, 565312440, 6898944364, 19203981732, 234361022432, 652370066464, 7961375818340, 22161378278060, 270452416801144, 752834491387592, 9187420795420572, 25574211328900084
Offset: 1

Views

Author

Bruno Berselli, Feb 19 2013

Keywords

Comments

a(n+2)/a(n) tends to A156164.
a(n) is congruent to {0,2,4} (mod 5, 6 and 10).

Links

Crossrefs

Cf. nonnegative integers n such that k*n*(n+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), A222390 (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), this sequence (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].

Programs

  • Magma
    m:=22; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(4*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2))));
  • Magma
    I:=[0,4,12,152,424]; [n le 5 select I[n] else Self(n-1)+34*Self(n-2)-34*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    LinearRecurrence[{1, 34, -34, -1, 1}, {0, 4, 12, 152, 424}, 23]
CoefficientList[Series[4 x (1 + x)^2 / ((1 - x) (1 - 6 x + x^2) (1 + 6 x + x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 18 2013 *)
  • Maxima
    makelist(expand(-1/2+((3+sqrt(2)*(-1)^n)*(3-2*sqrt(2))^(2*floor(n/2))+(3-sqrt(2)*(-1)^n)*(3+2*sqrt(2))^(2*floor(n/2)))/12), n, 1, 23);
  • PARI
    x='x+O('x^30); concat([0], Vec(4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)))) \\ G. C. Greubel, Jul 15 2018

Formula

G.f.: 4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)).
a(n) = a(-n+1) = a(n-1)+34*a(n-2)-34*a(n-3)-a(n-4)+a(n-5).
a(n) = -1/2+((3+t*(-1)^n)*(3-2*t)^(2*floor(n/2))+(3-t*(-1)^n)*(3+2*t)^(2*floor(n/2)))/12, where t=sqrt(2).
Showing 1-3 of 3 results.

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