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.

A105040 Nonnegative k such that 7*k^2 + 7*k + 1 is a square.

Original entry on oeis.org

0, 15, 111, 3936, 28320, 999855, 7193295, 253959360, 1827068736, 64504677711, 464068265775, 16383934179360, 117871512438240, 4161454776879855, 29938900091047311, 1056993129393303936, 7604362751613578880, 268472093411122320015, 1931478200009757988335
Offset: 1

Views

Author

Gerald McGarvey, Apr 03 2005

Keywords

Examples

			3936 = 254*15 + 111 + 15, 28320 = 254*111 + 111 + 15, 999855 = 254*3936 + 111, 7193295 = 254*28320 + 15.

Links

Programs

  • Mathematica
    LinearRecurrence[{1,254,-254,-1,1},{0,15,111,3936,28320},20] (* Harvey P. Dale, Jul 25 2018 *)
  • PARI
    for(n=0,7193295,if(issquare(7*n*(n+1)+1),print1(n,",")))

Formula

a(n) = ((7-2*sqrt(7)*(-1)^n)*(8-3*sqrt(7))^((2n-(-1)^n+1)/2)+(7+2*sqrt(7)*(-1)^n)*(8+3*sqrt(7))^((2n-(-1)^n+1)/2)-14)/28. [Bruno Berselli, Jun 13 2012].
G.f.: -3*x^2*(5*x^2+32*x+5)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)). [Colin Barker, Jul 22 2012]
a(n) = A253460(n) - 1. - Michel Marcus, Mar 12 2024

Extensions

More terms from Colin Barker, Jun 13 2012

A253459 Indices of centered square numbers (A001844) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 21, 148, 5208, 37465, 1322685, 9515836, 335956656, 2416984753, 85331667813, 613904611300, 21673907667720, 155929354285321, 5505087215932941, 39605442083860108, 1398270478939299168, 10059626359946181985, 355155196563366055605, 2555105489984246363956
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also positive integers x in the solutions to 4*x^2 - 7*y^2 - 4*x + 7*y = 0, the corresponding values of y being A253460.

Examples

			21 is in the sequence because the 21st centered square number is 841, which is also the 16th centered heptagonal number.

Links

Crossrefs

Cf. A001844, A069099, A253460, A253599.

Programs

  • PARI
    Vec(x*(20*x^3+127*x^2-20*x-1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+254*a(n-2)-254*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(20*x^3+127*x^2-20*x-1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).

A253599 Centered square numbers (A001844) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 841, 43513, 54236113, 2807177521, 3498988573081, 181102250526121, 225733748749491361, 11683630587634972513, 14562987063325697070313, 753757743549580366157401, 939516547177660272044661361, 48627927055673997154643575441, 60611970510056587727363585953081
Offset: 1

Views

Author

Colin Barker, Jan 05 2015

Keywords

Examples

			841 is in the sequence because it is the 21st centered square number and the 16th centered heptagonal number.

Links

Crossrefs

Cf. A001844, A069099, A253459, A253460.

Programs

  • Mathematica
    LinearRecurrence[{1,64514,-64514,-1,1},{1,841,43513,54236113,2807177521},20] (* Harvey P. Dale, Mar 26 2023 *)
  • PARI
    Vec(-x*(x^4+840*x^3-21842*x^2+840*x+1)/((x-1)*(x^2-254*x+1)*(x^2+254*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+64514*a(n-2)-64514*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+840*x^3-21842*x^2+840*x+1) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)).
Showing 1-3 of 3 results.

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

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