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.

A253674 Indices of centered octagonal numbers (A016754) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 10, 40, 931, 3871, 91180, 379270, 8934661, 37164541, 875505550, 3641745700, 85790609191, 356853914011, 8406604195120, 34968041827330, 823761420512521, 3426511245164281, 80720212606031890, 335763133984272160, 7909757073970612651, 32901360619213507351
Offset: 1

Views

Author

Colin Barker, Jan 08 2015

Keywords

Comments

Also positive integers y in the solutions to 3*x^2 - 8*y^2 - 3*x + 8*y = 0, the corresponding values of x being A253673.

Examples

			10 is in the sequence because the 10th centered octagonal number is 361, which is also the 16th centered triangular number.

Links

Crossrefs

Cf. A005448, A016754, A253673, A253675.

Programs

  • Mathematica
    LinearRecurrence[{1,98,-98,-1,1},{1,10,40,931,3871},30] (* Harvey P. Dale, Oct 01 2015 *)
  • PARI
    Vec(-x*(x^2-5*x+1)*(x^2+14*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^2-5*x+1)*(x^2+14*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)).

A253675 Centered triangular numbers (A005448) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 361, 6241, 3463321, 59923081, 33254804881, 575381414521, 319312633001041, 5524812282304561, 3066039868821187801, 53049246959306977201, 29440114501108412261161, 509378863778453312776441, 282683976373603105710477121, 4891055796951461749972406281
Offset: 1

Views

Author

Colin Barker, Jan 08 2015

Keywords

Examples

			361 is in the sequence because it is the 16th centered triangular number and the 10th centered octagonal number.

Links

Crossrefs

Cf. A005448, A016754, A253673, A253674.

Programs

  • Mathematica
    LinearRecurrence[{1,9602,-9602,-1,1},{1,361,6241,3463321,59923081},20] (* Harvey P. Dale, Dec 09 2017 *)
  • PARI
    Vec(-x*(x^4+360*x^3-3722*x^2+360*x+1)/((x-1)*(x^2-98*x+1)*(x^2+98*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+9602*a(n-2)-9602*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+360*x^3-3722*x^2+360*x+1) / ((x-1)*(x^2-98*x+1)*(x^2+98*x+1)).

A254895 Indices of octagonal numbers (A000567) that are also centered square numbers (A001844).

Original entry on oeis.org

1, 13, 53, 1241, 5161, 121573, 505693, 11912881, 49552721, 1167340733, 4855660933, 114387478921, 475805218681, 11208805593493, 46624055769773, 1098348560683361, 4568681660219041, 107626950141375853, 447684178645696213, 10546342765294150201
Offset: 1

Views

Author

Colin Barker, Feb 10 2015

Keywords

Comments

Also positive integers x in the solutions to 6*x^2 - 4*y^2 - 4*x + 4*y - 2 = 0, the corresponding values of y being A253673.

Examples

			13 is in the sequence because the 13th octagonal number is 481, which is also the 16th centered square number.

Links

Crossrefs

Cf. A000567, A001844, A253673, A254896.

Programs

  • PARI
    Vec(-x*(x^4+12*x^3-58*x^2+12*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+12*x^3-58*x^2+12*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)).

A254896 Octagonal numbers (A000567) that are also centered square numbers (A001844).

Original entry on oeis.org

1, 481, 8321, 4617761, 79897441, 44339739841, 767175219361, 425750177334721, 7366416376406081, 4088053158428250401, 70732329279075969601, 39253486001477883014881, 679171818371271083701921, 376911968498137474280636161, 6521407729268615666629875041
Offset: 1

Views

Author

Colin Barker, Feb 10 2015

Keywords

Examples

			481 is in the sequence because it is the 13th octagonal number and the 16th centered square number.

Links

Crossrefs

Cf. A000567, A001844, A253673, A254895.

Programs

  • Mathematica
    LinearRecurrence[{1,9602,-9602,-1,1},{1,481,8321,4617761,79897441},30] (* Harvey P. Dale, Feb 04 2017 *)
  • PARI
    Vec(-x*(x^4+480*x^3-1762*x^2+480*x+1)/((x-1)*(x^2-98*x+1)*(x^2+98*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+9602*a(n-2)-9602*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+480*x^3-1762*x^2+480*x+1) / ((x-1)*(x^2-98*x+1)*(x^2+98*x+1)).
Showing 1-4 of 4 results.

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

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