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.

A113338 Positive integers of the form (18*m^2+1)/11.

Original entry on oeis.org

41, 59, 419, 473, 1193, 1283, 2363, 2489, 3929, 4091, 5891, 6089, 8249, 8483, 11003, 11273, 14153, 14459, 17699, 18041, 21641, 22019, 25979, 26393, 30713, 31163, 35843, 36329, 41369, 41891, 47291, 47849, 53609, 54203, 60323, 60953
Offset: 1

Views

Author

Zak Seidov, Jan 08 2006

Keywords

Comments

Here m=(11*(2*n-1)-9*(-1)^n)/4 for n>0.
(18*m^2 + 1)/11 is an integer for m=5+11k and m=6+11k, k=0,1,2,.... [ Corrected by Bruno Berselli, Jun 28 2010]

Links

Crossrefs

Cf. A179088, A179337, A179338, A179339, A179370.

Programs

Formula

From Bruno Berselli, Jun 29 2010: (Start)
G.f.: x*(41+18*x+278*x^2+18*x^3+41*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n) = (198*n*(n-1)-81*(2*n-1)*(-1)^n+83)/4. (End)

Extensions

Definition corrected, comment and cross-reference added by Bruno Berselli, Jul 13 2010

A179337 Positive integers of the form (6*m^2 + 1)/11.

Original entry on oeis.org

5, 35, 107, 197, 341, 491, 707, 917, 1205, 1475, 1835, 2165, 2597, 2987, 3491, 3941, 4517, 5027, 5675, 6245, 6965, 7595, 8387, 9077, 9941, 10691, 11627, 12437, 13445, 14315, 15395, 16325, 17477, 18467, 19691, 20741, 22037, 23147, 24515
Offset: 1

Views

Author

Bruno Berselli, Jul 11 2010

Keywords

Comments

Here m = (11*(2*n-1) - (-1)^n)/4 for n > 0.

Links

Crossrefs

Cf. A113338, A179088, A179338, A179339, A179370.

Programs

  • Magma
    [(66*n*(n-1)-3*(2*n-1)*(-1)^n+17)/4: n in [1..40]]; // Vincenzo Librandi, Nov 16 2011
  • Mathematica
    LinearRecurrence[{1,2,-2,-1,1},{5,35,107,197,341},40] (* Vincenzo Librandi, Nov 16 2011 *)

Formula

a(n) = (66*n*(n-1) - 3*(2*n-1)*(-1)^n + 17)/4.
G.f.: x*(5 + 30*x + 62*x^2 + 30*x^3 + 5*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

A179370 Positive integers of the form (7*m^2+1)/11.

Original entry on oeis.org

16, 23, 163, 184, 464, 499, 919, 968, 1528, 1591, 2291, 2368, 3208, 3299, 4279, 4384, 5504, 5623, 6883, 7016, 8416, 8563, 10103, 10264, 11944, 12119, 13939, 14128, 16088, 16291, 18391, 18608, 20848, 21079, 23459, 23704, 26224, 26483
Offset: 1

Views

Author

Bruno Berselli, Jul 12 2010

Keywords

Comments

Here m=(11*(2*n-1)-9*(-1)^n)/4 for n>0.

Links

Crossrefs

Cf. A113338, A179088, A179337, A179338, A179339.

Formula

a(n) = (154*n*(n-1)-63*(2*n-1)*(-1)^n+65)/8.
G.f.: x*(16+7*x+108*x^2+7*x^3+16*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).

A179339 Positive integers of the form (30*m^2+1)/11.

Original entry on oeis.org

11, 221, 461, 1091, 1571, 2621, 3341, 4811, 5771, 7661, 8861, 11171, 12611, 15341, 17021, 20171, 22091, 25661, 27821, 31811, 34211, 38621, 41261, 46091, 48971, 54221, 57341, 63011, 66371, 72461, 76061, 82571, 86411, 93341, 97421
Offset: 1

Views

Author

Bruno Berselli, Jul 11 2010 - Dec 10 2010

Keywords

Comments

Here m = (11*(2*n-1)+3*(-1)^n)/4 for n>0.
More generally, (t*((11*(2*n-1)+k*(-1)^n)/4)^2 +1)/11 = ( 22*t*n*(n-1) +t*k*(2*n-1)*(-1)^n+(t*(k^2+121)+16)/22 )/8 for any natural number t == 2, 6, 7, 8, 10 (mod 11) and k = -5, -1, -9, 3, 7, respectively.

Links

Crossrefs

Cf. A113338, A179088, A179337, A179338, A179370.

Programs

  • Mathematica
    LinearRecurrence[{1,2,-2,-1,1},{11,221,461,1091,1571},40] (* Harvey P. Dale, Mar 03 2023 *)

Formula

a(n) = (330*n*(n-1)+45*(2*n-1)*(-1)^n+89)/4.
G.f.: x*(11+210*x+218*x^2+210*x^3+11*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
Showing 1-4 of 4 results.

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