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 1-5 of 5 results.

A179088 Positive integers of the form (2*m^2+1)/11.

Original entry on oeis.org

3, 9, 41, 59, 123, 153, 249, 291, 419, 473, 633, 699, 891, 969, 1193, 1283, 1539, 1641, 1929, 2043, 2363, 2489, 2841, 2979, 3363, 3513, 3929, 4091, 4539, 4713, 5193, 5379, 5891, 6089, 6633, 6843, 7419, 7641, 8249, 8483, 9123, 9369, 10041, 10299, 11003, 11273
Offset: 1

Views

Author

Bruno Berselli, Jun 29 2010 - Jul 09 2010

Keywords

Comments

Here m = (11*(2*n - 1) - 5*(-1)^n)/4 for n>0.
All terms of A113338 are in the sequence.

Links

Crossrefs

Cf. A113338, A179337-A179370.

Programs

  • Magma
    [i/11: m in [1..250] | i mod 11 eq 0 where i is 2*m^2+1]; // Bruno Berselli, Jun 26 2011
  • Maple
    t1:=[]; for n from 0 to 1000 do i:=2*n^2+1; if i mod 11 = 0 then t1:=[op(t1),i/11]; fi; od: # N. J. A. Sloane, Jul 05 2010
  • Mathematica
    Select[Table[(2m^2 + 1)/11, {m, 0, 300}], IntegerQ] (* Harvey P. Dale, Dec 17 2010 *)
  • PARI
    for(m=1, 250, i=2*m^2+1; if(Mod(i,11)==0, print1(i/11", "))); \\ Bruno Berselli, Jun 26 2011

Formula

a(n) = (22*n*(n - 1) - 5*(2*n - 1)*(-1)^n + 7)/4.
G.f.: x*(3 + 6*x + 26*x^2 + 6*x^3 + 3*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(2*n)+a(2*n+1)= 11*A000290(2*n)+6; a(2*n-1)+a(2*n) = 11*A000290(2*n-1)+1.

Extensions

Edited by N. J. A. Sloane, Jul 05 2010: offset changed to 1 (since this is a list), and terms verified.

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).

A179338 Positive integers of the form (10*m^2+1)/11.

Original entry on oeis.org

1, 91, 131, 401, 481, 931, 1051, 1681, 1841, 2651, 2851, 3841, 4081, 5251, 5531, 6881, 7201, 8731, 9091, 10801, 11201, 13091, 13531, 15601, 16081, 18331, 18851, 21281, 21841, 24451, 25051, 27841, 28481, 31451, 32131, 35281, 36001, 39331
Offset: 1

Views

Author

Bruno Berselli, Jul 11 2010

Keywords

Comments

Here m=(11*(2*n-1)+7*(-1)^n)/4 for n>0.

Links

Crossrefs

Cf. A113338, A179088, A179337, A179339, A179370.

Programs

  • Magma
    [(110*n*(n-1)+35*(2*n-1)*(-1)^n+39)/4: n in [1..50]]; // Vincenzo Librandi, Jul 10 2014
  • Mathematica
    Select[(10Range[300]^2+1)/11,IntegerQ] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{1,91,131,401,481},50] (* Harvey P. Dale, Jul 10 2014 *)
  • PARI
    isok(n) = (((11*n-1) % 10 == 0) && issquare((11*n-1)/10)) \\ Michel Marcus, Jun 07 2013

Formula

a(n) = (110*n*(n-1)+35*(2*n-1)*(-1)^n+39)/4.
G.f.: x*(1+90*x+38*x^2+90*x^3+1*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 1-5 of 5 results.

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

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