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.

A146512 Numbers congruent to {1, 3} mod 12.

Original entry on oeis.org

1, 3, 13, 15, 25, 27, 37, 39, 49, 51, 61, 63, 73, 75, 85, 87, 97, 99, 109, 111, 121, 123, 133, 135, 145, 147, 157, 159, 169, 171, 181, 183, 193, 195, 205, 207, 217, 219, 229, 231, 241, 243, 253, 255, 265, 267, 277, 279, 289, 291, 301, 303, 313, 315, 325, 327
Offset: 1

Views

Author

Artur Jasinski, Oct 30 2008

Keywords

Comments

Positive integers k such that Hypergeometric[k/4,(4-k)/4,1/2,3/4] = 2*cos(Pi/6).

Examples

			G.f. = x + 3*x^2 + 13*x^3 + 15*x^4 + 25*x^5 + 27*x^6 + 37*x^7 + 39*x^8 + ...

Links

Crossrefs

Cf. A146507, A146509, A146510, A146511.

Programs

  • Mathematica
    Select[Range[300],MemberQ[{1,3},Mod[#,12]]&] (* Ray Chandler, Dec 06 2016 *)
  • PARI
    {a(n) = 6*n - 9 + n%2*4}; /* Michael Somos, Dec 06 2016 */

Formula

a(2k-1) = 12*(k-1)+1, a(2k) = 12*(k-1)+3, where k>0.
With offset 0, a(n) = 8*floor(n/2) + 2*n + 1, or a(n) = 6*n - 1 + 2*(-1)^n. - Gary Detlefs, Mar 13 2010
a(n) = 12*n-a(n-1)-20 (with a(1)=1). - Vincenzo Librandi, Nov 26 2010
G.f.: x * (1 + 2*x + 9*x^2) / (1 - x - x^2 + x^3). - Michael Somos, Dec 06 2016
a(n) = a(n-1)+a(n-2)-a(n-3). - Wesley Ivan Hurt, May 03 2021
E.g.f.: 9 + (6*x - 7)*exp(x) - 2*exp(-x). - David Lovler, Sep 07 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)+1)*(2*Pi + 2*arccosh(26) - 4*sqrt(3)*arccoth(sqrt(3)) + 3*(sqrt(3)-1)*log(3))/48. - Amiram Eldar, Sep 26 2022

Extensions

Formula and crossrefs corrected by Ray Chandler, Dec 06 2016

A146509 Numbers that are congruent to {1, 5} mod 18.

Original entry on oeis.org

1, 5, 19, 23, 37, 41, 55, 59, 73, 77, 91, 95, 109, 113, 127, 131, 145, 149, 163, 167, 181, 185, 199, 203, 217, 221, 235, 239, 253, 257, 271, 275, 289, 293, 307, 311, 325, 329, 343, 347, 361, 365, 379, 383, 397, 401, 415, 419, 433, 437, 451, 455, 469, 473, 487
Offset: 1

Views

Author

Artur Jasinski, Oct 30 2008

Keywords

Comments

Positive integers k such that Hypergeometric[k/6,(6-k)/6,1/2,3/4] = 2Cos[2Pi/9].

Links

Crossrefs

Cf. A146507, A146511, A146512.

Programs

Formula

a(2k-1) = 18*(k-1)+1, a(2k) = 18*(k-1)+5, where k>0.
G.f.: x*(1+4*x+13*x^2)/((1+x)*(1-x)^2). - Vincenzo Librandi, Jul 11 2012
a(n) = (18*n - 5*(-1)^n - 21)/2. - Bruno Berselli, Jul 12 2012 [Corrected by David Lovler, Sep 24 2022]
a(1)=1, a(n) = 18*n -a(n-1) -30. - Vincenzo Librandi, Jul 12 2012
E.g.f.: 13 + ((18*x - 21)*exp(x) - 5*exp(-x))/2. - David Lovler, Sep 05 2022

Extensions

Crossrefs corrected by Ray Chandler, Dec 06 2016

A146507 Numbers congruent to {1, 13} mod 42.

Original entry on oeis.org

1, 13, 43, 55, 85, 97, 127, 139, 169, 181, 211, 223, 253, 265, 295, 307, 337, 349, 379, 391, 421, 433, 463, 475, 505, 517, 547, 559, 589, 601, 631, 643, 673, 685, 715, 727, 757, 769, 799, 811, 841, 853, 883, 895, 925, 937, 967, 979
Offset: 1

Views

Author

Artur Jasinski, Oct 30 2008

Keywords

Comments

Positive integers k such that Hypergeometric[k/14,(14-k)/14,1/2,3/4] = 2*cos(2Pi/7).

Links

Crossrefs

Cf. A146509, A146510, A146511, A146512.

Programs

  • Mathematica
    Select[Range[1000],MemberQ[{1,13},Mod[#,42]]&]  (* Ray Chandler, Dec 06 2016 *)
LinearRecurrence[{1,1,-1},{1,13,43},50] (* Harvey P. Dale, Apr 15 2020 *)

Formula

a(2k-1) = 42*(k-1)+1, a(2k) = 42*(k-1)+13, where k>0.
G.f.: x*(1 + 12*x + 29*x^2)/((1 - x)^2*(1 + x)). - Ilya Gutkovskiy, Dec 06 2016
E.g.f.: 29 + ((42*x - 49)*exp(x) - 9*exp(-x))/2. - David Lovler, Sep 10 2022

Extensions

Description, formula and crossrefs corrected by Ray Chandler, Dec 06 2016

A146510 Numbers congruent to {1, 4} mod 15.

Original entry on oeis.org

1, 4, 16, 19, 31, 34, 46, 49, 61, 64, 76, 79, 91, 94, 106, 109, 121, 124, 136, 139, 151, 154, 166, 169, 181, 184, 196, 199, 211, 214, 226, 229, 241, 244, 256, 259, 271, 274, 286, 289, 301, 304, 316, 319, 331, 334, 346, 349, 361, 364, 376, 379, 391, 394, 406
Offset: 1

Views

Author

Artur Jasinski, Oct 30 2008

Keywords

Comments

Positive integers k such that Hypergeometric[k/5,(5-k)/5,1/2,3/4] = 2Cos[Pi/5].

Links

Crossrefs

Cf. A146507, A146509, A146511, A146512.

Programs

  • Mathematica
    Select[Range[500],MemberQ[{1,4},Mod[#,15]]&] (* Harvey P. Dale, Jan 21 2016 *)

Formula

a(2k-1) = 15*(k-1)+1, a(2k) = 15*(k-1)+4, where k>0.
G.f.: x*(1 + 3*x + 11*x^2)/((1 - x)^2*(1 + x)). - Ilya Gutkovskiy, Dec 06 2016
E.g.f.: 11 + ((30*x - 35)*exp(x) - 9*exp(-x))/4. - David Lovler, Sep 08 2022

Extensions

Typo in name corrected by N. J. A. Sloane, Jan 21 2016
Formula and crossrefs corrected by Ray Chandler, Dec 06 2016
Showing 1-4 of 4 results.

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