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 1-6 of 6 results.

A113801 Numbers that are congruent to {1, 13} mod 14.

Original entry on oeis.org

1, 13, 15, 27, 29, 41, 43, 55, 57, 69, 71, 83, 85, 97, 99, 111, 113, 125, 127, 139, 141, 153, 155, 167, 169, 181, 183, 195, 197, 209, 211, 223, 225, 237, 239, 251, 253, 265, 267, 279, 281, 293, 295, 307, 309, 321, 323, 335, 337, 349, 351, 363, 365, 377, 379
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Comments

If 14k+1 is a perfect square..(0,12,16,52,60,120..) then the square root of 14k+1 = a(n) - Gary Detlefs, Feb 22 2010
More generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in our case, a(n)^2-1==0 (mod 14). Also a(n)^2-1==0 (mod 28). - Bruno Berselli, Oct 26 2010 - Nov 17 2010

Links

Crossrefs

Cf. A000217, A113802, A113803, A113804, A113805, A113806, A113807, A008589, A045472 (primes), A195145 (partial sums), A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887.

Programs

  • Haskell
    a113801 n = a113801_list !! (n-1)
a113801_list = 1 : 13 : map (+ 14) a113801_list
-- Reinhard Zumkeller, Jan 07 2012
  • Mathematica
    LinearRecurrence[{1,1,-1},{1,13,15},60] (* or *) Select[Range[500], MemberQ[{1,13},Mod[#,14]]&] (* Harvey P. Dale, May 11 2011 *)
  • PARI
    a(n)=n\2*14-(-1)^n \\ Charles R Greathouse IV, Sep 15 2015

Formula

a(n) = 14*(n-1)-a(n-1), n>1. - R. J. Mathar, Jan 30 2010
From Bruno Berselli, Oct 26 2010: (Start)
a(n) = -a(-n+1) = (14*n+5*(-1)^n-7)/2.
G.f.: x*(1+12*x+x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2)+14 for n>2.
a(n) = 14*A000217(n-1)+1 - 2*sum[i=1..n-1] a(i) for n>1. (End)
a(0)=1, a(1)=13, a(2)=15, a(n)=a(n-1)+a(n-2)-a(n-3). - Harvey P. Dale, May 11 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/14)*cot(Pi/14). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 5*exp(-x))/2. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/14).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/14)*cosec(Pi/14). (End)

Extensions

Corrected and extended by Giovanni Teofilatto, Nov 14 2008
Replaced the various formulas by a correct one - R. J. Mathar, Jan 30 2010

A113807 Permutation of natural numbers generated by 7-rowed array shown below.

Original entry on oeis.org

1, 13, 2, 15, 12, 3, 27, 16, 11, 4, 29, 26, 17, 10, 5, 41, 30, 25, 18, 9, 6, 43, 40, 31, 24, 19, 8, 7, 55, 44, 39, 32, 23, 20, 14, 57, 54, 45, 38, 33, 22, 21, 69, 58, 53, 46, 37, 34, 28, 71, 68, 59, 52, 47, 36, 35, 83, 72, 67, 60, 51, 48, 42, 85, 82, 73, 66, 61, 50, 49
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Comments

For such arrays A_k see the k-family of 2k-periodic sequences P_k defined in a comment on A203571. There the k residue classes mod n have been defined. The present array is A_7 if the last class, starting with 7, is taken as first class [0] after adding a 0 in front. Then one obtains a permutation of the nonnegative integers. However, each complete residue class also includes its negative members. - Wolfdieter Lang, Feb 02 2012

Examples

			1 13 15 27 29 41 43 55 57 69 71 83 85 ... (A113801)
2 12 16 26 30 40 44 54 58 68 72 82 86 ... (A113802)
3 11 17 25 31 39 45 53 59 67 73 81 87 ... (A113803)
4 10 18 24 32 38 46 52 60 66 74 80 88 ... (A113804)
5  9 19 23 33 37 47 51 61 65 75 70 89 ... (A113805)
6  8 20 22 34 36 48 50 62 64 76 78 90 ... (A113806)
7 14 21 28 35 42 49 56 63 70 77 84 91 ... (A008589)

Crossrefs

Cf. A088520, A090243, A090298, A092260, A204453.

Extensions

A-numbers added for array rows by Wolfdieter Lang, Dec 15 2011
More terms from Ray Chandler, Dec 15 2011

A113804 Numbers that are congruent to 4 or 10 mod 14.

Original entry on oeis.org

4, 10, 18, 24, 32, 38, 46, 52, 60, 66, 74, 80, 88, 94, 102, 108, 116, 122, 130, 136, 144, 150, 158, 164, 172, 178, 186, 192, 200, 206, 214, 220, 228, 234, 242, 248, 256, 262, 270, 276, 284, 290, 298, 304, 312, 318, 326, 332, 340, 346, 354, 360
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Comments

Fourth row of the 7-rowed array A113807. - Giovanni Teofilatto, Oct 26 2009 [crossref added by Wolfdieter Lang, Dec 15 2011]

Links

Crossrefs

Cf. A047385, A113801, A113802, A113803, A113805, A113806, A008589, A113807.

Programs

  • Mathematica
    Select[Range[2,400,2],MemberQ[{4,10},Mod[#,14]]&] (* or *) LinearRecurrence[{1,1,-1},{4,10,18},60] (* Harvey P. Dale, Jan 08 2023 *)
  • PARI
    a(n)=7*n-((-1)^n+7)/2 \\ Charles R Greathouse IV, Dec 27 2011

Formula

From R. J. Mathar, Aug 13 2008: (Start)
a(n) = 7n - ((-1)^n + 7)/2.
G.f.: 2x*(2 + 3x + 2x^2)/((1-x)^2*(1+x)). (End)
a(n) = 14*n - a(n-1) - 14 (with a(1)=4). - Vincenzo Librandi, Aug 01 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
E.g.f.: 4 + ((14*x - 7)*exp(x) - exp(-x))/2. - David Lovler, Sep 04 2022
a(n) = 2*A047385(n). - Michel Marcus, Sep 05 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(Pi/7)/2.
Product_{n>=1} (1 + (-1)^n/a(n)) = tan(3*Pi/14). (End)

Extensions

More terms from Neven Juric, Apr 10 2008

A113803 Numbers that are congruent to {3, 11} mod 14.

Original entry on oeis.org

3, 11, 17, 25, 31, 39, 45, 53, 59, 67, 73, 81, 87, 95, 101, 109, 115, 123, 129, 137, 143, 151, 157, 165, 171, 179, 185, 193, 199, 207, 213, 221, 227, 235, 241, 249, 255, 263, 269, 277, 283, 291, 297, 305, 311, 319, 325, 333, 339, 347, 353, 361, 367
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Crossrefs

Third row A113807.
Cf. A008589, A113801, A113802, A113804, A113805, A113806.

Programs

  • Mathematica
    {3+#,11+#}&/@(14*Range[0,30])//Flatten (* Harvey P. Dale, Jun 28 2020 *)

Formula

a(n) = 14*n - a(n-1) - 14 (with a(1) = 3). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7-(-1)^n)/2.
O.g.f.: x*(3+8*x+3*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(3*Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cot(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = sin(Pi/7)*cosec(3*Pi/14). (End)

A113802 Numbers that are congruent to {2, 12} mod 14.

Original entry on oeis.org

2, 12, 16, 26, 30, 40, 44, 54, 58, 68, 72, 82, 86, 96, 100, 110, 114, 124, 128, 138, 142, 152, 156, 166, 170, 180, 184, 194, 198, 208, 212, 222, 226, 236, 240, 250, 254, 264, 268, 278, 282, 292, 296, 306, 310, 320, 324, 334, 338, 348, 352, 362, 366, 376, 380
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Crossrefs

Second row A113807.
Cf. A008589, A113801, A113803, A113804, A113805, A113806.

Programs

  • Mathematica
    Select[Range[400],MemberQ[{2,12},Mod[#,14]]&] (* Harvey P. Dale, Oct 30 2011 *)

Formula

a(n) = 14*n - a(n-1) - 14 (with a(1) = 2). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7-3*(-1)^n)/2.
O.g.f.: 2*x*(1+5*x+x^2)/((1+x)*(1-x)^2).
See the contribution of Bruno Berselli under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(Pi/7)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(Pi/7)*sin(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/7)*sin(Pi/14). (End)

A113806 Numbers that are congruent to {6, 8} mod 14.

Original entry on oeis.org

6, 8, 20, 22, 34, 36, 48, 50, 62, 64, 76, 78, 90, 92, 104, 106, 118, 120, 132, 134, 146, 148, 160, 162, 174, 176, 188, 190, 202, 204, 216, 218, 230, 232, 244, 246, 258, 260, 272, 274, 286, 288, 300, 302, 314, 316, 328, 330, 342, 344, 356, 358, 370
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Crossrefs

Cf. A008589, A113801, A113802, A113803, A113804, A113805, A113807.

Programs

  • Mathematica
    Flatten[# + {6, 8} &/@ (14 Range[0, 30])] (* Harvey P. Dale, Jan 11 2011 *)

Formula

a(n) = 14*n - a(n-1) - 14 (with a(1) = 6). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7+5*(-1)^n)/2.
O.g.f.: 2*x*(3+x+3*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/7)*cosec(3*Pi/14)/4. (End)
Showing 1-6 of 6 results.

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