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.

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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

A195145 Concentric 14-gonal numbers.

Original entry on oeis.org

0, 1, 14, 29, 56, 85, 126, 169, 224, 281, 350, 421, 504, 589, 686, 785, 896, 1009, 1134, 1261, 1400, 1541, 1694, 1849, 2016, 2185, 2366, 2549, 2744, 2941, 3150, 3361, 3584, 3809, 4046, 4285, 4536, 4789, 5054, 5321, 5600, 5881, 6174, 6469, 6776, 7085, 7406
Offset: 0

Views

Author

Omar E. Pol, Sep 17 2011

Keywords

Comments

Also concentric tetradecagonal numbers or concentric tetrakaidecagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 14, ..., and the same line from 1, in the direction 1, 29, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Main axis, perpendicular to A024966 in the same spiral.
Partial sums of A113801. - Reinhard Zumkeller, Jan 07 2012

Links

Crossrefs

A144555 and A195314 interleaved.
Cf. A024966, A032527, A032528, A077221, A113801, A195045, A195046, A195142, A195143, A195146, A195147, A195148, A195149.
Column 14 of A195040. - Omar E. Pol, Sep 28 2011

Programs

  • Haskell
    a195145 n = a195145_list !! n
a195145_list = scanl (+) 0 a113801_list
-- Reinhard Zumkeller, Jan 07 2012
  • Magma
    [(14*n^2+5*(-1)^n-5)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011
  • Mathematica
    LinearRecurrence[{2, 0, -2, 1}, {0, 1, 14, 29}, 50] (* Amiram Eldar, Jan 16 2023 *)

Formula

G.f.: -x*(1+12*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = (14*n^2 + 5*(-1)^n - 5)/4;
a(n) = a(-n) = -a(n-1) + 7*n^2 - 7*n + 1. (End)
Sum_{n>=1} 1/a(n) = Pi^2/84 + tan(sqrt(5/7)*Pi/2)*Pi/(2*sqrt(35)). - Amiram Eldar, Jan 16 2023
E.g.f.: (7*x*(x + 1)*cosh(x) + (7*x^2 + 7*x - 5)*sinh(x))/2. - Stefano Spezia, Nov 30 2024

A175886 Numbers that are congruent to {1, 12} mod 13.

Original entry on oeis.org

1, 12, 14, 25, 27, 38, 40, 51, 53, 64, 66, 77, 79, 90, 92, 103, 105, 116, 118, 129, 131, 142, 144, 155, 157, 168, 170, 181, 183, 194, 196, 207, 209, 220, 222, 233, 235, 246, 248, 259, 261, 272, 274, 285, 287, 298, 300, 311, 313, 324, 326, 337, 339, 350
Offset: 1

Views

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Keywords

Comments

Cf. property described by Gary Detlefs in A113801: 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 this case, a(n)^2-1 == 0 (mod 13).

Links

Crossrefs

Cf. A000217, A091998, A113801, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A175887.
Cf. A195045 (partial sums).

Programs

  • Haskell
    a175886 n = a175886_list !! (n-1)
a175886_list = 1 : 12 : map (+ 13) a175886_list
-- Reinhard Zumkeller, Jan 07 2012
  • Magma
    [n: n in [1..350] | n mod 13 in [1, 12]]; // Bruno Berselli, Feb 29 2012
  • Magma
    [(26*n+9*(-1)^n-13)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013
  • Mathematica
    Select[Range[1, 350], MemberQ[{1, 12}, Mod[#, 13]]&] (* Bruno Berselli, Feb 29 2012 *)
CoefficientList[Series[(1 + 11 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{1,1,-1},{1,12,14},60] (* Harvey P. Dale, Oct 23 2015 *)
  • PARI
    a(n)=(26*n+9*(-1)^n-13)/4 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: x*(1+11*x+x^2)/((1+x)*(1-x)^2).
a(n) = (26*n+9*(-1)^n-13)/4.
a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
a(n) = a(n-2)+13.
a(n) = 13*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n>1.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/13)*cot(Pi/13). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((26*x - 13)*exp(x) + 9*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/13).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/13)*cosec(Pi/13). (End)

A175887 Numbers that are congruent to {1, 14} mod 15.

Original entry on oeis.org

1, 14, 16, 29, 31, 44, 46, 59, 61, 74, 76, 89, 91, 104, 106, 119, 121, 134, 136, 149, 151, 164, 166, 179, 181, 194, 196, 209, 211, 224, 226, 239, 241, 254, 256, 269, 271, 284, 286, 299, 301, 314, 316, 329, 331, 344, 346, 359, 361, 374, 376, 389, 391, 404
Offset: 1

Views

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Keywords

Comments

Cf. property described by Gary Detlefs in A113801: 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 this case, a(n)^2-1==0 (mod 15).

Links

Crossrefs

Cf. A000217, A019693, A019976, A113801, A175886, A091998, A175885, A090771, A056020, A047522, A047336, A007310, A047209, A005408, A001651.
Cf. A132240 (primes).

Programs

  • Haskell
    a175887 n = a175887_list !! (n-1)
a175887_list = 1 : 14 : map (+ 15) a175887_list
-- Reinhard Zumkeller, Jan 07 2012
  • Magma
    [n: n in [1..450] | n mod 15 in [1,14]];
  • Magma
    [(30*n+11*(-1)^n-15)/4: n in [1..55]]; // Vincenzo Librandi, Aug 19 2013
  • Mathematica
    Select[Range[1, 450], MemberQ[{1,14}, Mod[#, 15]]&]
CoefficientList[Series[(1 + 13 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* Vincenzo Librandi, Aug 19 2013 *)
  • PARI
    a(n)=(30*n+11*(-1)^n-15)/4 \\ Charles R Greathouse IV, Sep 28 2015

Formula

G.f.: x*(1+13*x+x^2)/((1+x)*(1-x)^2).
a(n) = (30*n+11*(-1)^n-15)/4.
a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
a(n) = 15*A000217(n-1) -2*sum(a(i), i=1..n-1) +1 for n>1.
a(n) = A047209(A047225(n+1)).
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/15)*cot(Pi/15) = A019693 * A019976 / 10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((30*x - 15)*exp(x) + 11*exp(-x))/4. - David Lovler, Sep 05 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = (Pi/15)*cosec(Pi/15).
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*cos(Pi/15). (End)

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)

A113805 Numbers that are congruent to {5, 9} mod 14.

Original entry on oeis.org

5, 9, 19, 23, 33, 37, 47, 51, 61, 65, 75, 79, 89, 93, 103, 107, 117, 121, 131, 135, 145, 149, 159, 163, 173, 177, 187, 191, 201, 205, 215, 219, 229, 233, 243, 247, 257, 261, 271, 275, 285, 289, 299, 303, 313, 317, 327, 331, 341, 345, 355, 359, 369
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Crossrefs

Cf. A113801, A113802, A113803, A113804, A113806, A008589, A113807.

Programs

  • Mathematica
    Flatten[Table[14n + {5, 9}, {n, 0, 28}]] (* Alonso del Arte, Dec 15 2011 *)

Formula

a(n) = 14*n - a(n-1) - 14 (with a(1) = 5). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7+3*(-1)^n)/2.
O.g.f.: x*(5+4*x+5*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/7)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 4*sin(Pi/7)*sin(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/7). (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)

A219191 Numbers of the form k*(7*k+1), where k = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

0, 6, 8, 26, 30, 60, 66, 108, 116, 170, 180, 246, 258, 336, 350, 440, 456, 558, 576, 690, 710, 836, 858, 996, 1020, 1170, 1196, 1358, 1386, 1560, 1590, 1776, 1808, 2006, 2040, 2250, 2286, 2508, 2546, 2780, 2820, 3066, 3108, 3366, 3410, 3680, 3726, 4008
Offset: 1

Views

Author

Bruno Berselli, Nov 14 2012

Keywords

Comments

Equivalently, numbers m such that 28*m+1 is a square.
Also, integer values of h*(h+1)/7.
Let F(r) = Product_{n >= 1} 1 - q^(14*n-r). The sequence terms are the exponents in the expansion of F(0)*F(6)*F(8) = 1 - q^6 - q^8 + q^26 + q^30 - q^60 - q^66 + + - - ... (by the triple product identity).- Peter Bala, Dec 25 2024

Links

Crossrefs

Cf. numbers of the form k*(i*k+1) with k in A001057: i=0, A001057; i=1, A110660; i=2, A000217; i=3, A152749; i=4, A074378; i=5, A219190; i=6, A036498; i=7, this sequence; i=8, A154260.
Cf. A113801 (square roots of 28*a(n)+1, see the comment).
Cf. similar sequences listed in A219257.
Subsequence of A011860.

Programs

  • Magma
    k:=7; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];
  • Magma
    I:=[0,6,8,26,30]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A := proc (q) local n; for n from 0 to q do if type(sqrt(28*n+1), integer) then print(n) fi; od; end: A(4100); # Peter Bala, Dec 25 2024
  • Mathematica
    Rest[Flatten[{# (7 # - 1), # (7 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (3 + x + 3 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,6,8,26,30},50] (* Harvey P. Dale, Sep 14 2022 *)

Formula

G.f.: 2*x^2*(3+x+3*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+5)/8.
a(n) = 2*A057570(n) = (1/7)*A047335(n)*A047274(n+1).
Sum_{n>=2} 1/a(n) = 7 - cot(Pi/7)*Pi. - Amiram Eldar, Mar 17 2022
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