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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A047279 Numbers that are congruent to {0, 1, 2, 6} mod 7.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 9, 13, 14, 15, 16, 20, 21, 22, 23, 27, 28, 29, 30, 34, 35, 36, 37, 41, 42, 43, 44, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 65, 69, 70, 71, 72, 76, 77, 78, 79, 83, 84, 85, 86, 90, 91, 92, 93, 97, 98, 99, 100, 104, 105, 106, 107, 111, 112
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047322, A047336, A047352, A047361.

Programs

  • Magma
    [n : n in [0..100] | n mod 7 in [0, 1, 2, 6]]; // Wesley Ivan Hurt, May 21 2016
  • Maple
    A047279:=n->(14*n-17+3*(I^(2*n)+(1+I)*I^(-n)+(1-I)*I^n))/8: seq(A047279(n), n=1..100); # Wesley Ivan Hurt, May 21 2016
  • Mathematica
    LinearRecurrence[{1,0,0,1,-1},{0,1,2,6,7},80] (* Harvey P. Dale, Jun 15 2015 *)

Formula

G.f.: x^2*(1+x+4*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14n-17+3*(i^(2n)+(1+i)*i^(-n)+(1-i)*i^n))/8 where i = sqrt(-1).
a(2n) = A047336(n), a(2n-1) = A047352(n).
a(n) = A047361(n+1) - 1. a(2-n) = - A047322(n). (End)

Extensions

More terms from Wesley Ivan Hurt, May 21 2016

A047291 Numbers that are congruent to {0, 1, 4, 6} mod 7.

Original entry on oeis.org

0, 1, 4, 6, 7, 8, 11, 13, 14, 15, 18, 20, 21, 22, 25, 27, 28, 29, 32, 34, 35, 36, 39, 41, 42, 43, 46, 48, 49, 50, 53, 55, 56, 57, 60, 62, 63, 64, 67, 69, 70, 71, 74, 76, 77, 78, 81, 83, 84, 85, 88, 90, 91, 92, 95, 97, 98, 99, 102, 104, 105, 106, 109, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047336, A047345.

Programs

  • Magma
    I:=[0, 1, 4, 6, 7]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Apr 26 2012
  • Maple
    A047291:=n->(-13-(-1)^n+(3-I)*(-I)^n+(3+I)*I^n+14*n)/8: seq(A047291(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,120], MemberQ[{0,1,4,6}, Mod[#,7]]&] (* Vincenzo Librandi, Apr 26 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,4,6,7},100] (* G. C. Greubel, Jun 01 2016 *)
  • PARI
    x='x+O('x^100); concat(0, Vec(x^2*(1+3*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

Formula

From Colin Barker, Mar 13 2012: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
G.f.: x^2*(1 + 3*x + 2*x^2 + x^3)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = (-13 - (-1)^n + (3-i)*(-i)^n + (3+i)*i^n + 14*n)/8 where i=sqrt(-1). - Colin Barker, May 14 2012
a(2k) = A047336(k), a(2k-1) = A047345(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (4 - sin(x) + 3*cos(x) + (7*x - 6)*sinh(x) + 7*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, Jun 01 2016

A318958 A(n, k) is a square array read in the decreasing antidiagonals, for n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 0, 0, -1, -1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 3, 2, 2, 0, 1, 3, 3, 4, 3, 3, 0, 3, 4, 6, 6, 7, 6, 6, 0, 2, 5, 6, 8, 8, 9, 8, 8, 0, 4, 6, 9, 10, 12, 12, 13, 12, 12, 0, 3, 7, 9, 12, 13, 15, 15, 16, 15, 15, 0, 5, 8, 12, 14, 17, 18, 20, 20, 21, 20, 20
Offset: 0

Views

Author

Paul Curtz, Sep 06 2018

Keywords

Examples

			The array starts:
[n\k][0,   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, ...]
[0]   0,   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ... = A000004
[1]   0,  -1,  1,  0,  2,  1,  3,  2,  4,  3,  5,  4, ... = A028242(n-2)
[2]  -1,   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ... = A023443(n)
[3]   0,   0,  3,  3,  6,  6,  9,  9, 12, 12, 15, 15, ... = 3*A004526(n)
[4]   0,   2,  4,  6,  8, 10, 12, 14, 16, 18, 20, 22, ... = A005843(n)
[5]   2,   3,  7,  8, 12, 13, 17, 18, 22, 23, 27, 28, ... = A047221(n+1)
[6]   3,   6,  9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ... = A008585(n+1)
[7]   6,   8, 13, 15, 20, 22, 27, 29, 34, 36, 41, 43, ... = A047336(n+2)
[8]   8,  12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, ... = A008586(n+2)
Successive columns: A198442(n-2), A198442(n-1), A004652(n), A198442(n+1), A198442(n+2), A079524(n), ... .
First subdiagonal: 0, 0, 3, 6, ... = A242477(n).
First upperdiagonal: 0, 1, 2, 6, 10, ... = A238377(n-1).
Array written as a triangle:
0;
0,  0;
0, -1, -1;
0,  1,  0, 0;
0,  0,  1, 0, 0;
0,  2,  2, 3, 2, 2;
etc.

Crossrefs

Cf. A000004, A004526, A004652, A005843, A008585, A008586, A023443, A028242, A047221, A047336, A079524, A198442, A238377, A242477.

Programs

  • Maple
    A := proc(n, k) option remember; local h;
h := n -> `if`(n<3, [0, 0, -1][n+1], iquo(n^2-4*n+3, 4));
if k = 0 then h(n) elif k = 1 then h(n+1) else A(n, k-2) + n fi end: # Peter Luschny, Sep 08 2018
  • Mathematica
    h[n_] := If[n < 3, {0, 0, -1}[[n + 1]], Quotient[n^2 - 4 n + 3, 4]];
A[n_, k_] := A[n, k] = If[k == 0, h[n], If[k == 1, h[n+1], A[n, k-2] + n]];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 22 2019, after Peter Luschny *)

Formula

Let h(n) = 0, 0, -1, A198442(1), A198442(2), A198442(3), ... Then A(n, 0) = h(n), A(n, 1) = h(n+1) and A(n, k) = A(n, k-2) + n otherwise.

A371858 Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^7) dx.

Original entry on oeis.org

1, 0, 3, 4, 3, 7, 6, 0, 5, 5, 2, 6, 6, 7, 9, 6, 4, 8, 2, 9, 4, 5, 3, 0, 6, 4, 0, 6, 5, 1, 2, 4, 8, 8, 7, 4, 8, 3, 6, 4, 2, 5, 6, 7, 2, 6, 4, 2, 7, 3, 3, 7, 5, 8, 1, 0, 2, 8, 3, 3, 2, 6, 8, 8, 1, 5, 2, 5, 9, 3, 1, 0, 0, 7, 4, 8, 6, 2, 5, 4, 8, 5, 5, 5, 2, 0, 7, 5, 8, 9, 3, 8, 1, 8, 2, 0, 0, 0, 5, 9, 6, 0
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 09 2024

Keywords

Examples

			1.0343760552667964829453064065124887483642567...

Links

Crossrefs

Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^k) dx: A019669 (k=2), A248897 (k=3), A093954 (k=4), A352324 (k=5), A019670 (k=6), this sequence (k=7), A352125 (k=8).
Cf. A019674, A047336, A121598.

Programs

  • Mathematica
    RealDigits[(1/7) Pi Csc[Pi/7], 10, 102][[1]]

Formula

Equals (1/7) * Pi * csc(Pi/7).
Equals A019674 * A121598.
Equals Product_{k>=2} (1 + (-1)^k/A047336(k)). - Amiram Eldar, Nov 22 2024
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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