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.

A047345 Numbers that are congruent to {0, 4} mod 7.

Original entry on oeis.org

0, 4, 7, 11, 14, 18, 21, 25, 28, 32, 35, 39, 42, 46, 49, 53, 56, 60, 63, 67, 70, 74, 77, 81, 84, 88, 91, 95, 98, 102, 105, 109, 112, 116, 119, 123, 126, 130, 133, 137, 140, 144, 147, 151, 154, 158, 161, 165, 168, 172, 175, 179, 182, 186, 189, 193
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Nonnegative k such that k or 5*k + 1 is divisible by 7. - Bruno Berselli, Feb 13 2018
Maximum number of 2's possible in an infinite Minesweeper grid with n mines. The pattern of mines (x) that generates these 2's looks like "...xx.xx.xx...". - Dmitry Kamenetsky, Apr 14 2018

Links

Crossrefs

Cf. A030123.
Cf. A005009, A030308.

Programs

Formula

a(n) = ceiling(7*(n-1)/2).
a(n) = 7*n - a(n-1) - 10 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 7*n/2 - 13/4 + (-1)^n/4.
G.f.: x^2*(4 + 3*x) / ((1 + x)*(x - 1)^2). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k), with b(0) = 4, b(k) = A005009(k-1) = 7*2^(k-1), and k>0. - Philippe Deléham, Oct 17 2011.
a(n) = 4*(n - 1) - floor((n - 1)/2). - Wesley Ivan Hurt, Jun 14 2013
a(n) = 2*(n - 1) + floor((3*n - 2 - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014
E.g.f.: 3 + ((14*x - 13)*exp(x) + exp(-x))/4. - David Lovler, Aug 31 2022

A047355 Numbers that are congruent to {0, 3} mod 7.

Original entry on oeis.org

0, 3, 7, 10, 14, 17, 21, 24, 28, 31, 35, 38, 42, 45, 49, 52, 56, 59, 63, 66, 70, 73, 77, 80, 84, 87, 91, 94, 98, 101, 105, 108, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 154, 157, 161, 164, 168, 171, 175, 178, 182, 185, 189, 192, 196, 199, 203
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers k such that k^2/7 + k*(k + 1)/7 = k*(2*k + 1)/7 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

Links

Crossrefs

Cf. A030123, A010702 (first differences).
Cf. A030308, A125176.

Programs

Formula

a(n) = a(n-2) + 7 = a(n-1) + a(n-2) - a(n-3). - Henry Bottomley, Jan 19 2001
From Bruno Berselli, Sep 12 2011: (Start)
G.f.: x^2*(3 + 4*x)/((1 + x)*(1 - x)^2).
a(n) = (14*n - (-1)^n - 15)/4. (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*A125176(k+2). - Philippe Deléham, Oct 17 2011
a(n) = 2*n - 2 + floor((3*n - 3)/2). - Wesley Ivan Hurt, Jan 30 2014
E.g.f.: 4 + ((14*x - 15)*exp(x) - exp(-x))/4. - David Lovler, Aug 31 2022

A256680 Minimal most likely sum for a roll of n 4-sided dice.

Original entry on oeis.org

0, 1, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65, 67, 70, 72, 75, 77, 80, 82, 85, 87, 90, 92, 95, 97, 100, 102, 105, 107, 110, 112, 115, 117, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 150, 152, 155, 157, 160, 162
Offset: 0

Views

Author

Ran Pan, Apr 08 2015

Keywords

Comments

In fact ceiling(5n/2) and floor(5n/2) have the same probability.
a(n) equals A047215(n) except for n=1.

Examples

			For n=1, there are four equally likely outcomes, 1,2,3,4, and the smallest of these is 1, so a(1)=1.

Links

Crossrefs

Cf. A030123, A047215.

Programs

  • Magma
    [n le 1 select n else Floor(5*n/2): n in [0..70]]; // Vincenzo Librandi, Apr 08 2015
  • Maple
    a:= n-> iquo(5*n, 2) -`if`(n=1, 1, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 08 2015
  • Mathematica
    Join[{0, 1}, Table[Floor[5 n/2], {n, 2, 100}]]
  • PARI
    a(n)=if(n<2,n,5*n\2) \\ Charles R Greathouse IV, Apr 08 2015
  • PARI
    concat(0, Vec(-x*(x^3-x^2-4*x-1)/((x-1)^2*(x+1)) + O(x^100))) \\ Colin Barker, Apr 08 2015

Formula

a(n) = floor(5*n/2), for n>=2; a(0)=0 and a(1)=1.
From Colin Barker, Apr 08 2015: (Start)
a(n) = (-1+(-1)^n+10*n)/4 for n>1.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>4.
G.f.: -x*(x^3-x^2-4*x-1) / ((x-1)^2*(x+1)).
(End)
a(n)-a(n-1) = A010693(n-3), n>=3. - R. J. Mathar, Aug 08 2025

A263941 Minimal most likely sum for a roll of n 8-sided dice.

Original entry on oeis.org

1, 9, 13, 18, 22, 27, 31, 36, 40, 45, 49, 54, 58, 63, 67, 72, 76, 81, 85, 90, 94, 99, 103, 108, 112, 117, 121, 126, 130, 135, 139, 144, 148, 153, 157, 162, 166, 171, 175, 180, 184, 189, 193, 198, 202, 207, 211, 216, 220, 225
Offset: 1

Views

Author

Gianmarco Giordano, Oct 30 2015

Keywords

Examples

			For n=1, there are eight equally likely outcomes, 1,2,3,4,5,6,7,8 and the smallest of these is 1, so a(1)=1.

Links

Crossrefs

Cf. A030123, A130877, A256680.

Programs

  • Mathematica
    Join[{1}, Table[(18 n + (-1)^n - 1)/4, {n, 2, 50}]]
  • PARI
    a(n)=if(n<2,1,9*n\2);
vector(50,n,a(n))

Formula

G.f.: x*(1 + 8*x + 3*x^2 - 3*x^3)/((1 - x)^2*(1 + x)).
a(n) = floor(9*n/2) = (18*n + (-1)^n - 1)/4 with n>1, a(1)=1.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
a(n) = -A130877(-n+1) for n>1.

Extensions

Edited by Bruno Berselli, Oct 30 2015

A258589 Minimal most likely sum for a roll of n 12-sided dice.

Original entry on oeis.org

1, 13, 19, 26, 32, 39, 45, 52, 58, 65, 71, 78, 84, 91, 97, 104, 110, 117, 123, 130, 136, 143, 149, 156, 162, 169, 175, 182, 188, 195, 201, 208, 214, 221, 227, 234, 240, 247, 253, 260, 266, 273, 279, 286, 292, 299, 305, 312, 318, 325, 331, 338, 344, 351, 357
Offset: 1

Views

Author

Gianmarco Giordano, Nov 06 2015

Keywords

Examples

			For n=1, there are twelve equally likely outcomes, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and the smallest of these is 1, so a(1)=1.

Links

Crossrefs

Cf. A030123, A256680, A263941, A258588.

Programs

  • Mathematica
    Join[{1}, Table[(26 n + (-1)^n - 1)/4, {n, 2, 50}]]
  • PARI
    a(n)=if(n<2, 1, 13*n\2);
vector(50, n, a(n))
  • PARI
    a(n) = if(n<2,n, (26*n + (-1)^n - 1)/4);
vector(50, n, a(n))
  • PARI
    Vec(-x*(5*x^3-5*x^2-12*x-1)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Nov 06 2015

Formula

a(n) = floor(13*n/2) = (26*n + (-1)^n - 1)/4 with n>1, a(1)=1.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
G.f.: -x*(5*x^3-5*x^2-12*x-1) / ((x-1)^2*(x+1)). - Colin Barker, Nov 06 2015

A258588 Minimal most likely sum for a roll of n 10-sided dice.

Original entry on oeis.org

1, 11, 16, 22, 27, 33, 38, 44, 49, 55, 60, 66, 71, 77, 82, 88, 93, 99, 104, 110, 115, 121, 126, 132, 137, 143, 148, 154, 159, 165, 170, 176, 181, 187, 192, 198, 203, 209, 214, 220, 225, 231, 236, 242, 247, 253, 258, 264, 269, 275
Offset: 1

Views

Author

Gianmarco Giordano, Nov 06 2015

Keywords

Examples

			For n=1, there are ten equally likely outcomes, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and the smallest of these is 1, so a(1)=1.

Links

Crossrefs

Cf. A030123, A256680, A263941.

Programs

  • Mathematica
    Join[{1}, Table[(22 n + (-1)^n - 1)/4, {n, 2, 50}]]
  • PARI
    a(n)=if(n<2, 1,11*n\2);
vector(50, n, a(n))

Formula

G.f.: x*(1 + 10*x + 4*x^2 - 4*x^3)/((1 - x)^2*(1 + x)).
a(n) = floor(11*n/2) = (22*n + (-1)^n - 1)/4 with n>1, a(1)=1.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
Showing 1-6 of 6 results.

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

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