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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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A183572 a(n) = n + floor(sqrt(2*n-1)).

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92
Offset: 1

Views

Author

Clark Kimberling, Jan 05 2011

Keywords

Links

Crossrefs

Cf. A074148 (complement), A103128.

Programs

A354698 T(n,k) is the number of points with integer coordinates strictly inside the triangle with vertices (0,0), (n,0), (n,k), where T(n,k) is a triangle read by rows, 2 <= k <= n.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 2, 4, 6, 6, 2, 4, 7, 10, 10, 3, 6, 9, 12, 15, 15, 3, 7, 9, 14, 17, 21, 21, 4, 7, 12, 16, 19, 24, 28, 28, 4, 9, 13, 16, 22, 27, 31, 36, 36, 5, 10, 15, 20, 25, 30, 35, 40, 45, 45, 5, 10, 15, 22, 25, 33, 37, 43, 49, 55, 55, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 66
Offset: 2

Views

Author

Hugo Pfoertner, Jun 06 2022

Keywords

Comments

T(n,n) = T(n,n-1) because all grid points with m = n lie on a side of the triangle and thus not strictly inside.

Examples

			The triangle begins:
  0;
  1,  1;
  1,  3,  3;
  2,  4,  6,  6;
  2,  4,  7, 10, 10;
  3,  6,  9, 12, 15, 15;
  3,  7,  9, 14, 17, 21, 21;
  4,  7, 12, 16, 19, 24, 28, 28;
  4,  9, 13, 16, 22, 27, 31, 36, 36;
  5, 10, 15, 20, 25, 30, 35, 40, 45, 45;
  5, 10, 15, 22, 25, 33, 37, 43, 49, 55, 55

Crossrefs

Cf. A000217 (right diagonal), A074148 (3rd diagonal).
Cf. A004526 (column 2), A117571 (column 3).

Programs

  • PARI
    T(n, m) = sum(i=1, n-1, sum(j=1, m-1, (i/j > n/m))); \\ Michel Marcus, Jun 07 2022

A376133 Triangle T read by rows: T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 and T(n, k) = T(n, k-1) + (-1)^k * 2 * (n+1-k) for k >= 2.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 6, 12, 8, 10, 9, 17, 11, 15, 13, 14, 24, 16, 22, 18, 20, 19, 31, 21, 29, 23, 27, 25, 26, 40, 28, 38, 30, 36, 32, 34, 33, 49, 35, 47, 37, 45, 39, 43, 41, 42, 60, 44, 58, 46, 56, 48, 54, 50, 52, 51, 71, 53, 69, 55, 67, 57, 65, 59, 63, 61, 62, 84, 64, 82, 66, 80, 68, 78, 70, 76, 72, 74
Offset: 1

Views

Author

Werner Schulte, Sep 11 2024

Keywords

Comments

Row n consists of the next n odd/even natural numbers if n is odd/even. So the sequence yields a permutation of the natural numbers.

Examples

			Row n=5: Next (1,3,5,7 see rows 1 and 3) five odd numbers are 9,11,13,15 and 17; with "9+8-6+4-2" we get 9,17,11,15,13 for row 5.
Row n=8: Next (2,4,..,24 see rows 2, 4 and 6) eight even numbers are 26,28,..,40; with "26+14-12+10-8+6-4+2" we get 26,40,28,38,30,36,32,34 for row 8.
Triangle T(n, k) for 1 <= k <= n starts:
n\ k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   2   4
   3 :   3   7   5
   4 :   6  12   8  10
   5 :   9  17  11  15  13
   6 :  14  24  16  22  18  20
   7 :  19  31  21  29  23  27  25
   8 :  26  40  28  38  30  36  32  34
   9 :  33  49  35  47  37  45  39  43  41
  10 :  42  60  44  58  46  56  48  54  50  52
  11 :  51  71  53  69  55  67  57  65  59  63  61
  12 :  62  84  64  82  66  80  68  78  70  76  72  74
  etc.

Crossrefs

Cf. A061925 (column 1), A074148 (column 2), A074149 (row sums), A236283 (main diagonal).

Programs

  • Maple
    T := (n, k) -> ((-1)^k*(2 + 4*(n - k)) + 2*n^2 + (-1)^n + 5)/4:
seq(seq(T(n, k), k = 1..n), n = 1..12);  # Peter Luschny, Sep 13 2024
  • PARI
    T(n,k)=(2*n*n+(-1)^k*4*(n-k)+5+2*(-1)^k+(-1)^n)/4

Formula

T(n, k) = (2*n*n + (-1)^k * 4 * (n - k) + 5 + 2 * (-1)^k + (-1)^n) / 4.
T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 = A061925(n-1).
T(n, 2) = (2*n*n + 4*n - 1 + (-1)^n) / 4 = A074148(n) for n > 1.
T(n, k) = T(n, k-2) - (-1)^k * 2 for 3 <= k <= n.
G.f.: x*y*(1 + 2*x*y + 2*x^5*y^2 + x^6*y^3 - x^4*y*(3 + y + y^2) - x^2*(1 + y + 3*y^2) + 2*x^3*(1 + y^3))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Sep 12 2024
Previous Showing 41-43 of 43 results.

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