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 1-3 of 3 results.

A134478 Triangle read by rows, T(0,0) = 1; n-th row = (n+1) terms of n, n+1, n+2, ...

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 12, 7, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
Offset: 0

Views

Author

Gary W. Adamson, Oct 27 2007

Keywords

Comments

Apart from the irregular choice of T(0,0) the same as A051162. - R. J. Mathar, Mar 28 2012

Examples

			First few rows of the triangle:
  1;
  1, 2;
  2, 3, 4;
  3, 4, 5,  6;
  4, 5, 6,  7,  8;
  5, 6, 7,  8,  9, 10;
  6, 7, 8,  9, 10, 11, 12;
  7, 8, 9, 10, 11, 12, 13, 14;
  ...

Links

Crossrefs

Cf. A051162, A134479 (row sums), A126804 (row products).

Programs

  • Mathematica
    Join[{1},Flatten[Table[Range[n,2n],{n,10}]]] (* Harvey P. Dale, Nov 21 2014 *)
  • PARI
    concat([1], for(n=1,10, for(k=0,n, print1(n+k, ", ")))) \\ G. C. Greubel, Sep 24 2017

A158822 Triangle read by rows, matrix triple product A000012 * A145677 * A000012.

Original entry on oeis.org

1, 3, 1, 6, 3, 2, 10, 6, 5, 3, 15, 10, 9, 7, 4, 21, 15, 14, 12, 9, 5, 28, 21, 20, 18, 15, 11, 6, 36, 28, 27, 25, 22, 18, 13, 7, 45, 36, 35, 33, 30, 26, 21, 15, 8, 55, 45, 44, 42, 39, 35, 30, 24, 17, 9, 66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10
Offset: 0

Views

Author

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

Keywords

Examples

			First few rows of the triangle =
   1;
   3,  1;
   6,  3,  2;
  10,  6,  5,  3;
  15, 10,  9,  7,  4;
  21, 15, 14, 12,  9,  5;
  28, 21, 10, 18, 15, 11,  6;
  36, 28, 27, 25, 22, 18, 13,  7;
  45, 36, 35, 33, 30, 26, 21, 15,  8;
  55, 45, 44, 42, 39, 35, 30, 24, 17,  9;
  66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10;
  78, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11;
  91, 78, 77, 75, 72, 68, 63, 57, 50, 42, 33, 23, 12;
  ...

Links

Crossrefs

Cf. A000012, A000096, A000217, A006527, A034828, A055998, A145677, A134479.

Programs

  • Mathematica
    T[n_, k_]:= If[k==0, Binomial[n+2, 2], (n+1-k)*(n+k)/2];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2021 *)
  • Sage
    def A158822(n,k):
    if (k==0): return binomial(n+2, 2)
    else: return (n-k+1)*(n+k)/2
flatten([[A158822(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 26 2021

Formula

Triangle read by rows, A000012 * A145677 * A000012; where A000012 = an infinite lower triangular matrix: (1; 1,1; 1,1,1; ...), with all 1's.
From G. C. Greubel, Dec 26 2021: (Start)
T(n, k) = (n+1-k)*(n+k)/2 with T(n, 0) = binomial(n+2, 2).
Sum_{k=0..n} T(n, k) = (1/3)*(n+1)*(n^2 + 2*n + 3) = A006527(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = binomial(n+2, 2) + A034828(n+1).
T(n, 1) = A000217(n).
T(n, 2) = A000096(n-1).
T(n, 3) = A055998(n-2).
T(2*n, n) = A134479(n). (End)

Extensions

Definition corrected by Michael Somos, Nov 05 2011

A134480 A134478 * A000012.

Original entry on oeis.org

1, 3, 2, 9, 7, 4, 18, 15, 11, 6, 30, 26, 21, 15, 8, 45, 40, 34, 27, 19, 10, 63, 57, 50, 42, 33, 23, 12, 84, 77, 69, 60, 50, 39, 27, 14, 108, 100, 91, 81, 70, 58, 45, 31, 16, 135, 126, 116, 105, 93, 80, 66, 51, 35, 18
Offset: 0

Views

Author

Gary W. Adamson, Oct 27 2007

Keywords

Comments

Row sums = A134481: (1, 5, 20, 50, 100, 175, ...).
Left border = A134479.

Examples

			First few rows of the triangle:
   1;
   3,  2;
   9,  7,  4;
  18, 15, 11,  6;
  30, 26, 21, 15,  8;
  45, 40, 34, 27, 19, 10;
  63, 57, 50, 42, 33, 23, 12;
  ...

Crossrefs

Cf. A134478, A134481, A134479.

Formula

A134478 * A000012 as infinite lower triangular matrices. Triangle read by rows, partial sums of A134478 terms starting from the right.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.