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 1-4 of 4 results.

A131414 A130302 + A130303 - A000012.

Original entry on oeis.org

1, 5, 1, 10, 3, 1, 16, 5, 3, 1, 23, 7, 5, 3, 1, 31, 9, 7, 5, 3, 1, 40, 11, 9, 7, 5, 3, 1, 50, 13, 11, 9, 7, 5, 3, 1, 61, 15, 13, 11, 9, 7, 5, 3, 1, 73, 17, 15, 13, 11, 9, 7, 5, 3, 1
Offset: 1

Views

Author

Gary W. Adamson, Jul 08 2007

Keywords

Comments

Left column = A052905: (1, 5, 10, 16, 23, 31,...). Row sums = A095794: (1, 6, 14, 25, 39,...).

Examples

			First few rows of the triangle are:
1;
5, 1;
10, 3, 1;
16, 5, 3, 1;
23, 7, 5, 3, 1;
31, 9, 7, 5, 3, 1;
...

Crossrefs

Cf. A130302, A130303, A052905, A095794.

Formula

A130302 + A130303 - A000012 as infinite lower triangular matrices.

A130302 A000012 * A130296.

Original entry on oeis.org

1, 3, 1, 6, 2, 1, 10, 3, 2, 1, 15, 4, 3, 2, 1, 21, 5, 4, 3, 2, 1, 28, 6, 5, 4, 3, 2, 1, 36, 7, 6, 5, 4, 3, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, May 20 2007

Keywords

Comments

Row sums = n^2. A130303 = A130296 * A000012.

Examples

			First few rows of the triangle are:
1;
3, 1;
6, 2, 1;
10, 3, 2, 1;
15, 4, 3, 2, 1;
21, 5, 4, 3, 2, 1;
...

Crossrefs

Cf. A130296, A130303.

Formula

A000012 * A130296 as infinite lower triangular matrices. Triangular series as the left border; (1,2,3...) in all other columns.

A158860 Triangle T(n,k)= ( 1 +T(n-1,k)*T(n,k-1) ) / T(n-1,k-1) initialized by T(n,0)=3n-2, T(n,k)=0 if k>=n, read by rows 0<=k

Original entry on oeis.org

1, 4, 1, 7, 2, 1, 10, 3, 2, 1, 13, 4, 3, 2, 1, 16, 5, 4, 3, 2, 1, 19, 6, 5, 4, 3, 2, 1, 22, 7, 6, 5, 4, 3, 2, 1, 25, 8, 7, 6, 5, 4, 3, 2, 1, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1
Offset: 1

Views

Author

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

Keywords

Comments

Row sums are in A052905.

Examples

			1;
4, 1;
7, 2, 1;
10, 3, 2, 1;
13, 4, 3, 2, 1;
16, 5, 4, 3, 2, 1;
19, 6, 5, 4, 3, 2, 1;
22, 7, 6, 5, 4, 3, 2, 1;
25, 8, 7, 6, 5, 4, 3, 2, 1;
28, 9, 8, 7, 6, 5, 4, 3, 2, 1;

References

  • H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162.

Crossrefs

Cf. A130303

Programs

  • Maple
    A158860 := proc(n,k)
    option remember;
    if k = 0 then
        3*n-2 ;
    elif k >= n then
        0 ;
    else
        (1+procname(n-1,k)*procname(n,k-1))/procname(n-1,k-1) ;
    end if;
end proc: # R. J. Mathar, Jul 11 2012
  • Mathematica
    Clear[e, n, k];
e[n_, 0] := 3*n - 2;
e[n_, k_] := 0 /; k >= n;
e[n_, k_] := (e[n - 1, k]*e[n, k - 1] + 1)/e[n - 1, k - 1];
Table[Table[e[n, k], {k, 0, n - 1}], {n, 1, 10}];
Flatten[%]

Formula

T(n,k) = n-k, k>=1. - R. J. Mathar, Jul 11 2012

A158858 Triangle T(n,k) =3^(k-1)*e(n,k) read by rows, where e(n,k)= (e(n - 1, k)*e(n, k - 1) + 1)/e(n - 1, k - 1).

Original entry on oeis.org

3, 5, 1, 7, 2, 27, 9, 3, 54, 9, 11, 4, 81, 18, 243, 13, 5, 108, 27, 486, 81, 15, 6, 135, 36, 729, 162, 2187, 17, 7, 162, 45, 972, 243, 4374, 729, 19, 8, 189, 54, 1215, 324, 6561, 1458, 19683, 21, 9, 216, 63, 1458, 405, 8748, 2187, 39366, 6561
Offset: 1

Views

Author

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

Keywords

Examples

			{3},
{5, 1},
{7, 2, 27},
{9, 3, 54, 9},
{11, 4, 81, 18, 243},
{13, 5, 108, 27, 486, 81},
{15, 6, 135, 36, 729, 162, 2187},
{17, 7, 162, 45, 972, 243, 4374, 729},
{19, 8, 189, 54, 1215, 324, 6561, 1458, 19683},
{21, 9, 216, 63, 1458, 405, 8748, 2187, 39366, 6561}

References

  • H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162.

Crossrefs

A130303

Programs

  • Mathematica
    Clear[e, n, k];
e[n_, 0] := 2*n + 1;
e[n_, k_] := 0 /; k >= n;
e[n_, k_] := (e[n - 1, k]*e[n, k - 1] + 1)/e[n - 1, k - 1];
Table[Table[3^k*e[n, k], {k, 0, n - 1}], {n, 1, 10}];
Flatten[%]

Formula

Row sums are (5-(-1)^n)*3^n/4-3*n/2.
T(n,k) = 3^(k-1)*e(n,k) where e(n,k)= ( 1+e(n-1,k)*e(n,k-1) )/e(n-1,k-1) and e(n,1)=2*n+1 define a triangle of fractions.

Extensions

Edited by the Associate Editors of the OEIS, Apr 22 2009
Showing 1-4 of 4 results.

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