A095872 -id:A095872 - cp's OEIS frontend

A095871 Triangle read by rows: T(n,k)=(n+1)(3(n+1)-1)/2-k(3k-1)/2.

1, 5, 4, 12, 11, 7, 22, 21, 17, 10, 35, 34, 30, 23, 13, 51, 50, 46, 39, 29, 16, 70, 69, 65, 58, 48, 35, 19, 92, 91, 87, 80, 70, 57, 41, 22, 117, 116, 112, 105, 95, 82, 66, 47, 25, 145, 144, 140, 133, 123, 110, 94, 75, 53, 28, 176, 175, 171, 164, 154, 141, 125, 106, 84, 59

Offset: 1

Gary W. Adamson, Jun 10 2004, Jul 28 2008

Octagonal pyramidal number triangle, read by rows.
The triangle is generated from the product B*A of the infinite lower triangular matrices A =
  1 0 0 0...
  1 1 0 0...
  1 1 1 0...
  1 1 1 1...
and B =
  1 0 0 0...
  1 4 0 0...
  1 4 7 0...
  1 4 7 10...
T(n,0)=A000326(n+1)
T(n,2)=A059845(n+2)
T(n,n)=3*n+1

			Column 3 = A059845: 7, 17, 30, 46, 65...; while rightmost terms of rows are 1, 4, 7, 10...
First few rows of the triangle =
  1;
  5, 4;
  12, 11, 7;
  22, 21, 17, 10;
  35, 34, 30, 23, 13;
  51, 50, 46, 39, 29, 16;
  70, 69, 65, 58, 48, 35, 19;
  ...

Cf. A095872, A000326, A059845, A002414 (row sums)

T(n, k) = local(i); if(k>n,0,(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2)
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print()) \\ Lambert Klasen

Triangle read by rows, T(n,k) = sum {j=k..n} 3*j - 2 = A000012 * ((3*j - 2) * 0^(n-k)) * A000012; 1<=k<=n. E.g. T(5,3) = 30 = (7 + 10 + 13).

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 21 2005

A095873 Triangle T(n,k) = (2k-1)(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 4, 9, 9, 24, 25, 16, 45, 60, 49, 25, 72, 105, 112, 81, 36, 105, 160, 189, 180, 121, 49, 144, 225, 280, 297, 264, 169, 64, 189, 300, 385, 432, 429, 364, 225, 81, 240, 385, 504, 585, 616, 585, 480, 289, 100, 297, 480, 637, 756, 825

Offset: 1

Gary W. Adamson, Jun 10 2004

Matrix square of A158405.

			[1 0 0 / 1 3 0 / 1 3 5]^2 = [1 0 0 / 4 9 0 / 9 24 25]. Delete the zeros and
read by rows:
1;
4, 9;
9, 24, 25;
16,45, 60, 49;
25,72,105,112, 81;

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966.

Cf. A094728, A095871, A095872.

A095873 := proc(n,k)
        (2*k-1)*(n+k-1)*(n-k+1) ;
end proc:
seq(seq(A095873(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Oct 30 2011

Table[(2k-1)(n+k-1)(n-k+1),{n,10},{k,n}]//Flatten (* Harvey P. Dale, May 03 2018 *)

T(n,k) = (2*k-1)*A094728(n,k).

Sum_{k=1..n} T(n,k)= n*(n+1)*(3*n^2+n-1)/6 = A103220(n). - R. J. Mathar, Oct 30 2011

Definition in closed form by R. J. Mathar, Oct 30 2011

cp's OEIS Frontend

A095871 Triangle read by rows: T(n,k)=(n+1)(3(n+1)-1)/2-k(3k-1)/2.

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A095873 Triangle T(n,k) = (2k-1)(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

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cp's OEIS Frontend

A095871 Triangle read by rows: T(n,k)=(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A095873 Triangle T(n,k) = (2*k-1)*(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A095871 Triangle read by rows: T(n,k)=(n+1)(3(n+1)-1)/2-k(3k-1)/2.

A095873 Triangle T(n,k) = (2k-1)(n+k-1)*(n-k+1) read by rows, 1<=k<=n.