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

Author

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

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
  ...
		

Crossrefs

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

Programs

PARI

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

Formula

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).

Extensions

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