A101493 Triangle read by rows: T(n,k) = (n+1)*(2*(n+1)-1) - k*(2*k-1).

1, 6, 5, 15, 14, 9, 28, 27, 22, 13, 45, 44, 39, 30, 17, 66, 65, 60, 51, 38, 21, 91, 90, 85, 76, 63, 46, 25, 120, 119, 114, 105, 92, 75, 54, 29, 153, 152, 147, 138, 125, 108, 87, 62, 33, 190, 189, 184, 175, 162, 145, 124, 99, 70, 37, 231, 230, 225, 216, 203, 186, 165, 140, 111, 78, 41

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

Keywords

• nonn
• tabl

Comments

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 5 0 0 ...

1 5 9 0 ...

1 5 9 13 ...

...

T(n+0,0) = n*(2*n-1) = A000384(n) (Hexagonal numbers)

since T(n,n) = 4*n+1 = A016813(n).

T(n,n) = 4*n + 1 = A016813(n);

T(n+1,n) = 8*n + 6 = A017137(n);

T(n+2,n) = 12*n + 3 = A017557(n);

T(n,n)*T(n,0) = (n+1)*(2*n+1)*(4*n+1) = A079588(n).

Examples

			Triangle begins:
   1;
   6,  5;
  15, 14,  9;
  28, 27, 22, 13;
  45, 44, 39, 30, 17;
  66, 65, 60, 51, 38, 21;
		

Links

• Muniru A Asiru, Rows n = 0..150 of triangle, flattened

Crossrefs

Row sums give 10-gonal pyramidal numbers: n(n+1)(8n-5)/6 = A007585(n+1).

Cf. A101492 (for product A*B), A007585, A000384.

Programs

GAP

Flat(List([0..10],n->List([0..n],k->(n+1)*(2*n+1)-k*(2*k-1)))); # Muniru A Asiru, Mar 05 2019

PARI

T(n,k)=if(k>n,0,(n+1)*(2*(n+1)-1)-k*(2*k-1))
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())