A103217 Hexagonal numbers triangle read by rows: T(n,k)=(n+1-k)*(2*(n+1-k)-1).

1, 6, 1, 15, 6, 1, 28, 15, 6, 1, 45, 28, 15, 6, 1, 66, 45, 28, 15, 6, 1, 91, 66, 45, 28, 15, 6, 1, 120, 91, 66, 45, 28, 15, 6, 1, 153, 120, 91, 66, 45, 28, 15, 6, 1, 190, 153, 120, 91, 66, 45, 28, 15, 6, 1, 231, 190, 153, 120, 91, 66, 45, 28, 15, 6, 1, 276, 231, 190, 153, 120, 91, 66

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

Keywords

• easy
• nonn
• tabl

Comments

The triangle is generated by the product A*B = 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...

5 1 0 0...

9 5 1 0...

13 9 5 1...

...

The only prime hexagonal pyramidal number is 7. The only semiprime hexagonal pyramidal numbers are: 22, 95, 161. All greater hexagonal pyramidal numbers A002412 have at least 3 prime factors. Note that 7337 = 11 * 23 * 29 is a palindromic 3-brilliant number and 65941 = 23 * 47 * 61 is 3-brilliant. - Jonathan Vos Post, Jan 26 2005

Examples

			Triangle begins:
  1,
  6,1,
  15,6,1,
  28,15,6,1,
  45,28,15,6,1,
  66,45,28,15,6,1,
  91,66,45,28,15,6,1,
		

Links

• Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
• Eric Weisstein's World of Mathematics, Hexagonal Number.
• Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number.

Crossrefs

Row sums give A002412 (hexagonal pyramidal numbers).

Cf. A000384, A002412.

Programs

Mathematica

T[n_, k_] := (n + 1 - k)*(2*(n + 1 - k) - 1); Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Feb 10 2005 *)

PARI

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