A103217 - OEIS

%I #14 Mar 01 2023 05:40:20

%S 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,

%T 1,120,91,66,45,28,15,6,1,153,120,91,66,45,28,15,6,1,190,153,120,91,

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

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

%C The triangle is generated by the product A*B = B*A of the infinite lower triangular matrices A =

%C    1 0 0 0...

%C    1 1 0 0...

%C    1 1 1 0...

%C    1 1 1 1...

%C    ...

%C   and B =

%C     1 0 0 0...

%C     5 1 0 0...

%C     9 5 1 0...

%C    13 9 5 1...

%C   ...

%C 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

%H Jonathan Vos Post, <a href="https://web.archive.org/web/20200219170305/http://www.magicdragon.com:80/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalPyramidalNumber.html">Hexagonal Pyramidal Number</a>.

%e Triangle begins:

%e   1,

%e   6,1,

%e   15,6,1,

%e   28,15,6,1,

%e   45,28,15,6,1,

%e   66,45,28,15,6,1,

%e   91,66,45,28,15,6,1,

%t 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 *)

%o (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())

%Y Row sums give A002412 (hexagonal pyramidal numbers).

%Y Cf. A000384, A002412.

%K easy,nonn,tabl

%O 0,2

%A Lambert Klasen (lambert.klasen(AT)gmx.de) and _Gary W. Adamson_, Jan 25 2005