A101468 Triangle read by rows: T(n,k)=(n+1-k)*(3*k+1).

1, 2, 4, 3, 8, 7, 4, 12, 14, 10, 5, 16, 21, 20, 13, 6, 20, 28, 30, 26, 16, 7, 24, 35, 40, 39, 32, 19, 8, 28, 42, 50, 52, 48, 38, 22, 9, 32, 49, 60, 65, 64, 57, 44, 25, 10, 36, 56, 70, 78, 80, 76, 66, 50, 28, 11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, 12, 44, 70, 90, 104, 112, 114

Offset

0,2

Comments

The triangle is generated from the product A*B

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...

...

Row sums give pentagonal pyramidal numbers A002411 T(n+0,0)= 1*n=A000027(n) T(n+0,1)= 4*n=A008586(n) T(n+1,2)= 7*n=A008589(n) T(n+2,3)=10*n=A008592(n) ...

so for example T(n+1,n-0)=6*n+2=A016933(n) T(n+1,n-1)=9*n+3=A017197(n) T(n+2,n-1)=12*n+4=A017569(n)

T(n,0)*T(n,1) = A033996(n) (8 times triangular numbers)

T(n,n)*T(n,0) = A000567(n+1) (Octagonal numbers) etc.

Links

Table of n, a(n) for n=0..72.

Example

Triangle begins:
1,
2,  4,
3,  8,  7,
4,  12, 14, 10,
5,  16, 21, 20, 13,
6,  20, 28, 30, 26, 16,
7,  24, 35, 40, 39, 32, 19,
8,  28, 42, 50, 52, 48, 38, 22,
9,  32, 49, 60, 65, 64, 57, 44, 25,
10, 36, 56, 70, 78, 80, 76, 66, 50, 28,
11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, etc.
[Bruno Berselli, Feb 10 2014]

Mathematica

t[n_, k_] := If[n < k, 0, (3*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Robert G. Wilson v, Jan 21 2005 *)

Prog

(PARI) T(n, k)=if(k>n, 0, (n-k+1)*(3*k+1)) for(i=0, 10, for(j=0, i, print1(T(i, j), ", ")); print())

Crossrefs

Cf. A095871 (product B*A), A002411.

Sequence in context: A324931 A168521 A244982 * A188866 A235486 A066194

Adjacent sequences: A101465 A101466 A101467 * A101469 A101470 A101471

Keyword

nonn,tabl

Author

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

Status

approved