A101447 Triangle read by rows: T(n,k) = (2*k+1)*(n+1-k), 0 <= k < n.

1, 2, 3, 3, 6, 5, 4, 9, 10, 7, 5, 12, 15, 14, 9, 6, 15, 20, 21, 18, 11, 7, 18, 25, 28, 27, 22, 13, 8, 21, 30, 35, 36, 33, 26, 15, 9, 24, 35, 42, 45, 44, 39, 30, 17, 10, 27, 40, 49, 54, 55, 52, 45, 34, 19, 11, 30, 45, 56, 63, 66, 65, 60, 51, 38, 21, 12, 33, 50, 63, 72, 77, 78, 75, 68, 57, 42, 23

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 19 2005

Keywords

• nonn
• tabl
• easy

Comments

The triangle is generated from the product of matrix A and matrix B, i.e., A * B where A = the infinite lower triangular matrix:

1 0 0 0 0 ...

1 1 0 0 0 ...

1 1 1 0 0 ...

1 1 1 1 0 ...

1 1 1 1 1 ...

... and B = the infinite lower triangular matrix:

1 0 0 0 0 ...

1 3 0 0 0 ...

1 3 5 0 0 ...

1 3 5 7 0 ...

1 3 5 7 9 ...

...

Row sums give the square pyramidal numbers A000330.

T(n+0,0)=1*n=A000027(n+1); T(n+1,1)=3*n=A008585(n); T(n+2,2)=5*n=A008587(n); T(n+3,3)=7*n=A008589(n); etc. So T(n,0)*T(n,1)=3*n*(n+1)=A028896(n) (6 times triangular numbers). T(n,1)*T(n,2)/10=3*n*(n+1)/2=A045943(n) for n>0 T(n,2)*T(n,3)/10=7/2*n*(n+1)=A024966(n) for n>1 (7 times triangular numbers), etc.

From Gary W. Adamson, Apr 25 2010: (Start)

Consider the following array, signed as shown:

...

1, 3, 5, 7, 9, 11, ...

2, -6, 10, -14, 18, -22, ...

3, 9, 15, 21, 27, 33, ...

4, -12, 20, -28, 36, -44, ...

5, 15, 25, 35, 45, 55, ...

6, -18, 30, -42, 54, -66, ...

7, 21, 35, 49, 63, 77, ...

...

Let each term (+, -)k = (+, -) phi^(-k).

Consider the inverse terms of the Lucas series (1/1, 1/3, 1/4, 1/7, ...).

By way of example, let q = phi = 1.6180339...; then

...

1/1 = q^(-1) + q^(-3) + q^(-5) + q^(-7) + q^(-9) + ...

1/3 = q^(-2) - q^(-6) + q^(-10) - q^(-14) + q^(-18) + ...

1/4 = q^(-3) + q^(-9) + q^(-15) + q^(-21) + q^(-27) +...

1/7 = q^(-4) - q^(-12) + q^(-20) - q^(-28) + q^(-36) + ...

1/11 = q^(-5) + q^(-15) + q^(-25) + q^(-35) + q^(-45) + ...

...

Relating to the Pell series, the corresponding "Lucas"-like series is (2, 6, 14, 34, 82, 198, ...) such that herein, q = 2.414213... = (1 + sqrt(2)).

Then analogous to the previous set,

...

1/2 = q^(-1) + q^(-3) + q^(-5) + q^(-7) + ...

1/6 = q^(-2) - q^(-6) + q^(-10) - q^(-14) + q^(-18) + ...

... (End)

Examples

			From _Bruno Berselli_, Feb 10 2014: (Start)
Triangle begins:
   1;
   2,  3;
   3,  6,  5;
   4,  9, 10,  7;
   5, 12, 15, 14,  9;
   6, 15, 20, 21, 18, 11;
   7, 18, 25, 28, 27, 22, 13;
   8, 21, 30, 35, 36, 33, 26, 15;
   9, 24, 35, 42, 45, 44, 39, 30, 17;
  10, 27, 40, 49, 54, 55, 52, 45, 34, 19;
  11, 30, 45, 56, 63, 66, 65, 60, 51, 38, 21;
  etc.
(End)
		

Crossrefs

Cf. A094728 (triangle generated by B*A), A000330.

Programs

Mathematica

t[n_, k_] := If[n < k, 0, (2*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Robert G. Wilson v, Jan 20 2005 *)

PARI

T(n,k)=if(n