A046902 Clark's triangle: left border = 0 1 1 1..., right border = multiples of 6; other entries = sum of 2 entries above.

0, 1, 6, 1, 7, 12, 1, 8, 19, 18, 1, 9, 27, 37, 24, 1, 10, 36, 64, 61, 30, 1, 11, 46, 100, 125, 91, 36, 1, 12, 57, 146, 225, 216, 127, 42, 1, 13, 69, 203, 371, 441, 343, 169, 48, 1, 14, 82, 272, 574, 812, 784, 512, 217, 54, 1, 15, 96, 354, 846, 1386, 1596, 1296, 729, 271, 60

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• tabl
• nice

Examples

			Triangle begins as:
  0;
  1,  6;
  1,  7, 12;
  1,  8, 19,  18;
  1,  9, 27,  37,  24;
  1, 10, 36,  64,  61,  30;
  1, 11, 46, 100, 125,  91,  36;
  1, 12, 57, 146, 225, 216, 127,  42;
  1, 13, 69, 203, 371, 441, 343, 169,  48;
		

References

• J. E. Clark, Clark's triangle, Math. Student, 26 (No. 2, 1978), p. 4.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Clark's Triangle.

Crossrefs

Cf. A100206 (row sums), A185080 (central terms).

Cf. A008588, A003215.

Programs

Haskell

a046902 n k = a046902_tabl !! n !! k
a046902_row n = a046902_tabl !! n
a046902_tabl = [0] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [6])) [1,6]
-- Reinhard Zumkeller, Dec 26 2012

Magma

A046902:= func< n,k | n eq 0 select 0 else 6*Binomial(n, k-1) + Binomial(n-1, k) >;
[A046902(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 01 2024

Mathematica

Join[{0},Flatten[Table[6*Binomial[n,k-1]+Binomial[n-1,k],{n,10},{k,0,n}]]] (* Harvey P. Dale, Nov 04 2012 *)

SageMath

def A046902(n,k): return 6*binomial(n, k-1) + binomial(n-1, k) - int(n==0)
flatten([[A046902(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 01 2024

Formula

T(2*n, n) = A185080(n), for n >= 1.

Sum_{k=0..n} T(n, k) = A100206(n) (row sums).

T(n, k) = 6*binomial(n, k-1) + binomial(n-1, k), with T(0, 0) = 0. - Max Alekseyev, Nov 06 2005

From G. C. Greubel, Apr 01 2024: (Start)

T(n, n) = A008588(n).

T(n, n-1) = A003215(n-1), for n >= 1.

Sum_{k=0..n} (-1)^k*T(n, k) = 6*(-1)^n - 6*[n=0] + [n=1].

Sum_{k=0..floor(n/2)} T(n-k, k) = 7*Fibonacci(n) - 3*(1 - (-1)^n).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n), where b(n) = b(n-12) is the repeating pattern {0, 1, -5, -6, 5, 11, 0, -11, -5, 6, 5, -1}. (End)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000

More terms from Max Alekseyev, May 12 2005