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
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.
Programs
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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
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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