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A046902
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Clark's triangle: left border = 0 1 1 1..., right border = multiples of 6; other entries = sum of 2 entries above.
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4
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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REFERENCES
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J. E. Clark, Clark's triangle, Math. Student, 26 (No. 2, 1978), p. 4.
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LINKS
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FORMULA
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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
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)
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EXAMPLE
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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;
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MATHEMATICA
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Join[{0}, Flatten[Table[6*Binomial[n, k-1]+Binomial[n-1, k], {n, 10}, {k, 0, n}]]] (* Harvey P. Dale, Nov 04 2012 *)
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PROG
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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]
(Magma)
A046902:= func< n, k | n eq 0 select 0 else 6*Binomial(n, k-1) + Binomial(n-1, k) >;
(SageMath)
def A046902(n, k): return 6*binomial(n, k-1) + binomial(n-1, k) - int(n==0)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000
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STATUS
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approved
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