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

%I A046902 #27 Feb 16 2025 08:32:39
%S A046902 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,
%T A046902 91,36,1,12,57,146,225,216,127,42,1,13,69,203,371,441,343,169,48,1,14,
%U A046902 82,272,574,812,784,512,217,54,1,15,96,354,846,1386,1596,1296,729,271,60
%N A046902 Clark's triangle: left border = 0 1 1 1..., right border = multiples of 6; other entries = sum of 2 entries above.
%D A046902 J. E. Clark, Clark's triangle, Math. Student, 26 (No. 2, 1978), p. 4.
%H A046902 Harvey P. Dale, <a href="/A046902/b046902.txt">Table of n, a(n) for n = 0..10000</a>
%H A046902 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ClarksTriangle.html">Clark's Triangle</a>.
%F A046902 T(2*n, n) = A185080(n), for n >= 1.
%F A046902 Sum_{k=0..n} T(n, k) = A100206(n) (row sums).
%F A046902 T(n, k) = 6*binomial(n, k-1) + binomial(n-1, k), with T(0, 0) = 0. - _Max Alekseyev_, Nov 06 2005
%F A046902 From _G. C. Greubel_, Apr 01 2024: (Start)
%F A046902 T(n, n) = A008588(n).
%F A046902 T(n, n-1) = A003215(n-1), for n >= 1.
%F A046902 Sum_{k=0..n} (-1)^k*T(n, k) = 6*(-1)^n - 6*[n=0] + [n=1].
%F A046902 Sum_{k=0..floor(n/2)} T(n-k, k) = 7*Fibonacci(n) - 3*(1 - (-1)^n).
%F A046902 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)
%e A046902 Triangle begins as:
%e A046902   0;
%e A046902   1,  6;
%e A046902   1,  7, 12;
%e A046902   1,  8, 19,  18;
%e A046902   1,  9, 27,  37,  24;
%e A046902   1, 10, 36,  64,  61,  30;
%e A046902   1, 11, 46, 100, 125,  91,  36;
%e A046902   1, 12, 57, 146, 225, 216, 127,  42;
%e A046902   1, 13, 69, 203, 371, 441, 343, 169,  48;
%t A046902 Join[{0},Flatten[Table[6*Binomial[n,k-1]+Binomial[n-1,k],{n,10},{k,0,n}]]] (* _Harvey P. Dale_, Nov 04 2012 *)
%o A046902 (Haskell)
%o A046902 a046902 n k = a046902_tabl !! n !! k
%o A046902 a046902_row n = a046902_tabl !! n
%o A046902 a046902_tabl = [0] : iterate
%o A046902                (\row -> zipWith (+) ([0] ++ row) (row ++ [6])) [1,6]
%o A046902 -- _Reinhard Zumkeller_, Dec 26 2012
%o A046902 (Magma)
%o A046902 A046902:= func< n,k | n eq 0 select 0 else 6*Binomial(n, k-1) + Binomial(n-1, k) >;
%o A046902 [A046902(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 01 2024
%o A046902 (SageMath)
%o A046902 def A046902(n,k): return 6*binomial(n, k-1) + binomial(n-1, k) - int(n==0)
%o A046902 flatten([[A046902(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 01 2024
%Y A046902 Cf. A100206 (row sums), A185080 (central terms).
%Y A046902 Cf. A008588, A003215.
%K A046902 nonn,easy,tabl,nice
%O A046902 0,3
%A A046902 _N. J. A. Sloane_
%E A046902 More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000
%E A046902 More terms from _Max Alekseyev_, May 12 2005