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A143183
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Triangle T(n,k) = 1 + (2+n)*abs(n-2*k), read by rows, for 0 <= k <= n.
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2
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1, 4, 4, 9, 1, 9, 16, 6, 6, 16, 25, 13, 1, 13, 25, 36, 22, 8, 8, 22, 36, 49, 33, 17, 1, 17, 33, 49, 64, 46, 28, 10, 10, 28, 46, 64, 81, 61, 41, 21, 1, 21, 41, 61, 81, 100, 78, 56, 34, 12, 12, 34, 56, 78, 100, 121, 97, 73, 49, 25, 1, 25, 49, 73, 97, 121
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OFFSET
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0,2
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LINKS
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FORMULA
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T(n, k) = 1 + (2+n)*abs(n-2*k), for 0 <= k <= n.
T(n, k) = T(n, n-k).
Sum_{k=0..n} T(n, k) = (n+2)*A007590(n+1) + n + 1 (row sums).
T(2*n-1, n) = A005843(n+1), n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*((n^2 + 3*n + 3) - (-1)^(n/2)*(n + 2)). (End)
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EXAMPLE
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Triangle begins as:
1;
4, 4;
9, 1, 9;
16, 6, 6, 16;
25, 13, 1, 13, 25;
36, 22, 8, 8, 22, 36;
49, 33, 17, 1, 17, 33, 49;
64, 46, 28, 10, 10, 28, 46, 64;
81, 61, 41, 21, 1, 21, 41, 61, 81;
100, 78, 56, 34, 12, 12, 34, 56, 78, 100;
121, 97, 73, 49, 25, 1, 25, 49, 73, 97, 121;
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MAPLE
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1+(2+n)*abs(n-2*m) ;
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MATHEMATICA
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T[n_, m_]:= 1 + Abs[(n-m+1)^2 - (m+1)^2];
Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten
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PROG
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(Magma)
[1+(n+2)*Abs(n-2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 23 2024
(SageMath)
flatten([[1+(n+2)*abs(n-2*k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 23 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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