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A343155
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Irregular triangle T read by rows: T(n, k) is the sum of the consecutive integers placed along the k-th turn of the spiral of the n X n matrix defined in A126224.
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0
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1, 10, 36, 9, 78, 58, 136, 164, 25, 210, 318, 138, 300, 520, 356, 49, 406, 770, 654, 250, 528, 1068, 1032, 612, 81, 666, 1414, 1490, 1086, 394, 820, 1808, 2028, 1672, 932, 121, 990, 2250, 2646, 2370, 1614, 570, 1176, 2740, 3344, 3180, 2440, 1316, 169, 1378, 3278, 4122, 4102, 3410, 2238, 778
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OFFSET
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1,2
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LINKS
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FORMULA
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T(n, k) = 2*(2*k - n - 1)*(3 + 8*k*(k - n + 4*n) + n^2*0^(n+1-2*k) with 0 < k <= ceiling(n/2).
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EXAMPLE
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The triangle T(n, k) begins:
n\k| 1 2 3 4
---+-------------------
1 | 1
2 | 10
3 | 36 9
4 | 78 58
5 | 136 164 25
6 | 210 318 138
7 | 300 520 356 49
...
For n = 1 the matrix is
1
and T(1, 1) = 1.
For n = 2 the matrix is
1, 2
4, 3
and T(2, 1) = 1 + 2 + 3 + 4 = 4*5/2 = 10.
For n = 3 the matrix is
1, 2, 3
8, 9, 4
7, 6, 5
and T(3, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 8*9/2 = 36; T(3, 2) = 9.
For n = 4 the matrix is
1, 2, 3, 4
12, 13, 14, 5
11, 16, 15, 6
10, 9, 8, 7
and T(4, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 12*13/2 = 78; T(4, 2) = 13 + 14 + 15 + 16 = (13 + 16)*4/2 = 58.
...
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MATHEMATICA
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Table[2(2k-n-1)(3+8k(k-n-1)+4n)+n^2KroneckerDelta[n, 2k-1], {n, 14}, {k, Ceiling[n/2]}]//Flatten
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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