A343155 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.
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
Offset: 1
Examples
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.
...
Programs
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Mathematica
Table[2(2k-n-1)(3+8k(k-n-1)+4n)+n^2KroneckerDelta[n,2k-1],{n,14},{k,Ceiling[n/2]}]//Flatten
Formula
T(n, k) = 2*(2*k - n - 1)*(3 + 8*k*(k - n - 1) + 4*n) + n^2*0^(n+1-2*k) with 0 < k <= ceiling(n/2).
T(n, 1) = A033585(n-1) for n > 1.