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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.

%I A343155 #21 Dec 23 2024 15:11:18
%S A343155 1,10,36,9,78,58,136,164,25,210,318,138,300,520,356,49,406,770,654,
%T A343155 250,528,1068,1032,612,81,666,1414,1490,1086,394,820,1808,2028,1672,
%U A343155 932,121,990,2250,2646,2370,1614,570,1176,2740,3344,3180,2440,1316,169,1378,3278,4122,4102,3410,2238,778
%N 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.
%F A343155 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).
%F A343155 T(n, 1) = A033585(n-1) for n > 1.
%e A343155 The triangle T(n, k) begins:
%e A343155 n\k|   1    2    3    4
%e A343155 ---+-------------------
%e A343155 1  |   1
%e A343155 2  |  10
%e A343155 3  |  36    9
%e A343155 4  |  78   58
%e A343155 5  | 136  164   25
%e A343155 6  | 210  318  138
%e A343155 7  | 300  520  356   49
%e A343155 ...
%e A343155 For n = 1 the matrix is
%e A343155       1
%e A343155 and T(1, 1) = 1.
%e A343155 For n = 2 the matrix is
%e A343155       1, 2
%e A343155       4, 3
%e A343155 and T(2, 1) = 1 + 2 + 3 + 4 = 4*5/2 = 10.
%e A343155 For n = 3 the matrix is
%e A343155       1, 2, 3
%e A343155       8, 9, 4
%e A343155       7, 6, 5
%e A343155 and T(3, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 8*9/2 = 36; T(3, 2) = 9.
%e A343155 For n = 4 the matrix is
%e A343155       1,  2,  3,  4
%e A343155      12, 13, 14,  5
%e A343155      11, 16, 15,  6
%e A343155      10,  9,  8,  7
%e A343155 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.
%e A343155 ...
%t A343155 Table[2(2k-n-1)(3+8k(k-n-1)+4n)+n^2KroneckerDelta[n,2k-1],{n,14},{k,Ceiling[n/2]}]//Flatten
%Y A343155 Cf. A000007, A000290, A033585, A037270 (row sums), A110654, A126224.
%K A343155 nonn,tabf
%O A343155 1,2
%A A343155 _Stefano Spezia_, Apr 07 2021