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

1,2

Links

Table of n, a(n) for n=1..56.

Formula

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).

T(n, 1) = A033585(n-1) for n > 1.

Example

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.
...

Mathematica

Table[2(2k-n-1)(3+8k(k-n-1)+4n)+n^2KroneckerDelta[n, 2k-1], {n, 14}, {k, Ceiling[n/2]}]//Flatten

Crossrefs

Cf. A000007, A000290, A033585, A037270 (row sums), A110654, A126224.

Sequence in context: A000820 A126249 A125550 * A223305 A176575 A073613

Adjacent sequences: A343152 A343153 A343154 * A343156 A343157 A343158

Keyword

nonn,tabf

Author

Stefano Spezia, Apr 07 2021

Status

approved