A211990 Triangle read by rows: T(n,k) = total number of regions in the last k shells of n.

1, 1, 2, 1, 2, 3, 2, 3, 4, 5, 2, 4, 5, 6, 7, 4, 6, 8, 9, 10, 11, 4, 8, 10, 12, 13, 14, 15, 7, 11, 15, 17, 19, 20, 21, 22, 8, 15, 19, 23, 25, 27, 28, 29, 30, 12, 20, 27, 31, 35, 37, 39, 40, 41, 42, 14, 26, 34, 41, 45, 49, 51, 53, 54, 55, 56, 21, 35, 47

Offset: 1

Author

Omar E. Pol, Apr 27 2012

Keywords

• nonn
• tabl

Comments

For the definition of "region of n" see A206437. For the definition of "last section of n" see A135010.

Apparently differs from A027300 at the right border.

Examples

			For n = 5 and k = 2 we have that the 4th shell of 5 contains two regions: [2] and [4,2,1,1,1]. Then we can see that the 5th shell of 5 contains two regions: [3] and [5,2,1,1,1,1,1]. Therefore the total number of regions in the last two shells of 5 is T(5,2) = 2+2 = 4 (see illustration in the link section).
Triangle begins:
1;
1,   2;
1,   2,  3;
2,   3,  4,  5;
2,   4,  5,  6,  7;
4,   6,  8,  9, 10, 11;
4,   8, 10, 12, 13, 14, 15;
7,  11, 15, 17, 19, 20, 21, 22;
8,  15, 19, 23, 25, 27, 28, 29, 30;
12, 20, 27, 31, 35, 37, 39, 40, 41, 42;
14, 26, 34, 41, 45, 49, 51, 53, 54, 55, 56;
21, 35, 47, 55, 62, 66, 70, 72, 74, 75, 76, 77;
		

Links

• Omar E. Pol, Illustration of the seven regions of 5

Crossrefs

Mirror of triangle A211980. Column 1 is A187219. Right border gives A000041, n >= 1.

Cf. A027300, A135010, A138121, A182703, A186114, A206437, A212010, A212011.

Formula

T(n,k) = A000041(n) - A000041(n-k), if 1

T(n,k) = A000041(n), if k = n.

T(n,k) = Sum_{j=1..k} A187219(n-j+1,1).

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