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

1,3

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.

Links

Table of n, a(n) for n=1..69.

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

Formula

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

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

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

Example

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;

Crossrefs

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

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

Sequence in context: A025798 A161064 A070086 * A162618 A036576 A171729

Adjacent sequences: A211987 A211988 A211989 * A211991 A211992 A211993

Keyword

nonn,tabl

Author

Omar E. Pol, Apr 27 2012

Status

approved