A264116 Irregular triangle read by rows: T(n,k), n>=1, k>=1, of the alternating sum of entries 1 through k in the n-th row of A235791; the first element of column k is in row k(k+1)/2.

1, 2, 3, 2, 4, 3, 5, 3, 6, 4, 5, 7, 4, 5, 8, 5, 6, 9, 5, 7, 10, 6, 8, 7, 11, 6, 8, 7, 12, 7, 10, 9, 13, 7, 10, 9, 14, 8, 11, 9, 15, 8, 12, 10, 11, 16, 9, 13, 11, 12, 17, 9, 13, 11, 12, 18, 10, 15, 12, 13, 19, 10, 15, 12, 13, 20, 11, 16, 13, 15, 21, 11, 17, 14, 16, 15, 22, 12, 18, 14, 16, 15

Offset: 1

Author

Hartmut F. W. Hoft, Nov 03 2015

Keywords

• nonn
• tabf

Comments

The numbers in the n-th row of the triangle are the coordinates on the diagonal in the first quadrant of the polygons constructed by alternately adding and subtracting squares taken from the n-th row of A236104. The boundary from (0,n) to (n,0) of the final polygon is the Dyck path as defined in the n-th row of A237593. Therefore, using the arguments in A196020, A236104 and A071561, sigma(n) equals the area of its symmetric representation, for all n>=1.

The right border gives A240542.

For an image of the construction process of the Dyck path for sigma(15) see the image file in the Links section.

The length of the n-th row is A003056(n). - Omar E. Pol, Nov 03 2015

Examples

			The data in form of the irregular triangle T(n,k):
1;
2;
3,   2;
4,   3;
5,   3;
6,   4,  5;
7,   4,  5;
8,   5,  6;
9,   5,  7;
10,  6,  8,  7;
11,  6,  8,  7;
12,  7, 10,  9;
13,  7, 10,  9;
14,  8, 11,  9;
15,  8, 12, 10, 11;
16,  9, 13, 11, 12;
17,  9, 13, 11, 12;
18, 10, 15, 12, 13;
19, 10, 15, 12, 13;
20, 11, 16, 13, 15;
21, 11, 17, 14, 16, 15;
22, 12, 18, 14, 16, 15;
		

Links

• Hartmut F. W. Hoft, Construction process for sigma(15)
• Hartmut F. W. Hoft, Sigma(n) equals area of its symmetric representation

Crossrefs

Cf. A000217, A003056, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A071561.

Programs

Mathematica

a264116[n_, k_] := Sum[(-1)^(i+1)*Ceiling[(n+1)/i - (i+1)/2], {i,k}]
a264116[n_] := Map[a264116[n, #]&, Range[Floor[(Sqrt[8*n+1] - 1)/2]]]
Flatten[Map[a264116,Range[22]]] (* data *)

Formula

T(n, k) = Sum_{i=1..k} (-1)^(i+1) A235791(n,i), for n>=1 and 1<=k<=floor((sqrt(8n+1) - 1)/2).