OFFSET
1,2
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
LINKS
Hartmut F. W. Hoft, Construction process for sigma(15)
Hartmut F. W. Hoft, Sigma(n) equals area of its symmetric representation
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).
EXAMPLE
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;
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Hartmut F. W. Hoft, Nov 03 2015
STATUS
approved