

A264116


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


1



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
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OFFSET

1,2


COMMENTS

The numbers in the nth 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 nth row of A236104. The boundary from (0,n) to (n,0) of the final polygon is the Dyck path as defined in the nth 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 nth row is A003056(n).  Omar E. Pol, Nov 03 2015


LINKS

Table of n, a(n) for n=1..82.
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

Cf. A000217, A003056, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A071561.
Sequence in context: A318153 A157893 A331253 * A283368 A199474 A246694
Adjacent sequences: A264113 A264114 A264115 * A264117 A264118 A264119


KEYWORD

nonn,tabf


AUTHOR

Hartmut F. W. Hoft, Nov 03 2015


STATUS

approved



