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A264116
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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.
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1
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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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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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).
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,tabf
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AUTHOR
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Hartmut F. W. Hoft, Nov 03 2015
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STATUS
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approved
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