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A283368
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Irregular triangle read by rows: T(n,k) = number of heights for the horizontal elements of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259176.
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2
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1, 2, 3, 2, 4, 3, 5, 3, 6, 5, 4, 7, 5, 4, 8, 6, 5, 9, 7, 5, 10, 8, 7, 6, 11, 8, 7, 6, 12, 10, 9, 7, 13, 10, 9, 7, 14, 11, 9, 8, 15, 12, 11, 10, 8, 16, 13, 12, 11, 9, 17, 13, 12, 11, 9, 18, 15, 13, 12, 10, 19, 15, 13, 12, 10, 20, 16, 15, 13, 11, 21, 17, 16, 15, 14, 11
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OFFSET
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1,2
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COMMENTS
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The dot product of the n-th row of this triangle and the n-th row of triangle A259176 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value is the sum of the rectangles between the x-axis and the horizontal legs of the symmetric representation of sigma(n). This is the companion computation to A283367.
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LINKS
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FORMULA
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T(n,k) = n - sum_{i=1..k-1} f(n, 2*i) where f is defined in A237593.
A024916(n) = sum_{i=1..row(n)} (T(n,i))*S(n,i)) where S(n,i) refers to the triangle of A259176 and row(n) = floor(sqrt(8*n+1)-1)/2).
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EXAMPLE
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The first horizontal leg of the symmetric representation of sigma(15) is at y-coordinate 15 and has length 8, and row 15 has 5 entries so that T(15,1) = 15 and T(15,5) = 8.
The first 16 rows of the irregular triangle:
1
2
3 2
4 3
5 3
6 5 4
7 5 4
8 6 5
9 7 5
10 8 7 6
11 8 7 6
12 10 9 7
13 10 9 7
14 11 9 8
15 12 11 10 8
16 13 12 11 9
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MATHEMATICA
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(* function f[n, k] and its support functions are defined in A237593 *)
a283368[n_, k_] := n - Sum[f[n, 2i], {i, k-1}]
TableForm[Table[a283368[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a283368[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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