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A061857
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Triangle where the k-th item at n-th row (both starting from 1) tells in how many ways we can add 2 distinct integers from 1 to n in such way that the sum is divisible by k.
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8
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0, 1, 0, 3, 1, 1, 6, 2, 2, 1, 10, 4, 4, 2, 2, 15, 6, 5, 3, 3, 2, 21, 9, 7, 5, 4, 3, 3, 28, 12, 10, 6, 6, 4, 4, 3, 36, 16, 12, 8, 8, 5, 5, 4, 4, 45, 20, 15, 10, 9, 7, 6, 5, 5, 4, 55, 25, 19, 13, 11, 9, 8, 6, 6, 5, 5, 66, 30, 22, 15, 13, 10, 10, 7, 7, 6, 6, 5, 78, 36, 26, 18, 16, 12, 12, 9, 8, 7, 7
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OFFSET
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1,4
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COMMENTS
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Since the sum of two distinct integers from 1 to n can be as much as 2n-1, this triangular table cannot show all the possible cases. For larger triangles showing all solutions, see A220691 and A220693. - Antti Karttunen, Feb 18 2013 based on Robert Israel's mail May 07 2012.
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LINKS
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_Reinhard Zumkeller_, Rows n = 1..150 of triangle, flattened
Stackexchange, Question 142323
Index entries for sequences related to subset sums modulo m
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FORMULA
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Contribution from Robert Israel, May 08 2012: (Start)
Let n+1 = b mod k with 0<=b<k, q = (n+1-b)/k. Let k = c mod 2, c = 0 or 1.
If b = 0 or 1 then a(n,k) = q^2*k/2 + q*b - 2*q - b + 1 + c*q/2.
If b >= (k+3)/2 then a(n,k) = q^2*k/2 + q*b - 2*q + b - 1 - k/2 + c*(q+1)/2.
Otherwise a(n,k) = q^2*k/2 + q*b - 2*q + c*q/2. (End)
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EXAMPLE
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E.g. the second term on the sixth row is 6 because we have 6 solutions: {1+3, 1+5, 2+4, 2+6, 3+5, 4+6} and the third term on the same row is 5 because we have solutions {1+2,1+5,2+4,3+6,4+5}
0;
1, 0;
3, 1, 1;
6, 2, 2, 1;
10, 4, 4, 2, 2;
15, 6, 5, 3, 3, 2;
21, 9, 7, 5, 4, 3, 3;
28, 12, 10, 6, 6, 4, 4, 3;
36, 16, 12, 8, 8, 5, 5, 4, 4;
45, 20, 15, 10, 9, 7, 6, 5, 5, 4;
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MAPLE
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[seq(DivSumChoose2Triangle(j), j=1..120)]; DivSumChoose2Triangle := (n) -> nops(DivSumChoose2(trinv(n-1), (n-((trinv(n-1)*(trinv(n-1)-1))/2))));
DivSumChoose2 := proc(n, k) local a, i, j; a := []; for i from 1 to (n-1) do for j from (i+1) to n do if(0 = ((i+j) mod k)) then a := [op(a), [i, j]]; fi; od; od; RETURN(a); end;
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PROG
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(Haskell)
a061857 n k = length [()| i <- [2..n], j <- [1..i-1], mod (i + j) k == 0]
a061857_row n = map (a061857 n) [1..n]
a061857_tabl = map a061857_row [1..]
-- Reinhard Zumkeller, May 08 2012
(Scheme): (define (A061857 n) (A220691bi (A002024 n) (A002260 n))) - Antti Karttunen, Feb 18 2013. Needs A220691bi from A220691.
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CROSSREFS
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This is the lower triangular region of square array A220691. See A220693 for all nonzero solutions.
The left edge (first diagonal) of the triangle: A000217, the second diagonal is given by C(((n+(n mod 2))/2), 2)+C(((n-(n mod 2))/2), 2) = A002620, the third diagonal by A058212, the fourth by A001971, the central column by A042963? trinv is given at A054425. Cf. A061865.
Sequence in context: A162315 A109446 A088441 * A067433 A133567 A184049
Adjacent sequences: A061854 A061855 A061856 * A061858 A061859 A061860
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KEYWORD
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nonn,tabl
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AUTHOR
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Antti Karttunen, May 11 2001
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EXTENSIONS
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Offset corrected by Reinhard Zumkeller, May 08 2012
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STATUS
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approved
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