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A239970
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Least positive k such that triangular(k) + triangular(n+k) is a triangular number (A000217).
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2
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2, 5, 3, 6, 9, 4, 15, 18, 21, 5, 27, 30, 9, 36, 6, 42, 10, 48, 51, 14, 7, 60, 63, 15, 69, 72, 19, 8, 81, 26, 20, 13, 17, 24, 99, 9, 105, 14, 111, 114, 29, 120, 123, 126, 10, 132, 135, 34, 20, 144, 147, 35, 153, 45, 11, 29, 165, 33, 17, 174, 30, 44, 183, 186, 189, 12, 18, 23
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OFFSET
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0,1
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COMMENTS
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a(n) <= 3*n-3, because triangular(3*n-3) + triangular(4*n-3) = triangular(5*n-4).
In other words, smallest k>0 such that 8*k^2 + 4*(2*k + 1)*n + 4*n^2 + 8*k + 1 = m^2 has an integer solution. - Ralf Stephan, Apr 01 2014
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LINKS
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EXAMPLE
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n=5: the smallest solution k>0 to 8*k^2 + 64*k + 225 = m^2 is k=4, so a(5)=4.
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PROG
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(PARI) triangular(n) = n*(n+1)/2;
is_triangular(n) = issquare(8*n+1);
s=[]; for(n=0, 100, k=1; while(!is_triangular(triangular(k)+triangular(n+k)), k++); s=concat(s, k)); s \\ Colin Barker, Mar 31 2014
(Haskell)
a239970 n = head [k | k <- [1..],
a010054 (a000217 k + a000217 (n + k)) == 1]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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