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Least k such that there are n triangular numbers between triangular(k) and k^2.

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`%I #5 Dec 13 2013 17:27:03
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`%S 0,4,7,9,11,14,16,19,21,23,26,28,31,33,36,38,40,43,45,48,50,52,55,57,
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`%T 60,62,64,67,69,72,74,77,79,81,84,86,89,91,93,96,98,101,103,106,108,
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`%U 110,113,115,118,120,122,125,127,130,132,134,137,139,142,144,147
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`%N Least k such that there are n triangular numbers between triangular(k) and k^2.
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`%e Least k such that there are 2 triangular numbers between triangular(k) and k^2 is k=7: triangular(8)=36 and triangular(9)=45 are between triangular(7)=28 and 7^2=49. Thus a(2)=7.
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`%e Least k such that there are 3 triangular numbers between triangular(k) and k^2 is k=9: 55, 66, 78 are the triangular numbers between triangular(9)=45 and 9^2=81. Thus a(3)=9.
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`%t nn = 1000; tri = Table[n*(n+1)/2, {n, 0, Sqrt[2]*nn}]; sq = Range[nn]^2; t = Table[Length[Select[tri, tri[[n]] < # < sq[[n]] &]], {n, nn}]; Join[{0}, Table[Position[t, n, 1, 1][[1, 1]], {n, Max[t]}]] (* _T. D. Noe_, Dec 13 2013 *)
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`%Y Cf. A000217, A000290.
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`%K nonn
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`%O 0,2
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`%A _Alex Ratushnyak_, Dec 11 2013
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