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 A136107 Number of representations of n as the difference of two positive triangular numbers. 12
 0, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 1, 2, 4, 2, 1, 4, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 5, 2, 2, 2, 3, 3, 4, 2, 2, 4, 3, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 3, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 3, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 3, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS a(n) is also the number of partitions of n into consecutive parts greater than 1. - Omar E. Pol, Feb 07 2022 a(n) is the number of solutions of the equations 2(x-1)y-(x-3)x=2(n+1) for 0y is not counted by Spezia. - R. J. Mathar, Feb 12 2022 LINKS Robert G. Wilson v, Table of n, a(n) for n = 1..54000. FORMULA G.f.: Sum_{n>=1} x^((n^2+3*n)/2)/(1-x^n). - Vladeta Jovovic, May 13 2008 a(n) = A001227(n) - A010054(n). - Max Alekseyev, May 13 2009 EXAMPLE a(2) = 1 because 3 - 1 = 2, a(5) = 2 because 6 - 1 = 15 - 10 = 5, a(9) = 3 because 10 - 1 = 15 - 6 = 45 - 36 = 9, etc. For n = 21 the four partitions of 21 into consecutive parts are , [11, 10], [8, 7, 6] and [6, 5, 4, 3, 2, 1]. The last partition contains 1 as a part, hence there are only three partitions of 21 into consecutive parts whose parts are greater than 1, so a(21) = 3. - Omar E. Pol, Feb 07 2022 MATHEMATICA f[n_] := Block[{c = 0, k = 1}, While[k < n, If[ IntegerQ[ Sqrt[8 n + 4 k (k + 1) + 1]], c++ ]; k++ ]; c]; Table[f@n, {n, 105}] CROSSREFS Cf. A000217, A001227, A010054, A136108, A351153, A351284, A351285. Sequence in context: A318831 A303710 A263280 * A178691 A329313 A329312 Adjacent sequences: A136104 A136105 A136106 * A136108 A136109 A136110 KEYWORD nonn AUTHOR John W. Layman and Robert G. Wilson v, Dec 12 2007 STATUS approved

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Last modified November 28 08:51 EST 2023. Contains 367411 sequences. (Running on oeis4.)