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A136107
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Number of representations of n as the difference of two positive triangular numbers.
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16
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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
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OFFSET
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1,5
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COMMENTS
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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 0<x<=y, x-values in A351284; y-values in A351285. Also the number of times n+1 appears in A351153. - Stefano Spezia, Feb 12 2022
Equivalence with Stefano Spezia solutions: The equation 2(x-1)y-(x-3)x=2(n+1) can be rewritten (y+1-x/2)(x+1)=n; proof by solving both for y. So solutions factorize n, and since x+1 must be an integer and y+1-x/2 must be an integer, x must be even. So (x+1)|n means we are looking for odd divisors of n, which is the A001227 term of the Alekseyev formula. The correction by A010054 in the Alekseyev formula means: if n is a triangular number, the solution x=y+1 where x>y is not counted by Spezia. - R. J. Mathar, Feb 12 2022
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LINKS
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Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube
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FORMULA
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EXAMPLE
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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 [21], [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
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MATHEMATICA
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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}]
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PROG
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(PARI) a(n) = numdiv(n>>valuation(n, 2)) - ispolygonal(n, 3); \\ Michel Marcus, Jan 08 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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