A136107 Number of representations of n as the difference of two positive triangular numbers.

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

Offset: 1

Author

John W. Layman and Robert G. Wilson v, Dec 12 2007

Keywords

• nonn

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 0A351284; 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

Examples

			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
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..54000.
• 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
• Robert Dougherty-Bliss and Natalya Ter-Saakov, The Comma Sequence is Finite in Other Bases, arXiv:2408.03434 [math.NT], 2024.

Crossrefs

Cf. A000217, A001227, A010054, A136108, A351153, A351284, A351285.

Programs

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}]

PARI

a(n) = numdiv(n>>valuation(n, 2)) - ispolygonal(n, 3); \\ Michel Marcus, Jan 08 2024

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