A321441 -id:A321441 - cp's OEIS frontend

A321440 Number of partitions of n into consecutive parts, all singletons except the largest.

1, 1, 2, 3, 3, 4, 5, 4, 5, 7, 5, 6, 8, 5, 8, 10, 5, 8, 10, 7, 10, 11, 7, 8, 13, 9, 9, 14, 7, 12, 15, 6, 12, 13, 11, 15, 14, 8, 10, 19, 10, 12, 18, 8, 16, 19, 9, 12, 17, 14, 16, 16, 10, 15, 21, 15, 14, 20, 7, 16, 25, 7, 20, 21, 14, 18, 18, 14, 12, 26, 16, 17

Offset: 0

Allan C. Wechsler, Nov 09 2018

nonn

Number of representations of n as the difference of two distinct triangular numbers, plus any multiple of the order of the larger triangular number.
From Jeremy Lovejoy, Nov 10 2022: (Start)
For n > 0, a(n) is also equal to the Hurwitz class number H(8n-1).
a(n) is also equal to the number of partitions y of n having no repeated even parts and smallest part odd, counted according to the weight w(y) = (-1)^(the number of even parts)*(the number of occurrences of the smallest part). For example, the partitions of 6 having no repeated even parts and smallest part odd are [5,1], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,1,1,1,1], and [1,1,1,1,1,1], which are counted with weights 1,-2,2,-1,3,-4, and 6, giving a(6) = 1-2+2-1+3-4+6 = 5. (End)

			Here are the derivations of the terms given. Partitions are listed as strings of digits.
n = 0: (empty partition)
n = 1: 1
n = 2: 11, 2
n = 3: 111, 12, 3
n = 4: 1111, 22, 4
n = 5: 11111, 122, 23, 5
n = 6: 111111, 123, 222, 33, 6
n = 7: 1111111, 1222, 34, 7
n = 8: 11111111, 2222, 233, 44, 8
n = 9: 111111111, 12222, 1233, 234, 333, 45, 9
n = 10: 1111111111, 1234, 22222, 55, (10)

Chai Wah Wu, Table of n, a(n) for n = 0..10000
C. Alfes, K. Bringmann, and J. Lovejoy, Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 3, 385-406.
Dandan Chen and Rong Chen, Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions, arXiv:2107.04809 [math.NT], 2021.
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

See comment by Emeric Deutsch at A001227 (partitions into consecutive parts, all singletons); the partitions considered in the present sequence are a superset of those described by Deutsch.

Cf. A259825, A321441, A321443.

from sympy.utilities.iterables import partitions
def A321440(n):
    return 1 if n == 0 else sum(1 for s,p in partitions(n,size=True) if len(p)-1 == max(p)-min(p) == s-p[max(p)]) # Chai Wah Wu, Nov 09 2018

from _future_ import division
def A321440(n): # a faster program based on the characterization in the comments
    if n == 0:
        return 1
    c = 0
    for i in range(n):
        mi = i*(i+1)//2 + n
        for j in range(i+1,n+1):
            k = mi - j*(j+1)//2
            if k < 0:
                break
            if not k % j:
                c += 1
    return c # Chai Wah Wu, Nov 09 2018

From Jeremy Lovejoy, Nov 10 2022: (Start)
G.f.: 1 + Sum_{n>=0} x^(n+1)*Product_{k=1..n} (1-x^(2*k))/Product_{k=1..n+1} (1-x^(2*k-1)).
G.f.: 1 + Sum_{n>=1} (-1)^(n+1)*x^(n^2)/((1-x^(2*n-1))*Product_{k=1..n} (1-x^(2*k-1))). (End)

More terms from Chai Wah Wu, Nov 09 2018

A321443 Number of "bilaterally symmetric hexagonal partitions" of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 5, 3, 5, 4, 7, 2, 8, 4, 7, 5, 9, 2, 11, 5, 8, 5, 10, 4, 13, 6, 8, 5, 13, 4, 16, 4, 8, 8, 14, 5, 16, 5, 11, 7, 16, 2, 17, 9, 12, 8, 13, 4, 20, 8, 14, 7, 15, 5, 22, 7, 11, 8, 20, 4, 23, 8, 10, 11, 20, 7, 20, 4, 17, 9, 24, 5, 22, 7, 13, 13, 16

Offset: 0

Allan C. Wechsler, Nov 09 2018

nonn

A bilaterally symmetric hexagonal partition is one whose parts are consecutive integers, of which all have multiplicity 2 except the largest part, which may have any multiplicity (including 1).

This is a restriction of the concept of hexagonal partition presented in A321441. The nomenclature is suggested by presenting such partitions as hexagonal patches of the triangular lattice A2.

			Here are the derivations of the terms up through n = 10. Partitions are abbreviated as strings of digits.
n = 0: (empty partition)
n = 1: 1
n = 2: 11, 2
n = 3: 111, 3
n = 4: 1111, 112, 22, 4
n = 5: 11111, 5
n = 6: 111111, 1122, 222, 33, 6
n = 7: 1111111, 223, 7
n = 8: 11111111, 11222, 2222, 44, 8
n = 9: 111111111, 11223, 333, 9
n = 10: 1111111111, 112222, 22222, 2233, 334, 55, (10)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

A321441 counts hexagonal partitions in general. A321440 counts a different special kind of hexagonal partition. A116513 counts hexagonal "diagrams", of which these partitions are a sort of projection.

def A321443(n):
    if n == 0:
        return 1
    c = 0
    for i in range(n):
        mi = i*(i+1) + n
        for j in range(i+1,n+1):
            k = mi - j*j
            if k < 0:
                break
            if not k % j:
                c += 1
    return c # Chai Wah Wu, Nov 10 2018

More terms from Chai Wah Wu, Nov 10 2018

cp's OEIS Frontend

A321440 Number of partitions of n into consecutive parts, all singletons except the largest.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Python

Formula

Extensions

A321443 Number of "bilaterally symmetric hexagonal partitions" of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions