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A321440 Number of partitions of n into consecutive parts, all singletons except the largest. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of representations of n as the difference of two distinct triangular numbers, plus any multiple of the order of the larger triangular number.

LINKS

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

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)

EXAMPLE

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)

PROG

(Python)

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

CROSSREFS

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. A321441, A321443.

Sequence in context: A094727 A089308 A305579 * A115729 A115728 A188553

Adjacent sequences:  A321437 A321438 A321439 * A321441 A321442 A321443

KEYWORD

nonn

AUTHOR

Allan C. Wechsler, Nov 09 2018

EXTENSIONS

More terms from Chai Wah Wu, Nov 09 2018

STATUS

approved

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Last modified August 3 14:46 EDT 2020. Contains 336198 sequences. (Running on oeis4.)