A307608 -id:A307608 - cp's OEIS frontend

A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			A000217(4) = 10 = 1 + 3 + 6, so a(4) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000217, A034706, A072964, A196010, A288126, A298858, A307608.

a(n) = [x^(n*(n+1)/2)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).

A360762 a(n) is the least n-gonal number that is the sum of two or more consecutive nonzero n-gonal numbers in more than one way, or -1 if no such number exists.

Original entry on oeis.org

9, 12880, 20449, 10764222, 794629045, 33205080888, 5985, 13925100

Offset: 2

Ilya Gutkovskiy, Feb 19 2023

			For n = 2: 9 = 2 + 3 + 4 = 4 + 5.
For n = 3: 12880 = 91 + ... + 903 = 300 + ... + 990.
For n = 4: 20449 = 7^2 + ... + 39^2 = 38^2 + ... + 48^2.
For n = 5: 10764222 = 1617 + ... + 115787 = 31032 + ... + 126005.
From _Michael S. Branicky_, Feb 19 2023: (Start)
n-th term and indices of n-gonal numbers summing to it:
a(2) = 9: 2..4, 4..5,
a(3) = 12880: 13..42, 24..44,
a(4) = 20449: 7..39, 38..48,
a(5) = 10764222: 33..278, 144..290,
a(6) = 794629045: 1305..1505, 5321..5334,
a(7) = 33205080888: 616..3422, 3235..4192,
a(8) = 5985: 1..18, 11..19,
a(9) = 13925100: 103..235, 220..282. (End)

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A057145, A062681, A307608, A307614, A343777, A360777.

a(6)-a(9) from Michael S. Branicky, Feb 19 2023

cp's OEIS Frontend

A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

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