bibkey: sloane2016a001108
authors: N. J. A. Sloane; Jaroslav Krizek
year: 2016
title: "OEIS A001108, a(n)-th triangular number is a square"
doi: null
url: https://oeis.org/A001108
claim: "%N a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1. %C The squares of NSW numbers (A008843) interleaved with twice squares from A084703, where A008843(n) = A002315(n)^2 and A084703(n) = A001542(n)^2. Conjecture: Also numbers n such that sigma(n) = A000203(n) and sigma(n-th triangular number) = A074285(n) are both odd numbers. - _Jaroslav Krizek_, Aug 05 2016"
strata_touched:
  - D5/S3/Arith/KrizekTriangularSquareSigmaParity
license: citation-only
triage: anchor

OEIS A001108

The NAME of A001108 states:

a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1.

The Krizek COMMENT CONJECTURE is printed after this context sentence:

The squares of NSW numbers (A008843) interleaved with twice squares from A084703, where A008843(n) = A002315(n)^2 and A084703(n) = A001542(n)^2. Conjecture: Also numbers n such that sigma(n) = A000203(n) and sigma(n-th triangular number) = A074285(n) are both odd numbers. - Jaroslav Krizek, Aug 05 2016

The AUTHOR line is:

N. J. A. Sloane

The AUTHOR line carries no date; the year 2016 records the date of Krizek’s conjecture. The divisor-sum parity characterization is proved for positive indices n >= 1. The boundary n = 0 is excluded and disclosed because T_0 = 0 is a square while Mathlib’s sigma(0) = 0 is even. The classical sigma-parity fact is re-proved privately; its forward direction also occurs as a private lemma in the landed Lagneau module from pull request 7579.

Verified locator

• URL: https://oeis.org/A001108
• NAME (verbatim): a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1.
• COMMENT context and conjecture (verbatim): The squares of NSW numbers (A008843) interleaved with twice squares from A084703, where A008843(n) = A002315(n)^2 and A084703(n) = A001542(n)^2. Conjecture: Also numbers n such that sigma(n) = A000203(n) and sigma(n-th triangular number) = A074285(n) are both odd numbers. - Jaroslav Krizek, Aug 05 2016
• AUTHOR (verbatim): N. J. A. Sloane