Cyril Zhang - cp's OEIS frontend

User: Cyril Zhang

Cyril Zhang has authored 2 sequences.

A175039 Minimum number of integer-sided squares needed to tile an n-row staircase (a figure with n unit squares in the n-th row, and the leftmost squares of each row vertically aligned).

1, 3, 3, 7, 6, 7, 7, 11, 12, 13, 12, 15, 14, 15, 15, 20, 20, 23, 22, 23, 24, 25, 24, 29, 28, 29, 28

Offset: 1

Cyril Zhang, Apr 04 2010

a(n) >= n, since the rightmost squares in each row must be covered by distinct tiles.
a(n) = n iff n = 2^k - 1.
a(n) = n+1 iff n = 2^k - 2^m - 1.
a(2*k) <= 2*a(k) + 1, a(2*k+1) <= 2*a(k) + 1 for k >= 1. - Jinyuan Wang, Jul 17 2019
a(n) <= A003817(n). - Austin Shapiro, Dec 29 2022

			See link for diagrams of tilings.

Canadian Mathematical Society, 2010 Canadian Mathematical Olympiad, Problem 1
C. Zhang, Diagrams of tilings [BROKEN LINK]

Solutions for a(n) = n: A000225. Solutions for a(n) = n+1: A030130, excluding 0.

A143800 In acoustics, using 12-tone equal temperament, the rounded number of semitones in the interval perceived when a vibrating string is divided into n congruent segments.

Original entry on oeis.org

0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 56, 57, 58, 58, 59, 59, 60, 61, 61, 62, 62, 63, 63, 63, 64, 64, 65, 65, 66, 66, 66, 67, 67, 67, 68, 68, 68, 69, 69, 69, 70, 70, 70, 71, 71, 71, 71, 72, 72, 72, 73, 73, 73, 73, 74

Offset: 1

Cyril Zhang, Sep 01 2008

In music, these are known as harmonics.
Observe that log_2(n) produces irrational numbers for all n that are not powers of 2, and that dividing a string in half produces an octave interval.
Therefore the only harmonics that are perfectly in tune (equal to an interval in 12-TET) are the octaves, which correspond to all harmonics n that are powers of 2.

			For n = 3, a(3) = round(log_2(3)*12) = round(19.0195500086539...) = 19 Therefore dividing a string in three equal parts will result in a tone approximately 19 semitones higher, or an octave and a perfect fifth.

Wikipedia, Harmonic series (music)

a:= n-> round(12*log[2](n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Nov 07 2019

a(n) = round(log_2(n)*12).

cp's OEIS Frontend

User: Cyril Zhang

A175039 Minimum number of integer-sided squares needed to tile an n-row staircase (a figure with n unit squares in the n-th row, and the leftmost squares of each row vertically aligned).

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