A103922 -id:A103922 - cp's OEIS frontend

A010774 Decimal expansion of 12th root of 2.

1, 0, 5, 9, 4, 6, 3, 0, 9, 4, 3, 5, 9, 2, 9, 5, 2, 6, 4, 5, 6, 1, 8, 2, 5, 2, 9, 4, 9, 4, 6, 3, 4, 1, 7, 0, 0, 7, 7, 9, 2, 0, 4, 3, 1, 7, 4, 9, 4, 1, 8, 5, 6, 2, 8, 5, 5, 9, 2, 0, 8, 4, 3, 1, 4, 5, 8, 7, 6, 1, 6, 4, 6, 0, 6, 3, 2, 5, 5, 7, 2, 2, 3, 8, 3, 7, 6, 8, 3, 7, 6, 8, 6, 3, 9, 4, 5, 5, 6

Offset: 1

N. J. A. Sloane

This number figures in our standard 12-tone tuning of music today.

It represents the frequency ratio of a semitone in equal temperament. The equal-tempered chromatic scale divides the octave, which has a ratio of 2:1, into twelve parts of equal ratio: [2^(n/12), 2^((n+1)/12)), 0 <= n <= 11. - Daniel Forgues, Feb 28 2013

			2^(1/12) = 1.059463094359295264561825294946341700779204317494...

D. Coulter, Digital Audio Processing. Berkeley, California: Focal Press (2000) p. 30
Ian Stewart, Professor Stewart's Incredible Numbers, London, Profile Books, 2015, pp. 217-228.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Étienne Ghys, Musique..., Images des Mathématiques, CNRS, Mar 15 2020.
Wikipedia, Twelfth root of two
Index entries for sequences based on music
Index entries for algebraic numbers, degree 12

Cf. A103922, A010768.

Cf. A214832, A254531.

RealDigits[N[2^(1/12), 100]][[1]] (* Alonso del Arte, Jan 06 2011 *)

sqrtn(2,12) \\ Charles R Greathouse IV, Apr 14 2014

Equals Product_{k>=0} (1 + (-1)^k/(12*k + 11)). - Amiram Eldar, Jul 29 2020

Equals sqrt(A010768). - Hugo Pfoertner, May 31 2024

A008951 Array read by columns: number of partitions of n into parts of 2 kinds.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 1, 5, 7, 2, 7, 12, 5, 11, 19, 9, 1, 15, 30, 17, 2, 22, 45, 28, 5, 30, 67, 47, 10, 42, 97, 73, 19, 1, 56, 139, 114, 33, 2, 77, 195, 170, 57, 5, 101, 272, 253, 92, 10, 135, 373, 365, 147, 20, 176, 508, 525, 227, 35, 1, 231, 684, 738, 345, 62, 2, 297

Offset: 0

N. J. A. Sloane

Fine-Riordan array S_n(m) = a(n,m) with extra row for n=0 added.
Row n of this triangle has length floor(1/2 + sqrt(2*(n+1))), n>=0. This is sequence {A002024(n+1)} = [1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,...].
Written as a triangle this becomes A103923.
a(n,m) also gives the number of partitions of n-t(m), where t(m):=A000217(m) (triangular numbers), with two kinds of parts 1,2,..m. See the column o.g.f.'s in table A103923.
In general, column m is asymptotic to exp(Pi*sqrt(2*n/3)) * 6^(m/2) * n^((m-2)/2) / (4*sqrt(3) * m! * Pi^m), equivalently to 6^(m/2) * n^(m/2) / (m! * Pi^m) * p(n), where p(n) is the partition function A000041. - Vaclav Kotesovec, Aug 28 2015

			Array begins:
m\n 0 1 2 3 4 .5 .6 .7 .8 ...
0 | 1 1 2 3 5 .7 11 15 22 ... (A000041)
1 | . 1 2 4 7 12 19 ... (A000070)
2 | . . . 1 2 .5 .9 ... (A000097)
3 | . . . . . .. .1 ... (A000098)
[1]; [1,1]; [2,2]; [3,4,1]; [5,7,2]; [7,12,5]; [11,19,9,1]...
a(3,1) = 4 because the partitions (3), (1,2) and (1^3) have q values 1,2 and 1 which sum to 4.
a(3,1) = 4 because the exponents of part 1 in the above given partitions of 3 are 0,1,3 and they sum to 4.
a(3,1) = 4 because the partitions of 3-t(1)=2 with two kinds of part 1, say 1 and 1' and one kind of part 2 are (2),(1^2), (1'^2) and (11').

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Alois P. Heinz, Columns n = 0..500, flattened
William K. Keith, Restricted k-color partitions, arXiv preprint arXiv:1408.4089 [math.CO], 2014.
Wolfdieter Lang, First 20 rows and comments.

The first column (m=0) gives A000041(n). Columns m=1..10 are A000070 (partial sums of partition numbers), A000097, A000098, A000710, A103924-A103929.

a:= proc(n, m) option remember; `if`(n<0, 0,
      `if`(m=0, combinat[numbpart](n), a(n-m, m-1) +a(n-m, m)))
    end:
seq(seq(a(n,m), m=0..round(sqrt(2*n+2))-1), n=0..20);  # Alois P. Heinz, Nov 16 2012

a[n_, 0] := PartitionsP[n]; a[n_, m_] /; (n >= m*(m+1)/2) := a[n, m] = a[n-m, m-1] + a[n-m, m]; a[n_, m_] = 0; Flatten[ Table[ a[n, m], {n, 0, 18}, {m, 0, Floor[1/2 + Sqrt[2*(n+1)]] - 1}]](* Jean-François Alcover, May 02 2012, after recurrence formula *)
DeleteCases[Flatten@Transpose@Table[ConstantArray[0, m (m + 1)/2]~Join~Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@m], {n, 0, 21 - m (m + 1)/2}] , {m, 0, 6}], 0](* Robert Price, Jul 28 2020 *)

Riordan gives formula.
a(n, m) is the sum over partitions of n of Product_{j=1..m} k(j), where k(j) is the number of parts of size j (exponent of j in a given partition of n), if m>=1. If m=0 then a(n, 0)=p(n):=A000041(n) (number of partitions of n). O is counted as a part for n=0 and only for this n.
a(n, m) is the sum over partitions of n of binomial(q(partition), m), with q the number of distinct parts of a given partition. m>=0.
a(n, m) = a(n-m, m-1) + a(n-m, m), n >= t(m):=m*(m+1)/2 = A000217(m) (triangular numbers), otherwise 0, with input a(n, 0) = p(n):=A000041(n).

More terms from Robert G Bearden (nem636(AT)myrealbox.com), Apr 27 2004

Correction, comments and Riordan formulas from Wolfdieter Lang, Apr 28 2005

A328229 Decimal expansion of 2^(7/12).

Original entry on oeis.org

1, 4, 9, 8, 3, 0, 7, 0, 7, 6, 8, 7, 6, 6, 8, 1, 4, 9, 8, 7, 9, 9, 2, 8, 0, 7, 3, 2, 0, 2, 9, 7, 9, 5, 7, 9, 6, 3, 0, 2, 1, 5, 1, 5, 5, 3, 7, 3, 1, 7, 4, 6, 8, 7, 6, 4, 4, 1, 0, 2, 3, 8, 4, 4, 1, 0, 5, 1, 5, 7, 7, 0, 9, 4, 1, 9, 9, 1, 8, 7, 2, 0, 6, 9, 1, 0, 4, 9, 7, 9, 6, 9, 7, 8, 2, 4, 3, 5, 4, 6, 9

Offset: 1

Daniel Hoyt, Oct 08 2019

2^(7/12) is the multiplier with respect to a base frequency to produce a perfect fifth interval in an equal tempered chromatic scale.

			1.498307076876681498...

Wikipedia, Twelfth root of two
Index entries for sequences based on music

Cf. A010774, A103922, A328228.

First@ RealDigits[2^(7/12), 10, 105] (* Michael De Vlieger, Oct 25 2019 *)

sqrtn(2^7, 12) \\ Michel Marcus, Oct 08 2019

Equals A010774^7. - Michel Marcus, Oct 08 2019

a(17) ff. corrected by Georg Fischer, Apr 04 2020

A341113 Numerators of continued fraction convergents to 2^(1/12).

Original entry on oeis.org

1, 1, 17, 18, 89, 196, 1461, 1657, 3118, 7893, 18904, 140221, 579788, 720009, 2019806, 2739815, 166408706, 169148521, 673854269, 843002790, 2359859849, 19721881582, 100969267759, 120691149341, 342351566441, 463042715782, 805394282223, 1268436998005, 7147579272248

Offset: 0

Seiichi Manyama, Feb 05 2021

			Convergents to [1; 16, 1, 4, 2, 7, 1, 1, 2, 2, ...]: 1, 17/16, 18/17, 89/84, 196/185, 1461/1379, 1657/1564, 3118/2943, 7893/7450, 18904/17843, ...

Eric Weisstein's World of Mathematics, Continued Fraction

For denominators see A341114.

Cf. A001333, A010774, A103922.

Join[{1}, Numerator[Convergents[2^(1/12), 28]]] (* Amiram Eldar, Apr 28 2021 *)

a(0) = 1, a(1) = 1, a(n) = A103922(n-1) * a(n-1) + a(n-2) for n >1.

A328228 Simple continued fraction expansion of 2^(7/12).

Original entry on oeis.org

1, 2, 147, 5, 1, 3, 5, 4, 4, 1, 1, 159, 6, 1, 1, 1, 4, 1, 2, 1, 2, 3, 1, 8, 15, 47, 1, 103, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 10, 3, 1, 2, 1, 2, 4, 1, 1, 1, 9, 28, 2, 4, 2, 2, 5, 1, 3, 1, 1, 2, 1, 1, 1, 52, 6, 2, 6, 1, 5, 94, 3, 6, 26, 1, 6, 5, 1, 3, 109

Offset: 0

Daniel Hoyt, Oct 08 2019

2^(7/12) is the multiplier with respect to a base frequency to produce a perfect fifth interval in an equal tempered chromatic scale.

This constant is of interest because it is close to the just intonation perfect fifth coefficient of 1.5 (continued fraction [1, 2]). It is the closest to just intonation of the chromatic scale divisions other than the octaves (2*frequency), and unison (1*frequency). The perfect fifth is the most consonant division of the chromatic scale.

Wikipedia, Twelfth root of two
Index entries for sequences based on music

Cf. A005664, A103922, A328229.

convert(2^(7/12), confrac,100); # Robert Israel, Oct 24 2019

ContinuedFraction[2^(7/12), 82] (* Michael De Vlieger, Oct 25 2019 *)

contfrac(sqrtn(2^7, 12)) \\ Michel Marcus, Oct 09 2019

A341114 Denominators of continued fraction convergents to 2^(1/12).

Original entry on oeis.org

0, 1, 16, 17, 84, 185, 1379, 1564, 2943, 7450, 17843, 132351, 547247, 679598, 1906443, 2586041, 157068903, 159654944, 636033735, 795688679, 2227411093, 18614977423, 95302298208, 113917275631, 323136849470, 437054125101, 760190974571, 1197245099672

Offset: 0

Seiichi Manyama, Feb 05 2021

Eric Weisstein's World of Mathematics, Continued Fraction
Wikipedia, Twelfth root of two
Index entries for sequences based on music

For numerators see A341113.

Cf. A002351, A010774, A103922.

Join[{0}, Denominator[Convergents[2^(1/12), 27]]] (* Amiram Eldar, Feb 05 2021 *)

a(0) = 0, a(1) = 1, a(n) = A103922(n-1) * a(n-1) + a(n-2) for n > 1.

cp's OEIS Frontend

A010774 Decimal expansion of 12th root of 2.

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A008951 Array read by columns: number of partitions of n into parts of 2 kinds.

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A328229 Decimal expansion of 2^(7/12).

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A341113 Numerators of continued fraction convergents to 2^(1/12).

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Mathematica

Formula

A328228 Simple continued fraction expansion of 2^(7/12).

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Crossrefs

Programs

Maple

Mathematica

PARI

A341114 Denominators of continued fraction convergents to 2^(1/12).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula