A060529 -id:A060529 - cp's OEIS frontend

A006008 Number of inequivalent ways to color vertices of a regular tetrahedron using <= n colors.

0, 1, 5, 15, 36, 75, 141, 245, 400, 621, 925, 1331, 1860, 2535, 3381, 4425, 5696, 7225, 9045, 11191, 13700, 16611, 19965, 23805, 28176, 33125, 38701, 44955, 51940, 59711, 68325, 77841, 88320, 99825, 112421, 126175, 141156, 157435, 175085, 194181, 214800, 237021, 260925, 286595, 314116, 343575

Offset: 0

N. J. A. Sloane, Clint. C. Williams (Clintwill(AT)aol.com)

Here "inequivalent" refers to the rotation group of the tetrahedron, of order 12, with cycle index (x1^4 + 8*x1*x3 + 3*x2^2)/12, which is also the alternating group A_4.
Equivalently, number of distinct tetrahedra that can be obtained by painting its faces using at most n colors. - Lekraj Beedassy, Dec 29 2007
Equals row sums of triangle A144680. - Gary W. Adamson, Sep 19 2008

J.-P. Delahaye, 'Le miraculeux "lemme de Burnside"', 'Le coloriage du tetraedre' pp 147 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
Martin Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Gugisch, A. Kerber, R. Laue, M. Meringer and C. Ruecker, Kombinatorische Chemie, eine Herausforderung für Mathematik und Infomatik, Spektrum 1/02, 64-67, 2002.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Eric Weisstein's World of Mathematics, Polyhedron Coloring.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A006550, A060529.
Cf. A144680. - Gary W. Adamson, Sep 19 2008
Cf. A000332(n+3) (unoriented), A000332 (chiral), A006003 (achiral).
Row 3 of A324999.

[(n^4 + 11*n^2 )/12: n in [0..40]]; // Vincenzo Librandi, Aug 12 2011

A006008 := n->1/12*n^2*(n^2+11);
A006008:=-z*(z+1)*(z**2-z+1)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Table[(n^4+11n^2)/12,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,5,15,36},40] (* Harvey P. Dale, Aug 11 2011 *)

apply( {A006008(n)=(n^4+11*n^2)/12}, [0..50]) \\ M. F. Hasler, Jan 26 2020

a(n) = (n^4 + 11*n^2)/12. (Replace all x_i's in the cycle index with n.)
Binomial transform of [1, 4, 6, 5, 2, 0, 0, 0, ...]. - Gary W. Adamson, Apr 23 2008
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0)=0, a(1)=1, a(2)=5, a(3)=15, a(4)=36. - Harvey P. Dale, Aug 11 2011
a(n) = C(n,1) + 3C(n,2) + 3C(n,3) + 2C(n,4). Each term indicates the number of tetrahedra with exactly 1, 2, 3, or 4 colors. - Robert A. Russell, Dec 03 2014
a(n) = binomial(n+3,4) + binomial(n,4). - Collin Berman, Jan 26 2016
a(n) = A000332(n+3) + A000332(n) = 2*A000332(n+3) - A006003(n) = 2*A000332(n) + A006003(n).
a(n) = A324999(3,n).
E.g.f.: (1/12)*exp(x)*x*(12 + 18*x + 6*x^2 + x^3). - Stefano Spezia, Jan 26 2020
Sum_{n>=1} 1/a(n) = (6 + 22*Pi^2 - 6*sqrt(11)*Pi*coth(sqrt(11)*Pi))/121. - Amiram Eldar, Aug 23 2022

A060233 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to six complementary pairs of ratios which generate simple musical tones (scale steps): 8/7 and 7/4, 6/5 and 5/3, 16/13 and 13/8, 5/4 and 8/5, 4/3 and 3/2 and 11/8 and 16/11.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 15, 19, 22, 24, 26, 31, 37, 41, 46, 50, 53, 72, 84, 87, 130, 137, 140, 171, 183, 217, 224, 270, 494, 764, 851, 1038, 1282, 1308, 1578, 2190, 2684, 3395, 4843, 5004, 5585, 6079, 8269, 14124, 14618, 17302, 20203, 22887, 31737

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 14 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to over 31737. The numerical value of each term represents a musical scale based on an equal division of the octave. 19, for example, signifies the scale which is formed by dividing the octave into 19 equal parts.

			6 = 4 + the previous term 2. Again, 48545 = 46625 + the previous terms (1578 + 270 + 72).

A054540, A060525, A060526, A060527, A060528, A060529, A001149, A000045.

Recurrence: The next term equals the current term plus one or more of the previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z), etc.

A061918 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 5/4 and 8/5 which generate two complementary tones of musical harmony, the Major 3rd (5/4) and the Minor 6th (8/5).

Original entry on oeis.org

1, 2, 3, 16, 19, 22, 25, 28, 59, 87, 146, 351, 497, 643, 2718, 3361, 4004, 8651, 12655, 21306, 55267, 76573, 97879, 489395, 1055363, 1153242, 1251121, 1349000, 1446879, 1544758, 1642637, 1740516, 1838395, 1936274, 5808822, 7647217

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 7647217. The numerical value of each term represents a musical scale based on an equal division of the octave. 19, for example, signifies the scale which is formed by dividing the octave into 19 equal parts. Among the terms listed, the self-accumulating nature (recurrence) in this sequence breaks down down five times, between the 3rd and 4th terms, between the 14th and 15th terms, between the 20th and 21st terms, between the 23rd and 24th terms and between the 24th and 25th terms. In later sequences, this pair of target ratios will appear in combination with other pairs of target ratios, resulting in new, different (and often recurrent), composite sequences. The examples of proper recurrence which do occur in this sequence are of the same type as is seen in sequences A054540, A060526, A060527, A060233.

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061416.

A061416 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 11/8 and 16/11 which generate two complementary musical tones.

Original entry on oeis.org

1, 2, 6, 7, 9, 11, 13, 24, 37, 505, 542, 579, 616, 653, 690, 727, 764, 801, 838, 875, 912, 949, 986, 1935, 2921, 4856, 11647, 16503, 148527, 181533, 214539, 219395, 235898, 252401, 268904, 285407, 301910, 318413, 334916, 351419, 367922, 384425

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 02 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 384425. The numerical value of each term represents a musical scale based on an equal division of the octave. 24, for example, signifies the scale of quartertones which is formed by dividing the octave into 24 equal parts. The recurrence in this sequence breaks down three times, between the 2nd and 3rd terms, between the 9th and 10th terms and between the 28th and 29th terms, but the sequence is of interest because shows the terms generated when this pair of target ratios stands alone. Later, in other sequences, this pair of target ratios will appear in combination with other pairs of target ratios, resulting in new, different, composite sequences. The examples of proper recurrence which do occur in this sequence are of the same type as is seen in sequences A054540, A060526, A060527, A060529 and A060233.

A054540, A060526, A060527, A060529, A060233, A001149 and A000045.

A061919 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 6/5 and 5/3 which generate two complementary musical harmonies, the Minor 3rd (6/5) and the Major 6th (5/3).

Original entry on oeis.org

1, 2, 3, 4, 11, 15, 19, 95, 232, 251, 270, 289, 308, 327, 346, 365, 384, 403, 422, 1285, 1707, 2129, 3836, 19180, 28981, 32817, 36653, 40489, 44325, 48161, 51997, 259985, 3591629, 3643626, 3695623, 3747620, 3799617, 3851614, 3903611, 3955608

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 3955608. The numerical value of each term represents a musical scale based on an equal division of the octave. 19, for example, signifies the scale formed by dividing the octave into 19 equal parts. Within the terms shown, the self-accumulating nature of this sequence breaks down five times, between the 4th and 5th terms, between the 7th and 8th terms, between the 8th and 9th terms, between the 23rd and 24th terms and between the 32nd and 33rd terms, but the sequence is of interest because it shows the terms generated when this pair of target ratios stands alone.

Later, in other sequences, this pair of target ratios will appear in combination with other pairs of target ratios, resulting in new, different (and often recurrent), composite sequences. The examples of proper recurrence which do occur in this sequence are of the same type which is seen in sequences A054540, A060526, A060527 and A060233.

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061416, A061918.

A061920 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the 7 pairs of complementary target ratios needed to express the 12 unsymmetrical steps of the untempered (Just Intonation) scale known as the Duodene: 3/2 and 4/3, 5/4 and 8/5, 6/5 and 5/3, 9/8 and 16/9, 10/9 and 9/5, 16/15 and 15/8 and 45/32 and 64/45.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 19, 22, 31, 34, 41, 53, 118, 171, 289, 323, 376, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, 12888, 16572, 20868, 25164, 44249, 48545, 52841, 57137, 69413, 73709, 78005, 151714, 229719, 307724, 537443, 714321

Offset: 1

Mark William Rankin (MarkRankin95511(AT)yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 714321. The numerical value of each term represents a musical scale based on an equal division of the octave. The term 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

			118 = 53 + [34 + 31]; Again, 69413 = 57137 + [4296 + 3684 + 2513 + 1783].

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061918, A061919.

Recurrence Rule: The next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z), etc.

A061921 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the 11 pairs of target ratios needed to express the 22 steps of the theoretical Hindu scale known as the 22 Srutis: 45/32 and 64/45, 27/20 and 40/27, 4/3 and 3/2, 81/64 and 128/81, 5/4 and 8/5, 6/5 and 5/3, 32/27 and 27/16, 9/8 and 16/9, 10/9 and 9/5, 16/15 and 15/8, 256/243 and 243/128.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 30, 32, 34, 37, 39, 40, 41, 53, 118, 171, 323, 335, 376, 388, 441, 494, 506, 559, 612, 1171, 1783, 2513, 3072, 3125, 3684, 4296, 12276, 16572, 20868, 40565, 44861, 48545, 52841, 57137, 61433, 69413, 73709

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 73709. The numerical value of each term represents a musical scale based on an equal division of the octave. The term 32, for example, signifies the scale which is formed by dividing the octave into 32 equal parts.

			118 = 53 + [34 + 31]; Again, 229719 = 78005 + [73709 + 69413 + 4296 + 3684 + 612].

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061416, A061918, A061919, A061920.

Recurrence rule: The next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z), etc.

cp's OEIS Frontend

A006008 Number of inequivalent ways to color vertices of a regular tetrahedron using <= n colors.

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A061918 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 5/4 and 8/5 which generate two complementary tones of musical harmony, the Major 3rd (5/4) and the Minor 6th (8/5).

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A061416 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 11/8 and 16/11 which generate two complementary musical tones.

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A061919 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 6/5 and 5/3 which generate two complementary musical harmonies, the Minor 3rd (6/5) and the Major 6th (5/3).

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