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A131979
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A graph substitution group based on an heptagon ( Church Music 7 tones; white piano keys) and a pentagon ( flats and sharps: black piano keys): the polygons are tied together with 5 connections beside chord connections 12 X 12 matrix substitution with polynomial: 8 + 4 x - 4 x^2 + 9 x^3 - 141 x^4 + 196 x^5 + 35 x^6 - 259x^7 + 265 x^8 - 156 x^9 + 58 x^10 - 12 x^11 + x^12.
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1
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1, 3, 5, 7, 8, 2, 3, 5, 7, 9, 2, 4, 5, 7, 10, 2, 4, 6, 7, 12, 1, 8, 9, 12, 1, 2, 4, 6, 2, 3, 5, 7, 9, 2, 4, 5, 7, 10, 2, 4, 6, 7, 12, 3, 8, 9, 12, 1, 2, 4, 6, 1, 3, 4, 6, 2, 4, 5, 7, 10, 2, 4, 6, 7, 12, 5, 9, 10, 11, 1, 2, 4, 6, 1, 3, 4, 6, 1, 3, 5, 6, 11, 2, 4, 6, 7, 12, 7, 10, 11, 12, 1, 3, 5, 7, 8, 1, 8, 9
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OFFSET
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1,2
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COMMENTS
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As D_6 is two like interconnected hexagons in an hexagonal prism, this figure is an unexpected asymmetry break to that: {6,6}->{7,5} This sequence has the virtue of tying music theory to both graph theory and and a spacial model in group theory. The sequence gives a type of mathematical "model" for 12 tone music theory. It is interesting to note that: binomial[12,8]=495 Dimension of E_8*E_8=496
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LINKS
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Table of n, a(n) for n=1..99.
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FORMULA
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12 Substitutions of the form: 1->{1, 3, 5, 7, 8}; 2->{1, 2, 4, 6}; 3->{2, 3, 5, 7, 9}; 4->{1, 3, 4, 6}; 5->{2, 4, 5, 7, 10}; 6->{1, 3, 5, 6, 11}; 7->{2,4, 6, 7, 12}; 8->{1, 8, 9, 12}; 9->{3, 8, 9, 12}; 10->{5,9, 10, 11}; 11->{6, 10, 11, 12}; 12->{7, 10, 11, 12};
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MATHEMATICA
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Clear[s] s[1] = {1, 3, 5, 7, 8}; s[2] = {1, 2, 4, 6}; s[3] = {2, 3, 5, 7, 9}; s[4] = {1, 3, 4, 6}; s[5] = {2, 4, 5, 7, 10}; s[6] = {1, 3, 5, 6, 11}; s[7] = {2, 4, 6, 7, 12}; s[8] = {1, 8, 9, 12}; s[9] = {3, 8, 9, 12}; s[10] = {5, 9, 10, 11}; s[11] = {6, 10, 11, 12}; s[12] = {7, 10, 11, 12}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]]; aa = p[4]
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CROSSREFS
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Cf. A131213.
Sequence in context: A201042 A142340 A185168 * A101496 A218490 A161696
Adjacent sequences: A131976 A131977 A131978 * A131980 A131981 A131982
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KEYWORD
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nonn,uned,obsc
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AUTHOR
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Roger L. Bagula, Oct 07 2007
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EXTENSIONS
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This is very unclear. Which numbers refer to vertices of the pentagon and which are the vertices of the 7-gon? Once this is straightened out, the entry needs to be edited in the same way that I edited A131213. - N. J. A. Sloane, Jan 25 2012
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STATUS
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approved
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