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A120771
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Expansion of ( 1-x^3+x^4+x^5-x^8 ) / ( 1-2*x^3-x^6+x^9 ).
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1, 0, 0, 1, 1, 1, 3, 2, 1, 6, 5, 3, 14, 11, 6, 31, 25, 14, 70, 56, 31, 157, 126, 70, 353, 283, 157, 793, 636, 353, 1782, 1429, 793, 4004, 3211, 1782
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Modified wave sequence based on the heptagon.
In any row p of three terms q, r, s, (D3)^(p+1) = (D3)*q + (D2)*r + (D1)*s, where D3, D2 and D1 are the heptagon diagonals 2.246979603...(Sin 3*Pi/7 / Sin Pi/7), 1.801937735...(Sin 2Pi/7 / Sin Pi/7) and 1, respectively. Example: when p = 3, M^3 * [1,0,0] = (6,5,3). Then (D3)^4 = 6*(D3) + 5*(D2) + 3*(D1); or, 25.491566... = 13.481877... + 9.009688... + 3.
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REFERENCES
| P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (0,0,2,0,0,1,0,0,-1).
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FORMULA
| In subsets of p = 3 terms, generated from M^p * [1,0,0],(p=0,1,2...); where M = the heptagon matrix [1,1,1; 1,1,0; 1,0,0].
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EXAMPLE
| With p = 0, 1, 2...we perform M^p * [1,0,0], getting column vectors of 3 terms in each row,
1, 0, 0;
1, 1, 1;
3, 2, 1;
...
which we append to form a continuous string: (1, 0, 0, 1, 1, 1, 3, 2, 1...)
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CROSSREFS
| Cf. A077998 (trisection), A006054 (trisection), A006356 (trisection), A038196.
Sequence in context: A031252 A194761 A129674 * A115094 A165958 A113655
Adjacent sequences: A120768 A120769 A120770 * A120772 A120773 A120774
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KEYWORD
| nonn,easy,uned
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2006
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