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A372533
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Expansion of g.f. A(x) satisfying A(x) = A(x^3) / A(x^2 - x^3 - x^4).
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1
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1, 1, 1, 4, 6, 9, 14, 33, 47, 63, 121, 203, 312, 469, 867, 1482, 1969, 3288, 6470, 9376, 13429, 24872, 42253, 61377, 97504, 175250, 277678, 413675, 700594, 1193566, 1842503, 2889154, 4901502, 8051064, 12455432, 20026636, 34079228, 54609720, 83775247, 140490784, 235393569
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OFFSET
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1,4
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = A(x^3) / A(x^2 - x^3 - x^4).
(2) A(x) = A(x^9) / ( A(x^2 - x^3 - x^4) * A(x^6 - x^9 - x^12) ).
(3) A(x) = A(x^27) / ( A(x^2 - x^3 - x^4) * A(x^6 - x^9 - x^12) * A(x^18 - x^27 - x^36) ).
(4) A(x) = A(x^(3^n)) / Product_{k=0..n-1) A(x^(2*3^k) - x^(3*3^k) - x^(4*3^k)) for n > 0.
lim_{n->oo} a(n+1)/a(n) = (sqrt(5) + 1)/2, the golden ratio (A001622).
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 33*x^8 + 47*x^9 + 63*x^10 + 121*x^11 + 203*x^12 + 312*x^13 + 469*x^14 + 867*x^15 + ...
where A(x^3) = A(x) * A(x^2 - x^3 - x^4).
RELATED SERIES.
A(x^2 - x^3 - x^4) = x^2 - x^3 - 2*x^5 - x^7 + 5*x^8 - 11*x^9 + 14*x^10 - x^11 + 18*x^12 - 26*x^13 + 13*x^14 - 58*x^15 + 53*x^16 - 160*x^17 + 360*x^18 + ...
A(x^6 - x^9 - x^12) = x^6 - x^9 - 2*x^15 - x^21 + 5*x^24 - 11*x^27 + 14*x^30 - x^33 + 18*x^36 - 26*x^39 + 13*x^42 - 58*x^45 + 53*x^48 - 160*x^51 + ...
where A(x^9) = A(x) * A(x^2 - x^3 - x^4) * A(x^6 - x^9 - x^12).
SPECIFIC VALUES.
A(sqrt(2) - 1) = 1.
A(3/5) = A(27/125) / A(9/625),
A(1/2) = A(1/8) / A(1/16),
A(1/3) = A(1/27) / A(5/81),
where
A(3/5) = 19.555114187191014691309930863886185708293476404730662191...
A(1/2) = 2.1550667641040265957227997815239238132154727134090997030...
A(1/3) = 0.5842757997708565788965210514702465577564629424946531777...
A(1/4) = 0.3534527422903162362456108551975145834593467179842488057...
A(1/5) = 0.2571938792455848507418753018381908621085405095830136782...
A(1/8) = 0.1437811945137175769699819845041308139100698704315447556...
A(1/16) = 0.066717744855340897966898686726339689305046583480519537...
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PROG
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(PARI) {a(n) = my(Ax = x); for(m=1, n, Ax = truncate(Ax);
Ax = subst(Ax, x, x^3 +x^3*O(x^m)) / subst(Ax, x, x^2 - x^3 - x^4 +x^3*O(x^m)) ); polcoeff(Ax, n)}
for(n=1, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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