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A078140
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Convolutory inverse of signed lower Wythoff sequence.
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40
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1, 3, 5, 9, 17, 30, 52, 90, 154, 262, 446, 758, 1285, 2176, 3683, 6230, 10533, 17803, 30085, 50831, 85873, 145063, 245037, 413891, 699082, 1180761, 1994293, 3368302, 5688920, 9608292, 16227841, 27407792, 46289925, 78180465, 132041227
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OFFSET
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1,2
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COMMENTS
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Suppose that r is a real number in the interval [3/2, 5/3). Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k). It appears that c(k) > 0 for all k >= 0. Indeed, it appears that C(r) is strictly increasing and that the limit L(r) of c(k+1)/c(k) as k -> oo exists. Following is a guide for selected numbers r.
** r ** C(r) L(r)
sqrt(7/3) A188135 A288238
Pi/2 A288229 A288239
sqrt(5/2) A288230 A288240
4^(1/3) A288231 A288241
(1 + sqrt(5))/2 A078140 A281112
3e/5 A288232 A288242
sqrt(8/3) A288233 A288935
-1 + sqrt(7) A288234 A289003
sqrt(e) A288235 A289005
-4/5 + sqrt(6) A288236 A289032
sqrt(11/4) A288237 A289033
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LINKS
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Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling, Another question about the golden ratio and other numbers, MathOverflow, Jan 17 2017.
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FORMULA
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a(n) = d*[w(n)*a(1)-w(n-1)*a(2)+...+d*w(2)*a(n-1)], where d=(-1)^n, with a(1)=1 and w=floor(n*tau), tau=(1+sqrt(5))/2.
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EXAMPLE
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a(5) = 17 = -[w(5)*a(1)-w(4)*a(2)+w(3)*a(3)-w(2)*a(4)] = -8*1+6*3-4*5+3*9. (a(1),a(2),...,a(n))(*)(w(1),-w(2),w(3),...,-d*w(n)) = (1,0,0,...,0), where (*) denotes convolution, w = lower Wythoff sequence, A000201.
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MATHEMATICA
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CoefficientList[Series[1/Sum[Floor[GoldenRatio*(k + 1)] (-x)^k, {k, 0, 50}],
{x, 0, 50}], x] (* Clark Kimberling, Dec 12 2016 *)
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CROSSREFS
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Cf. A000201, A077607, A281112, A279676.
Sequence in context: A298338 A018162 A077879 * A279780 A289260 A279595
Adjacent sequences: A078137 A078138 A078139 * A078141 A078142 A078143
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling, Nov 23 2002
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EXTENSIONS
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Comments added by Clark Kimberling, Jul 10 2017
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STATUS
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approved
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