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A084202
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G.f. A(x) defined by: A(x)^2 consists entirely of integer coefficients between 1 and 2 (A083952); A(x) is the unique power series solution with A(0)=1.
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17
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1, 1, 0, 1, 0, 1, -1, 2, -2, 4, -6, 10, -16, 27, -44, 75, -127, 218, -375, 650, -1130, 1974, -3460, 6086, -10736, 18993, -33685, 59882, -106683, 190446, -340611, 610243, -1095102, 1968200, -3542468, 6384518, -11521308, 20815942, -37651528, 68176596, -123574852, 224204708, -407153894
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OFFSET
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0,8
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COMMENTS
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Limit a(n)/a(n+1) -> r = -0.530852489019085 where A(r)=0.
Let A_n(x) be the power series formed from the first n terms of this sequence. Then a(0) = 1, a(n) = floor(1 - [x^n] (A_(n-1)(x))^2/2). Replacing 2 with a larger integer k generates the related sequences A084203-A084212. - Charlie Neder, Jan 16 2019
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LINKS
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MATHEMATICA
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a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Table[a[n], {n, 0, 42}]; CoefficientList[ Series[ Sqrt[ Sum[ a[i]*x^i, {i, 0, 42}]], {x, 0, 42}], x] (* Robert G. Wilson v, Nov 11 2007 *)
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PROG
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(PARI) /* Using Charlie Neder's formula */
{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = floor(1 - polcoeff( Ser(A)^2, #A-1)/2) ); A[n+1]}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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