The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A084202

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A084202

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.
(history; published version)

#17 by Paul D. Hanna at Thu Jan 17 17:37:01 EST 2019

STATUS

editing

approved

#16 by Paul D. Hanna at Thu Jan 17 17:36:59 EST 2019

LINKS

Paul D. Hanna, <a href="/A084202/b084202.txt">Table of n, a(n) for n = 0..1024</a>

STATUS

approved

editing

#15 by Paul D. Hanna at Thu Jan 17 12:27:54 EST 2019

STATUS

editing

approved

#14 by Paul D. Hanna at Thu Jan 17 12:26:09 EST 2019

PROG

(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]}

for(n=0, 50, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 17 2019

STATUS

proposed

editing

Discussion

Thu Jan 17

12:27

Paul D. Hanna: @ Charlie Neder: nice formula. I've added a Pari program using your formula. Thanks.

#13 by Michel Marcus at Wed Jan 16 12:35:43 EST 2019

STATUS

editing

proposed

#12 by Michel Marcus at Wed Jan 16 12:35:39 EST 2019

LINKS

N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="httphttps://arXivarxiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

STATUS

proposed

editing

#11 by Charlie Neder at Wed Jan 16 12:19:47 EST 2019

STATUS

editing

proposed

#10 by Charlie Neder at Wed Jan 16 12:19:38 EST 2019

COMMENTS

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

STATUS

approved

editing

#9 by Russ Cox at Fri Mar 30 18:36:33 EDT 2012

AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, May 19 2003

Discussion

Fri Mar 30

18:36

OEIS Server: https://oeis.org/edit/global/213

#8 by Russ Cox at Fri Mar 30 17:30:54 EDT 2012

MATHEMATICA

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 (rgwv(AT)rgwv.com), _, Nov 11 2007 *)

Discussion

Fri Mar 30

17:30

OEIS Server: https://oeis.org/edit/global/156