A380708 G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^2.

1, 1, 2, 3, 2, 7, 16, 32, 26, 119, 314, 687, 600, 2940, 8104, 18404, 16618, 84447, 238454, 553121, 509362, 2645367, 7582080, 17828384, 16631704, 87642628, 253770136, 602394756, 567132656, 3019835984, 8808836984, 21056808924, 19960043146, 107115901135, 314214037774, 755139832949, 719601214982

Offset: 0

Author

Paul D. Hanna, Feb 08 2025

Keywords

• nonn

Comments

Conjecture: for n > 0, a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 are the fibbinary numbers.

The values of a(n)/a(n-1) tend to a period-4 sequence of reals near [0.99304..., 5.66753..., 3.05528..., 2.50250...] (the values at n = 5000..5003).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 2*x^4 + 7*x^5 + 16*x^6 + 32*x^7 + 26*x^8 + 119*x^9 + 314*x^10 + 687*x^11 + 600*x^12 + 2940*x^13 + 8104*x^14 + 18404*x^15 + ...
RELATED SERIES.
1/A(x) = 1 - x - x^2 + 3*x^4 - 5*x^5 - 8*x^6 + 47*x^8 - 95*x^9 - 165*x^10 + 1132*x^12 - 2400*x^13 - 4324*x^14 + 32079*x^16 + ...
in which the coefficients of x^(4*n+3) are zero for n >= 0.
The absolute value of the series 1/A(x) begins
abs(1/A(x)) = 1 + x + x^2 + 3*x^4 + 5*x^5 + 8*x^6 + 47*x^8 + 95*x^9 + 165*x^10 + 1132*x^12 + 2400*x^13 + 4324*x^14 + 32079*x^16 + 69823*x^17 + 128363*x^18 + 996675*x^20 + 2204161*x^21 + 4104512*x^22 + ...
the square of which starts as
abs(1/A(x))^2 = 1 + 2*x + 3*x^2 + 2*x^3 + 7*x^4 + 16*x^5 + 32*x^6 + 26*x^7 + 119*x^8 + 314*x^9 + ...
where A(x) = 1 + x*abs(1/A(x))^2.
Compare the series A(x) to the series expansion of x/A(x)^2:
x/A(x)^2 = x - 2*x^2 - x^3 + 2*x^4 + 7*x^5 - 16*x^6 - 12*x^7 + 26*x^8 + 119*x^9 - 314*x^10 - 257*x^11 + 600*x^12 + 2940*x^13 + ...
the coefficients of which agree (in absolute value) with A(x) except at x^(4*n+3) for n >= 0.
Finally, compare the coefficients in A(x) to
x/(A(x)*A(-x)) = x - 3*x^3 + 7*x^5 - 32*x^7 + 119*x^9 - 687*x^11 + 2940*x^13 - 18404*x^15 + 84447*x^17 + ...
where x/(A(x)*A(-x)) = -i*(A(i*x) - A(-i*x))/2 and i^2 = -1.
SPECIFIC VALUES.
A(t) = 2 at t = 0.36331951384016303986306639613751667776159588776520452...
A(t) = 7/4 at t = 0.326966999214379107150878447476073620003819339260938...
A(t) = 5/3 at t = 0.310176414242953172528297628193378907011618213081175...
A(t) = 3/2 at t = 0.267595268495149277553658920442049656678199654428621...
A(t) = 4/3 at t = 0.209333706309773766820377034653096490187763605618596...
A(t) = 5/4 at t = 0.172077763804168042063566486393582731980710392315592...
A(t) = 6/5 at t = 0.146244889874475639762076320821916672953391854203454...
A(1/3) = 1.78546425081776353787146582798099929973363125010159...
A(1/4) = 1.44383991745968239677184978037034999614371014280141...
A(1/5) = 1.31105443576939701517901371991612546062464905386179...
A(1/6) = 1.23904680648959409260281764010309447249414731224581...
A(1/7) = 1.19385316740530832932821328815303856705214957870222...
A(1/8) = 1.16289039855382906980050154981133704244022138742405...
A(1/9) = 1.14037889673829352431661709747406163381220554084075...
A(1/10) = 1.1232896182662527402130760033096594937509712953868...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..2100

Crossrefs

Cf. A003714.

Programs

PARI

{a(n) = my(A=1); for(i=1,n, A = 1 + x*Ser(abs(Vec(1/(A +x*O(x^n)))))^2 ); polcoef(A,n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas, in which i^2 = -1.

(1) A(x) = 1 + x*abs( 1/A(x) )^2.

(2.a) A(x)*A(-x) = 2*i*x/(A(i*x) - A(-i*x)).

(2.b) (A(x) - A(-x))/2 = x/(A(i*x)*A(-i*x)).

(3.a) [x^(4*n+1)] A(x) = [x^(4*n+1)] x/A(x)^2 for n >= 0.

(3.b) [x^(4*n+2)] A(x) = [x^(4*n+2)] -x/A(x)^2 for n >= 0.

(3.c) [x^(4*n+3)] A(x) = [x^(4*n+3)] -x/(A(x)*A(-x)) for n >= 0.

(3.d) [x^(4*n+4)] A(x) = [x^(4*n+4)] x/A(x)^2 for n >= 0.