A380711 G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).

1, 1, 4, 13, 32, 147, 460, 1436, 5662, 17287, 60644, 209377, 688370, 2391256, 8105590, 27102666, 92744010, 312994179, 1067043874, 3659563265, 12430287670, 42225015449, 143808001426, 487301478188, 1658050374982, 5637187122368, 19153301908756, 65251831433398, 222042679730222, 755372323224172

Offset: 0

Author

Paul D. Hanna, Feb 18 2025

Keywords

• nonn

Comments

Conjecture: a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) for n >= 0.

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

Given radius of convergence r of series A(x), A(x) diverges at x = r, but abs(1/A(x)) at x = r equals 2 and abs(1/A(x)^3) at x = r equals 1/r, where r = 0.29395473764961622547646584308431424060367446826992230069820567167994719...

a(n) ~ c/r^n where 1/r = 3.4018842764560748576..., and c = 0.2869732827715974104746811524073635455389390484876881563355879896659794...

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + 460*x^6 + 1436*x^7 + 5662*x^8 + 17287*x^9 + 60644*x^10 + 209377*x^11 + 688370*x^12 + ...
where A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 34*x^3 + 106*x^4 + 462*x^5 + 1639*x^6 + 5800*x^7 + 22722*x^8 + 78754*x^9 + 289543*x^10 + ...
1/A(x) = 1 - x - 3*x^2 - 6*x^3 - x^4 - 51*x^5 - 84*x^6 - 42*x^7 - 891*x^8 - 627*x^9 - 2373*x^10 + ...
The absolute value of the series 1/A(x) begins
abs(1/A(x)) = 1 + x + 3*x^2 + 6*x^3 + x^4 + 51*x^5 + 84*x^6 + 42*x^7 + 891*x^8 + 627*x^9 + 2373*x^10 + 7848*x^11 + 15624*x^12 + ...
where abs(1/A(x)) = 2 - 1/A(x).
The absolute value of the series 1/A(x)^3 starts as
abs( 1/A(x)^3 ) = 1 + 3*x + 6*x^2 + x^3 + 51*x^4 + 84*x^5 + 42*x^6 + 891*x^7 + 627*x^8 + 2373*x^9 + 7848*x^10 + 15624*x^11 + ...
where abs(1/A(x)) = 1 + x*abs( 1/A(x)^3 ).
The occurrence of signs in the expansion of 1/A(x)^3 has no obvious pattern.
SPECIFIC VALUES.
A(t) = 6 at t = 0.2776546403334668208899822312116577117579321589899...
A(t) = 5 at t = 0.27378228956266390389083139456755472304789559095846856286...
A(t) = 4 at t = 0.26751987468975853019031596683845328283581047906415763868...
A(t) = 3 at t = 0.25570653476578627566868647080655632304757429284241743094...
A(t) = 2 at t = 0.22541634177918528190705637551445570310188162066848813268...
A(t) = 3/2 at t = 0.181930310644869474243648515956090159019218115295765171...
A(1/4) = 2.7078534198843535187257007342533795310245294411311514375...
A(1/5) = 1.6527957689077139045813143038292189120779186108157811947...
A(1/6) = 1.4039503414912111190464124769746901176157597824012670753...
A(1/7) = 1.2919470482512907310654123055517832107265014355362879392...
A(1/8) = 1.2281933933225341024142993760196501004863261649342668152...
A(1/9) = 1.1870632020801295908616256565906659737605022656656501307...
A(1/10) = 1.158356714849802903775203606108124940003741201462273033...
Let B(x) = abs(1/A(x)^3) then B(x) = (1 - 1/A(x))/x with
B(r) = 1/r = 3.4018842764560748576093421240750088532575559256068992507649...
B(1/4) = 2.5228151676796334019994272154634466465512689587018470209...
B(1/5) = 1.9748228461981367841496533002109555899836788562358152424...
B(1/6) = 1.7263445702594598152254927641049269014234569252848538480...
B(1/7) = 1.5818212832524217283358477460759387507949300014779006395...
B(1/8) = 1.4863678281494130346658952345106428057380981383649536590...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1030

Crossrefs

Cf. A380708, A380709, A380710.

Programs

PARI

{a(n) = my(A=1); for(i=1, n, A = 1 + x*A*Ser(abs(Vec(1/(A^3 +x*O(x^n))))) ); polcoef(H=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.

(1) A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).

(2) A(x) = 1 / (1 - x*abs( 1/A(x)^3 )).

(3) abs(1/A(x)) = 2 - 1/A(x).

(4) abs(1/A(x)) = 1 + x*abs( 1/A(x)^3 ).

(5) abs(1/A(x)) = abs( abs(1/A(x))/A(x) ) + x*abs( 1/A(x)^3 )/A(x).

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

(7) abs( 1/A(x)^3 ) = A(x) * (2 - abs(1/A(x))) * (abs(1/A(x)) - 1)/x.

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