This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380711 #7 Feb 19 2025 10:27:19
%S A380711 1,1,4,13,32,147,460,1436,5662,17287,60644,209377,688370,2391256,
%T A380711 8105590,27102666,92744010,312994179,1067043874,3659563265,
%U A380711 12430287670,42225015449,143808001426,487301478188,1658050374982,5637187122368,19153301908756,65251831433398,222042679730222,755372323224172
%N A380711 G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
%C A380711 Conjecture: a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) for n >= 0.
%C A380711 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.)
%C A380711 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...
%C A380711 a(n) ~ c/r^n where 1/r = 3.4018842764560748576..., and c = 0.2869732827715974104746811524073635455389390484876881563355879896659794...
%H A380711 Paul D. Hanna, <a href="/A380711/b380711.txt">Table of n, a(n) for n = 0..1030</a>
%F A380711 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380711 (1) A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
%F A380711 (2) A(x) = 1 / (1 - x*abs( 1/A(x)^3 )).
%F A380711 (3) abs(1/A(x)) = 2 - 1/A(x).
%F A380711 (4) abs(1/A(x)) = 1 + x*abs( 1/A(x)^3 ).
%F A380711 (5) abs(1/A(x)) = abs( abs(1/A(x))/A(x) ) + x*abs( 1/A(x)^3 )/A(x).
%F A380711 (6) abs( abs(1/A(x))/A(x) ) = 2 - 2*abs(1/A(x)) + abs(1/A(x))^2.
%F A380711 (7) abs( 1/A(x)^3 ) = A(x) * (2 - abs(1/A(x))) * (abs(1/A(x)) - 1)/x.
%F A380711 (8) A(x) = 1 + A(x)^2 * (abs(1/A(x)) - 1) * (2 - abs(1/A(x))).
%e A380711 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 + ...
%e A380711 where A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
%e A380711 RELATED SERIES.
%e A380711 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 + ...
%e A380711 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 + ...
%e A380711 The absolute value of the series 1/A(x) begins
%e A380711 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 + ...
%e A380711 where abs(1/A(x)) = 2 - 1/A(x).
%e A380711 The absolute value of the series 1/A(x)^3 starts as
%e A380711 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 + ...
%e A380711 where abs(1/A(x)) = 1 + x*abs( 1/A(x)^3 ).
%e A380711 The occurrence of signs in the expansion of 1/A(x)^3 has no obvious pattern.
%e A380711 SPECIFIC VALUES.
%e A380711 A(t) = 6 at t = 0.2776546403334668208899822312116577117579321589899...
%e A380711 A(t) = 5 at t = 0.27378228956266390389083139456755472304789559095846856286...
%e A380711 A(t) = 4 at t = 0.26751987468975853019031596683845328283581047906415763868...
%e A380711 A(t) = 3 at t = 0.25570653476578627566868647080655632304757429284241743094...
%e A380711 A(t) = 2 at t = 0.22541634177918528190705637551445570310188162066848813268...
%e A380711 A(t) = 3/2 at t = 0.181930310644869474243648515956090159019218115295765171...
%e A380711 A(1/4) = 2.7078534198843535187257007342533795310245294411311514375...
%e A380711 A(1/5) = 1.6527957689077139045813143038292189120779186108157811947...
%e A380711 A(1/6) = 1.4039503414912111190464124769746901176157597824012670753...
%e A380711 A(1/7) = 1.2919470482512907310654123055517832107265014355362879392...
%e A380711 A(1/8) = 1.2281933933225341024142993760196501004863261649342668152...
%e A380711 A(1/9) = 1.1870632020801295908616256565906659737605022656656501307...
%e A380711 A(1/10) = 1.158356714849802903775203606108124940003741201462273033...
%e A380711 Let B(x) = abs(1/A(x)^3) then B(x) = (1 - 1/A(x))/x with
%e A380711 B(r) = 1/r = 3.4018842764560748576093421240750088532575559256068992507649...
%e A380711 B(1/4) = 2.5228151676796334019994272154634466465512689587018470209...
%e A380711 B(1/5) = 1.9748228461981367841496533002109555899836788562358152424...
%e A380711 B(1/6) = 1.7263445702594598152254927641049269014234569252848538480...
%e A380711 B(1/7) = 1.5818212832524217283358477460759387507949300014779006395...
%e A380711 B(1/8) = 1.4863678281494130346658952345106428057380981383649536590...
%o A380711 (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)}
%o A380711 for(n=0, 40, print1(a(n), ", "))
%Y A380711 Cf. A380708, A380709, A380710.
%K A380711 nonn
%O A380711 0,3
%A A380711 _Paul D. Hanna_, Feb 18 2025