A389429 - OEIS

%I #38 Dec 30 2025 01:44:11

%S 1,1,2,4,5,8,17,27,44,82,139,228,394,678,1136,1940,3339,5666,9647,

%T 16514,28128,47871,81714,139350,237399,404887,690570,1177098,2006935,

%U 3422404,5834717,9947619,16961832,28919798,49306740,84069972,143340887,244393124,416692340

%N G.f. A(x) satisfies A(x)/(1 + A(x)) = -A(x^3)/A(-x^2).

%H Paul D. Hanna, <a href="/A389429/b389429.txt">Table of n, a(n) for n = 1..3000</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x)/(1 + A(x)) = -A(x^3) / A(-x^2).

%F (2) A(x) = A(x^3) / (-A(-x^2) - A(x^3)).

%F a(n) ~ c/r^n where r satisfies A(r^3) = -A(-r^2) with r = 0.58650914531033251827997644302682150107207378615964230152..., A(r^3) = 0.26806735363507509315369343646818659950279278045866922998..., and c = 0.38241331953495816830125769425310328874133486976840530596... - _Paul D. Hanna_, Dec 29 2025

%e G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 17*x^7 + 27*x^8 + 44*x^9 + 82*x^10 + 139*x^11 + 228*x^12 + ...

%e where A(x)/(1 + A(x)) = -A(x^3)/A(-x^2).

%e RELATED SERIES.

%e A(x)/(1 + A(x)) = -A(x^3) / A(-x^2) = x + x^3 + x^4 - x^5 + x^6 + 3*x^7 - x^8 + 4*x^9 + 5*x^10 - 3*x^11 + 6*x^12 + 2*x^13 - 5*x^14 + 11*x^15 + 7*x^16 + ...

%e SPECIFIC VALUES.

%e A(t) = 10  at t = 0.5644904095474885664719928595724120717360821454989809... where 10/11 = -A(t^3)/A(-t^2).

%e A(t) = 9   at t = 0.5620954635384541709262405101763991277002004655893228... where 9/10 = -A(t^3)/A(-t^2).

%e A(t) = 5   at t = 0.5433184616336794298242947425047846064595048972237614...

%e A(t) = 4   at t = 0.5330608569574098321256840259157131103693546718406133...

%e A(t) = 3   at t = 0.5164625265405386605340936837743512394620387649185260...

%e A(t) = 5/2 at t = 0.5036373138906460194764584383087021930518523843472935...

%e A(t) = 2   at t = 0.4851599311018346468322280837571727692719713561225392...

%e A(t) = 3/2 at t = 0.4563761784063827863023758584087985615635613692815020...

%e A(t) = 1   at t = 0.4059679106866480865701791986373883511810299895033377... where 1/2 = -A(t^3)/A(-t^2).

%e A(t) = 1/2 at t = 0.2994426175395975825714103375040380612329428044057102... where 1/3 = -A(t^3)/A(-t^2).

%e A(t) = 1/3 at t = 0.2345040778376040316032143395710174627413414769997438... where 1/4 = -A(t^3)/A(-t^2).

%e ...

%e A(1/2) = 2.38492453387320416779364322459649193508833259593892473056... where A(1/2) = A(1/8) / (-A(-1/4) - A(1/8)).

%e A(1/3) = 0.61675038380396975750634966844586085725260748647366648666... where A(1/3) = A(1/27) / (-A(-1/9) - A(1/27)).

%e A(1/4) = 0.36796354501638544328201060019644801104885558495522769025... where A(1/4) = A(1/64) / (-A(-1/16) - A(1/64)).

%e A(1/8) = 0.14570105813131515871513811641993180063743647712913489254...

%e A(1/27) = 0.0385182872170364526244520595503819637635228711677026175...

%e A(1/64) = 0.0158770132151023377664284254349090500444489716031669155...

%e A(-1/4) = -0.206793581631289953336418920701214800034553319376031943...

%e A(-1/9) = -0.100971896048967473697604856558278869175899345355052556...

%e A(-1/16) = -0.05902534524455741114671784512694200268463663231830399...

%o (PARI) {a(n) = my(A = x+x^2 + x*O(x^n), N=ceil(log(n+1)/log(2))); for(i=1,N,

%o A = subst(A,x,x^3) / (subst(-A,x,-x^2) - subst(A,x,x^3)) + x*O(x^n)); polcoef(A,n)}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A377100.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Dec 28 2025