A385915 G.f. satisfies A(x) = -(A(x^3) + A(x^4)) / A(-x^2).

1, 1, 1, 2, 0, 2, 2, 2, 4, 7, 4, 11, 4, 12, 8, 19, 14, 29, 23, 47, 32, 74, 44, 110, 83, 164, 135, 276, 196, 439, 304, 663, 489, 1051, 768, 1656, 1192, 2581, 1856, 4046, 2888, 6317, 4547, 9848, 7130, 15440, 11106, 24186, 17377, 37729, 27231, 59062, 42614, 92484, 66682, 144664, 104328, 226371, 163174, 354230

Offset

1,4

Comments

a(n) ~ c*d^n, where d = 1.25088706673007476636567388431275493535326837186841972110953..., and c = 0.52020182784673907154955829274303103782236665499908622597516... if n is even, or c = 0.29984063427406787235208627225075145443391314443990234248956... if n is odd.

Radius of convergence r of g.f. A(x) satisfies A(-r^2) = 0 and A(r^8) = -A(-r^6) where r = 0.79943267989338565357086513413379878916201504254400772696808...

Links

Paul D. Hanna, Table of n, a(n) for n = 1..4100

Formula

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

(1) A(x) = -(A(x^3) + A(x^4)) / A(-x^2).

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

Example

G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 4*x^9 + 7*x^10 + 4*x^11 + 11*x^12 + 4*x^13 + 12*x^14 + 8*x^15 + 19*x^16 + 14*x^17 + ...
where A(x) = -(A(x^3) + A(x^4)) / A(-x^2).
RELATED SERIES.
A(x^3) + A(x^4) = x^3 + x^4 + x^6 + x^8 + x^9 + 3*x^12 + 2*x^16 + 2*x^18 + 2*x^21 + 4*x^24 + 4*x^27 + 2*x^28 + 7*x^30 + 2*x^32 + 4*x^33 + ...
where A(x^3) + A(x^4) = -A(x)*A(-x^2).
Let B(x) satisfy A(B(x)) = x then
B(x) = x - x^2 + x^3 - 2*x^4 + 8*x^5 - 30*x^6 + 96*x^7 - 293*x^8 + 945*x^9 - 3274*x^10 + 11679*x^11 - 41637*x^12 + 148232*x^13 - 531931*x^14 + 1932116*x^15 + ...
where -x*A(-B(x)^2) = A(B(x)^3) + A(B(x)^4).
SPECIFIC VALUES.
The radius of convergence r satisfies A(-r^2) = 0 and A(r^8) = -A(-r^6) (see values given below).
Also, -A(r^2)*A(-r^4) = A(r^6) - A(-r^6) (verify using values given below).
Pertinent values of the form A(+-r^n) are as follows.
A(r^2) = 2.25568357277545334879615953002039818218094300680461...
A(r^3) = 1.13146138072432881698300088743197078335065492774082...
A(r^4) = 0.71641352894347364220461475600253964520851864903599...
A(r^6) = 0.35711522301431469636315144464034804067241658443788...
A(r^8) = 0.20089620831732266634625988585698697217669751459458...
A(r^9) = 0.15416703344989606171813799820914068395259643759003...
A(r^12) = 0.07313926779510423315813760034344399834214660198599...
A(-r^3) = -0.21985397321719823661610779049573983213878785101569...
A(-r^4) = -0.24738019022989345323378959578383752763617413768043...
A(-r^6) = -0.20089620831732266634625988585698697217669751459458...
A(-r^8) = -0.14204971726158381650811456142497985591356688167501...
A(-r^9) = -0.11730709739793756826670096033821084565818297459544...
A(-r^12) = -0.06376735751196287607391908207491894730221727143981...
...
A(1/2) = 1.076088418495368709480408335347599544142613316488183... where A(1/2) = -(A(1/8) + A(1/16)) / A(-1/4).
A(1/3) = 0.510506935002162613788658566406685686749792809989670... where A(1/3) = -(A(1/27) + A(1/81)) / A(-1/9).
A(1/4) = 0.336602030533756099597633504337162753515731187520108... where A(1/4) = -(A(1/64) + A(1/256)) / A(-1/16).
A(1/8) = 0.143075145485757320815392125895338831436230491529773...
A(1/16) = 0.066681035393498075099284704625662933959506411882465...
A(1/27) = 0.038463353126979968975084607088546188041484601759398...
A(1/64) = 0.015873074561121087188918170893728977022238322582751...
A(1/81) = 0.0125000229473609050663549233903594151185002805142699...
A(1/256) = 0.003921568859375695990570527967100959643728275680027...
A(-1/4) = -0.194924670941580390863804372991458359617278204823944...
A(-1/9) = -0.099828959373735016072583725604101146855506666073829...
A(-1/16) = -0.058807260874534978256299853660358029251607884417895...
...

Prog

(PARI) {a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, #binary(n+1), A = -(subst(A, x, x^3) + subst(A, x, x^4))/subst(A, x, -x^2) +x*O(x^n); ); polcoef(H=A, n)}
for(n=1, 100, print1(a(n), ", "))

Crossrefs

Cf. A385908.

Sequence in context: A325589 A291308 A207944 * A063088 A101276 A339434

Adjacent sequences: A385912 A385913 A385914 * A385916 A385917 A385918

Keyword

nonn

Author

Paul D. Hanna, Aug 21 2025

Status

approved