A389472 G.f. satisfies A(x) = A(x^2 + x^3)/x^2 - 1.

1, 1, 2, 3, 6, 9, 14, 24, 42, 72, 114, 179, 284, 453, 736, 1213, 2016, 3362, 5606, 9318, 15378, 25121, 40614, 65216, 104422, 167118, 267674, 429297, 689478, 1109149, 1788032, 2889897, 4683664, 7610640, 12394998, 20225081, 33050490, 54068421, 88519060, 144984819, 237508938

Offset

1,3

Comments

The radius of convergence r of the g.f. A(x) is r = (sqrt(5) - 1)/2.

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Formula

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

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

(2) A( (1 + R(x))*R(x)^2 ) = (1 + x)*R(x)^2 where A(R(x)) = x.

(3) A(x) = A( x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + x^9 )/(x^3 + x^4)^2 - (1 + x^2)/x^2.

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 24*x^8 + 42*x^9 + 72*x^10 + 114*x^11 + 179*x^12 + ...
where A(x) = A(x^2 + x^3)/x^2 - 1.
RELATED SERIES.
A(x^2 + x^3) = x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 6*x^7 + 9*x^8 + 14*x^9 + 24*x^10 + 42*x^11 + 72*x^12 + ...
SPECIFIC VALUES.
A(t) = 4 at t = 0.54630732655250591963600229046300146964975371294458448305... where A(t^2 + t^3) = 5*t^2.
A(t) = 3 at t = 0.52722270818896035199514400539574987808215313226299224871... where A(t^2 + t^3) = 4*t^2.
A(t) = 2 at t = 0.49272028663490084801295926284595098722341201634420367857... where A(t^2 + t^3) = 3*t^2.
A(t) = 1 at t = 0.40985335046493014523232307699710548484464368402693550468... where A(t^2 + t^3) = 2*t^2.
A(t) = 1/2 at t = 0.301419584866199287965112423742740765371185087430484838... where A(t^2 + t^3) = 3*t^2/2.
A(1/2) = 2.159559553026250611213337771695330430974181015965001... = 4*A(3/8) - 1.
A(2/5) = 0.933573329276106618842623562850786079989669037646701... = 25*A(28/125)/4 - 1.
A(1/3) = 0.607372832553248176009540249167895014820360005807581... = 9*A(4/27) - 1.
A(1/4) = 0.365019912926048482191422544055660211914898834378703... = 16*A(5/64) - 1.
A(1/5) = 0.263568927606249662991971930247996469975567607911732... = 25*A(6/125) - 1.

Prog

(PARI) {a(n) = my(A=x+x^2); for(i=0, n, A = subst(A, x, x^2 + x^3 + x^3*O(x^n))/x^2 - 1); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", "))

Crossrefs

Cf. A350432.

Sequence in context: A147007 A146937 A032231 * A200273 A303664 A190276

Adjacent sequences: A389469 A389470 A389471 * A389473 A389474 A389475

Keyword

nonn

Author

Paul D. Hanna, Oct 15 2025

Status

approved