OFFSET
1,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
G.f.: Sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..infinity).
Also (1-x)^2*Sum(x^k/(1-x-x^(k+1))^2, k=1..infinity). - Vladeta Jovovic, Apr 05 2005
a(n) = 1 + sum(k=2..[(n+3)/2], k * sum(s=1..[(n-1)/k], binomial(n-k*s-1, k-2) ) ). - Max Alekseyev, Apr 15 2005
a(n) ~ (2*sqrt(5)-4)/10 * n * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 02 2014
Equivalently, a(n) ~ n * phi^(n-3) / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
EXAMPLE
a(5) = 8 because we have 5, 14, 41, 23, 32, 122, 212 and 221.
MAPLE
G:= sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..70): Gser:=series(G, x=0, 44): seq(coeff(Gser, x^n), n=1..41); # Emeric Deutsch, Apr 13 2005
MATHEMATICA
nn=37; Drop[CoefficientList[Series[Sum[x^j/(1-x^(j+1)/(1-x))^2, {j, 1, nn}], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Mar 31 2014 *)
PROG
(PARI) a(n)=1+sum(k=2, (n+3)\2, k*sum(s=1, (n-1)\k, binomial(n-k*s-1, k-2))) (Alekseyev)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Apr 03 2005
EXTENSIONS
More terms from Emeric Deutsch and Max Alekseyev, Apr 13 2005
STATUS
approved