OFFSET
0,3
COMMENTS
n divides a(n) iff the binary representation of n ends with an even number of zeros (i.e. n is in A003159)
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
FORMULA
log(abs(a(n))) is asymptotic to c*n where c=0.80... [c = log(5)/2 = 0.8047189562... - Vaclav Kotesovec, Feb 15 2019]
a(n) ~ sqrt(n) * 5^(n/2) / sqrt(8*Pi) * ((sqrt(2 + sqrt(5)) + sqrt(38 + 25*sqrt(5)) / (16*n)) * sin(n*arctan(2)) - (sqrt(-2 + sqrt(5)) - sqrt(-38 + 25*sqrt(5)) / (16*n)) * cos(n*arctan(2))). - Vaclav Kotesovec, Feb 15 2019
From Seiichi Manyama, Jul 09 2024: (Start)
G.f.: x/(1 - 2*x + 5*x^2)^(3/2).
a(n+1) = binomial(n+2,2) * A343773(n). (End)
MATHEMATICA
RecurrenceTable[{-5 n a[n-2] + (2*n - 1) a[n-1] + (1 - n) a[n] ==
0, a[0] == 0, a[1] == 1}, a, {n, 0, 30}] (* Vaclav Kotesovec, Feb 15 2019 *)
nxt[{n_, a_, b_}]:={n+1, b, (b(2n+1)-5a(n+1))/n}; NestList[nxt, {1, 0, 1}, 40][[;; , 2]] (* Harvey P. Dale, Apr 22 2024 *)
PROG
(PARI) a(n)=if(n<2, if(n, 1, 0), 1/(n-1)*((2*n-1)*a(n-1)-5*n*a(n-2)))
CROSSREFS
KEYWORD
sign
AUTHOR
Benoit Cloitre, Feb 27 2005
STATUS
approved