OFFSET
0,3
COMMENTS
The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.
The Hankel sequence transform starting with the second term gives {1, -3, -2, 1, 3, 2, -1, -3, -2, ...}.
The periodic pattern in the continued fractions of the generating function starts after some prefix, this causes high complexity in the generating function and makes a nice combinatorial interpretation less likely, therefore the keyword "less" was considered.
FORMULA
G.f.: 1/(1-1*x/(1-1*x/(1+1*x/(1+1*x/(1-3*x/(1-(1/3)*x/(1-(2/3)*x/(1-(3/2)*x/(1+(1/2)*x/(...)))))))))), a continued fraction expansion. The coefficients of x are {-1, -1, 1, 1, -3, -(1/3), -(2/3), -(3/2), (1/2), 2, -3, ...}. The numerators will repeat {1, 2, 3} the denominators {1, 1, 2, 2, 3, 3} the sign repeats {-,-,-,-,+,+}.
G.f.: (2*x^2 - sqrt(-3*x^2 - 2*x + 1) + 3*x - 1)/(-5*x^3 + sqrt(-3*x^2 - 2*x + 1)*(x^2 + x - 1) + 4*x - 1)
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A376277(k).
D-finite with recurrence (-n+3)*a(n) +(7*n-25)*a(n-1) +2*(-6*n+25)*a(n-2) +10*(-n+4)*a(n-3) +2*(16*n-73)*a(n-4) +(n-5)*a(n-5) +21*(-n+5)*a(n-6)=0. - R. J. Mathar, Oct 24 2024
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec((2*x^2-sqrt(-3*x^2-2*x+1)+3*x-1)/(-5*x^3+sqrt(-3*x^2-2*x+1)*(x^2+x-1)+4*x-1))
CROSSREFS
KEYWORD
nonn,less
AUTHOR
Thomas Scheuerle, Sep 23 2024
STATUS
approved