OFFSET
0,3
REFERENCES
D. E. Knuth, personal communication.
LINKS
FORMULA
Sum_{n>=0} a(n)*q^n = 1/(1 - Sum_{n>=0} a(n)*q^(2^n) ).
As n increases, a(n+1)/a(n) approaches the value x = 1.874542... satisfying 1 = Sum_{j>=0} a(j)/x^(2^j). [David W. Wilson]
EXAMPLE
E.g. 4 = 1+1+1+1 = 2^1 + 1+1 = 2^1 + 2^1 = 2^2 = 2^1+1 = 1+2^1 + 1 = 1+1+2^1.
MAPLE
N:= 100: # to get all terms up to a(N)
F:= add(a[i]*q^i, i=0..N) - 1/(1 - add(a[i]*q^(2^i), i=0..ilog2(N))):
S:= series(F, q, N+1);
sol:= {}:
for i from 0 to N do
eq:= subs(sol, coeff(S, q, i));
sol:= sol union {a[i]=solve(eq, a[i])};
od:
subs(sol, [seq(a[i], i=0..N)]); # Robert Israel, Dec 12 2014
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved