OFFSET
0,5
COMMENTS
The intervals are: [1,2], [3,5], [6,9], [10,14], [15,20], [21,27], [28,35], [36,44], [45,54], [55,65], ... .
FORMULA
G.f.: Product_{n>=0} (1 + Sum_{k=(n*(n+1)/2)..(n*(n+3)/2)} x^k).
EXAMPLE
a(0)=1 by definition of the empty partition.
a(10)=6 because 10=9+1=8+2=7+3=6+4=6+3+1 (for example, you cannot take 5+5 or 7+2+1 because of the definition of a(n)).
MAPLE
f:= n-> 1+add(x^j, j=n*(n+1)/2..n*(n+3)/2):
a:= n-> coeff(mul(f(k), k=1..ceil((sqrt(9+8*n)-3)/2)), x, n):
seq(a(n), n=0..61);
PROG
(PARI) f(n, x) = (1+sum(j=n*(n+1)/2, n*(n+3)/2, x^j));
a(n) = polcoef(prod(k=1, ceil((sqrt(9+8*n)-3)/2), f(k, x)), n, x); \\ version 2.11.0 or newer; Michel Marcus, Apr 08 2019
(PARI) first(n) = v = Vecrev(Vec(a(n))); vector(n, i, v[i]) \\ using a(n) from above \\ David A. Corneth, Apr 08 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Igor Haladjian, Apr 06 2019
STATUS
approved