A227277 G.f.: Sum_{n>=0} x^n * (1+x)^A007814(n), where A007814(n) is the exponent of the highest power of 2 dividing n.

1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 6, 11, 12, 6, 4, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 7, 16, 22, 16, 9, 3, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 6, 11, 12, 6, 4, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..10000

Formula

a(4*A004772(n)) = 1 where A004772 lists numbers that are not congruent to 1 mod 4.

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 2*x^7 + x^8 + 4*x^9 + 4*x^10 + 3*x^11 + x^12 + 3*x^13 + 2*x^14 + 2*x^15 + x^16 +...
where
A(x) = 1 + x + x^2*(1+x) + x^3 + x^4*(1+x)^2 + x^5 + x^6*(1+x) + x^7 + x^8*(1+x)^3 + x^9 + x^10*(1+x) + x^11 + x^12*(1+x)^2 + x^13 + x^14*(1+x) + x^16*(1+x)^4 +...
AS AN IRREGULAR TRIANGLE.
When formatted into a triangle with rows of 2^n terms, we see patterns more clearly.
Sequence starts with 1, 1, followed by terms in the following triangle:
1;
2, 1;
3, 2, 2, 1;
4, 4, 3, 1, 3, 2, 2, 1;
5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1;
6, 11, 12, 6, 4, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1;
7, 16, 22, 16, 9, 3, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 6, 11, 12, 6, 4, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1, 5, 7, 6, 2, 3, 2, 2, 1, 4, 4, 3, 1, 3, 2, 2, 1; ...
in which row n+1 can be generated from row n by:
* T(n+1,k) = T(n,k) + binomial(n,k)  for 0<= k <=2^n, and
* T(n+1,2^n+k) = T(n,k)  for 0<= k <=2^n.
Note that the rows, when read in reverse, tend to have a limit (A227287):
[1, 2, 2, 3, 1, 3, 4, 4, 1, 2, 2, 3, 2, 6, 7, 5, 1, 2, 2, 3, 1, 3, 4, 4, 1, 2, 2, 4, 6, 12, 11, 6, 1, 2, 2, 3, 1, 3, 4, 4, 1, ...];
where the g.f. of the resulting sequence is:
Sum_{n>=0} x^(n - b(n)) * (1+x)^b(n), where b(n) = A007814(n).

Prog

(PARI) {a(n)=polcoeff(1+sum(m=1, n, x^m*(1+x+x*O(x^n))^valuation(m, 2)), n)}
for(n=0, 128, print1(a(n), ", "))

Crossrefs

Cf. A227287, A227318, A007814.

Sequence in context: A366790 A369459 A369467 * A071481 A324293 A336927

Adjacent sequences: A227274 A227275 A227276 * A227278 A227279 A227280

Keyword

nonn

Author

Paul D. Hanna, Jul 04 2013

Status

approved