A373184 -id:A373184 - cp's OEIS frontend

A084432 Expansion of 2/(1-x) + Sum_{k>=0} t^2(3-t)/(1+t)/(1-t)^2, where t=x^2^k.

2, 5, 4, 10, 6, 11, 8, 19, 10, 17, 12, 24, 14, 23, 16, 36, 18, 29, 20, 38, 22, 35, 24, 49, 26, 41, 28, 52, 30, 47, 32, 69, 34, 53, 36, 66, 38, 59, 40, 79, 42, 65, 44, 80, 46, 71, 48, 98, 50, 77, 52, 94, 54, 83, 56, 109, 58, 89, 60, 108, 62, 95, 64, 134, 66, 101, 68, 122, 70, 107, 72, 139, 74, 113

Offset: 1

Ralf Stephan, Jun 27 2003

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A069895, A373184, A373185.

for(n=1, 100, l=ceil(log(n)/log(2)); t=polcoeff(sum(k=0, l, 1/(1-x^2^k)^2) + O(x^(n+1)), n); print1(t", "))

a(1)=2, a(2*n) = a(n)+2*n+1, a(2*n+1) = 2*n+2.
Dirichlet g.f.: 2^s/(2^s-1) * (zeta(s)+zeta(s-1)). - Ralf Stephan, Jun 17 2007
From Seiichi Manyama, May 27 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x)^2 - 1 + A(x^2).
G.f.: A(x) = Sum_{k>=0} (1/(1 - x^(2^k))^2 - 1). (End)
Conjecture: a(n) is the number of integer solutions (x,y,z) to the Diophantine equation 2^x*(y+z) = n, where 0 <= x,y,z <= n. - Joseph M. Shunia, Aug 27 2024
Sum_{k=1..n} a(k) ~ 2*n^2/3 + 2*n. - Vaclav Kotesovec, Aug 28 2024

A373185 G.f. A(x) satisfies A(x) = 1/(1 - x)^2 - 1 + A(x^4).

Original entry on oeis.org

2, 3, 4, 7, 6, 7, 8, 12, 10, 11, 12, 17, 14, 15, 16, 24, 18, 19, 20, 27, 22, 23, 24, 32, 26, 27, 28, 37, 30, 31, 32, 45, 34, 35, 36, 47, 38, 39, 40, 52, 42, 43, 44, 57, 46, 47, 48, 66, 50, 51, 52, 67, 54, 55, 56, 72, 58, 59, 60, 77, 62, 63, 64, 89, 66, 67, 68, 87, 70, 71, 72, 92, 74, 75, 76, 97, 78, 79, 80, 108, 82, 83, 84, 107

Offset: 1

Seiichi Manyama, May 27 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A084432, A373184.

def A(k, n)
  ary = [0]
  (1..n).each{|i|
    j = i + 1
    j += ary[i / k] if i % k == 0
    ary << j
  }
  ary[1..-1]
end
p A(4, 90)

a(4*n+1) = 4*n+2, a(4*n+2) = 4*n+3, a(4*n+3) = 4*n+4 and a(4*n+4) = 4*n+5 + a(n+1) for n >= 0.

G.f.: A(x) = Sum_{k>=0} (1/(1 - x^(4^k))^2 - 1).

cp's OEIS Frontend

A084432 Expansion of 2/(1-x) + Sum_{k>=0} t^2(3-t)/(1+t)/(1-t)^2, where t=x^2^k.

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