A373188 Expansion of Sum_{k>=0} x^(4^k) / (1 - x^(4^k))^2.

1, 2, 3, 5, 5, 6, 7, 10, 9, 10, 11, 15, 13, 14, 15, 21, 17, 18, 19, 25, 21, 22, 23, 30, 25, 26, 27, 35, 29, 30, 31, 42, 33, 34, 35, 45, 37, 38, 39, 50, 41, 42, 43, 55, 45, 46, 47, 63, 49, 50, 51, 65, 53, 54, 55, 70, 57, 58, 59, 75, 61, 62, 63, 85, 65, 66, 67, 85, 69, 70, 71, 90, 73, 74, 75, 95, 77, 78, 79, 105, 81, 82, 83, 105, 85, 86

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

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

From Amiram Eldar, Oct 30 2025: (Start)

Multiplicative with a(2^e) = ceiling(2*(2^(e+1)-1)/3) = A000975(e+1), and a(p^e) = p^e for an odd prime p.

Dirichlet g.f.: zeta(s-1) * 4^s / (4^s-1).

Sum_{k=1..n} a(k) ~ (8/15) * n^2. (End)

Maple

f:= proc(n) option remember;
  if n mod 4 <> 0 then n else n + procname(n/4) fi
end proc:
f(4):= 5:
map(f, [$1..100]); # Robert Israel, Oct 29 2025

Mathematica

a[n_] := Module[{e = IntegerExponent[n, 2]}, n * Ceiling[2*(2^(e+1) - 1)/3] / 2^e]; Array[a, 100] (* Amiram Eldar, Oct 30 2025 *)

Prog

(PARI) a(n) = {my(e = valuation(n, 2)); (n >> e) * ceil(2*(2^(e+1) - 1)/3); } \\ Amiram Eldar, Oct 30 2025

Crossrefs

Cf. A373187, A373189.

Cf. A000975, A129527, A327625, A359100.

Sequence in context: A078903 A296206 A079228 * A067535 A076752 A393160

Adjacent sequences: A373185 A373186 A373187 * A373189 A373190 A373191

Keyword

nonn,mult,easy

Author

Seiichi Manyama, May 27 2024

Status

approved