A284556 Sequence c of the mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1), c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1).

0, 0, 1, 1, 0, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 2, 0, 3, 2, 3, 2, 4, 2, 4, 1, 3, 3, 4, 1, 4, 2, 2, 1, 3, 2, 5, 2, 5, 4, 5, 1, 6, 4, 6, 3, 6, 3, 5, 1, 4, 4, 6, 2, 7, 4, 5, 2, 5, 3, 6, 2, 4, 3, 3, 0, 4, 3, 5, 3, 7, 4, 7, 2, 7, 6, 9, 3, 9, 5, 6, 2, 7, 5, 10, 4, 10, 7, 9, 2, 9, 6, 9, 4, 8, 4, 6, 1, 5, 5, 8, 3, 10, 6, 8, 3, 9, 6, 11, 4, 9, 6, 7, 1, 7, 5, 8, 4, 9, 5

Offset

0,6

Links

Antti Karttunen, Table of n, a(n) for n = 0..8192

Formula

a(n) = A001222(A284554(n)).

Other identities. For all n >= 1:

a(n) = (A002487(n) - A102283(n))/2.

a(n) = A002487(n) - A000360(n-1).

A000360(n-1) - a(n) = A102283(n) ≡ n (mod 3).

Mathematica

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[PrimeOmega[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &@ a@ n], {n, 0, 118}] (* or *)
a[n_] := Which[n < 2, n, EvenQ@ n, a[n/2], True, a[(n - 1)/2] + a[(n + 1)/2]]; Table[(a[n] - JacobiSymbol[n, 3])/2, {n, 0, 118}] (* Michael De Vlieger, Apr 05 2017, after Alonso del Arte at A102283 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A284556 n) (cond ((<= n 1) 0) ((even? n) (A000360with_prep_0 (/ n 2))) (else (+ (A284556 (/ (- n 1) 2)) (A284556 (/ (+ n 1) 2))))))
(definec (A000360with_prep_0 n) (cond ((<= n 1) n) ((even? n) (A284556 (/ n 2))) (else (+ (A000360with_prep_0 (/ (- n 1) 2)) (A000360with_prep_0 (/ (+ n 1) 2))))))

Crossrefs

Cf. A000360, A001222, A002487, A102283, A284554, A284566 (odd bisection).

Sequence in context: A066882 A300831 A068347 * A025865 A085091 A345994

Adjacent sequences: A284553 A284554 A284555 * A284557 A284558 A284559

Keyword

nonn

Author

Antti Karttunen, Apr 05 2017

Status

approved