A093879 - OEIS

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A093879

First differences of A004001.

0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

All the terms are 0 or 1: it is easy to show that if {b(n)} = A004001, b(n)>=b(n-1) and b(n)<n, therefore the first differences form an infinite binary word. - Benoit Cloitre, Jun 05 2004

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..9999

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.

FORMULA

From Antti Karttunen, Jan 18 2016: (Start)

a(n) = A004001(n+1) - A004001(n).

Other identities. For all n >= 1:

a(A087686(n+1)-1) = 0.

a(A088359(n)-1) = 1.

a(n) = 1 if and only if A051135(n+1) = 1.

(End)

MATHEMATICA

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; t = Table[a[n], {n, 110}]; Drop[t, 1] - Drop[t, -1] (* Robert G. Wilson v, May 28 2004 *)

PROG

(PARI) {m=106; v=vector(m, j, 1); for(n=3, m, a=v[v[n-1]]+v[n-v[n-1]]; v[n]=a); for(n=2, m, print1(v[n]-v[n-1], ", "))}

(Scheme) (define (A093879 n) (- (A004001 (+ 1 n)) (A004001 n))) ;; Code for A004001 given in that entry. - Antti Karttunen, Jan 18 2016

(Magma)

h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n-Self(n-1)): n in [1..160]];

A093879:= func< n | h[n+1] - h[n] >;

[A093879(n): n in [1..120]]; // G. C. Greubel, May 19 2024

(SageMath)

@CachedFunction

def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1))

def A093879(n): return h(n+1) - h(n)

[A093879(n) for n in range(1, 101)] # G. C. Greubel, May 19 2024