A079559 Number of partitions of n into distinct parts of the form 2^j-1, j=1,2,....

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

Offset

0,1

Comments

Differences of the Meta-Fibonacci sequence for s=0. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

Fixed point of morphism 0-->0, 1-->110 - Joerg Arndt, Jun 07 2007

A006697(k) gives number of distinct subwords of length k, conjectured to be equal to A094913(k)+1. - M. F. Hasler, Dec 19 2007

Characteristic function for the range of A005187: a(A055938(n))=0; a(A005187(n))=1; if a(m)=1 then either a(m-1)=1 or a(m+1)=1. - Reinhard Zumkeller, Mar 18 2009

The number of zeros between successive pairs of ones in this sequence is A007814. - Franklin T. Adams-Watters, Oct 05 2011

Length of n-th run = abs(A088705) + 1. - Reinhard Zumkeller, Dec 11 2011

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Gary W. Adamson, Comments on A079559

Joerg Arndt, Matters Computational (The Fxtbook), section 1.26.5, Recursive generation and relation to a power series, page 74, figure 1.26-E and function A.

Benoit Cloitre, Fractal walk generated by the 130000 first terms (step of unit length moving to right if 0 left if 1) starting at (0,0)

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences

B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), R26.[alt source]

Thomas M. Lewis and Fabian Salinas, Optimal pebbling of complete binary trees and a meta-Fibonacci sequence, arXiv:2109.07328 [math.CO], 2021.

F. Ruskey and C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]

Index entries for characteristic functions

Formula

G.f.: Product_{n>=1} (1 + x^(2^n-1)).

a(n) = 1 if n=0, otherwise A043545(n+1)*a(n+1-A053644(n+1)). - Reinhard Zumkeller, Aug 19 2006

a(n) = p(n,1) with p(n,k) = p(n-k,2*k+1) + p(n,2*k+1) if k <= n, otherwise 0^n. - Reinhard Zumkeller, Mar 18 2009

Euler transform is sequence A111113 sequence offset -1. - Michael Somos, Aug 03 2009

G.f.: Product_{k>0} (1 - x^k)^-A111113(k+1). - Michael Somos, Aug 03 2009

a(n) = A108918(n+1) mod 2. - Joerg Arndt, Apr 06 2011

a(n) = A000035(A153000(n)), n >= 1. - Omar E. Pol, Nov 29 2009, Aug 06 2013

Example

a(11)=1 because we have [7,3,1].
G.f. = 1 + x + x^3 + x^4 + x^7 + x^8 + x^10 + x^11 + x^15 + x^16 + x^18 + ...
From Omar E. Pol, Nov 30 2009: (Start)
The sequence, displayed as irregular triangle, in which rows length are powers of 2, begins:
1;
1,0;
1,1,0,0;
1,1,0,1,1,0,0,0;
1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0;
1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0;
1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0,0;
(End)

Maple

g:=product(1+x^(2^n-1), n=1..15): gser:=series(g, x=0, 110): seq(coeff(gser, x, n), n=0..104); # Emeric Deutsch, Apr 06 2006
d := n -> if n=1 then 1 else A046699(n)-A046699(n-1) fi; # Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

Mathematica

row[1] = {1}; row[2] = {1, 0}; row[n_] := row[n] = row[n-1] /. 1 -> Sequence[1, 1, 0]; Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Jul 30 2012, after Omar E. Pol *)
CoefficientList[ Series[ Product[1 + x^(2^n - 1), {n, 6}], {x, 0, 104}], x] (* or *)
Nest[ Flatten[# /. {0 -> {0}, 1 -> {1, 1, 0}}] &, {1}, 6] (* Robert G. Wilson v, Sep 08 2014 *)

Prog

(PARI) w="1, "; for(i=1, 5, print1(w=concat([w, w, "0, "])))
(PARI) A079559(n, w=[1])=until(n<#w=concat([w, w, [0]]), ); w[n+1] \\ M. F. Hasler, Dec 19 2007
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, #binary(n+1), 1 + x^(2^k-1), 1 + x * O(x^n)), n))} /* Michael Somos, Aug 03 2009 */
(Haskell)
a079559 = p $ tail a000225_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Dec 11 2011
(Haskell)
a079559_list = 1 : f [1] where
   f xs = ys ++ f ys where ys = init xs ++ [1] ++ tail xs ++ [0]
-- Reinhard Zumkeller, May 05 2015
(Python)
def a053644(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1)
def a043545(n):
    x=bin(n)[2:]
    return int(max(x)) - int(min(x))
l=[1]
for n in range(1, 101): l+=[a043545(n + 1)*l[n + 1 - a053644(n + 1)], ]
print(l) # Indranil Ghosh, Jun 11 2017

Crossrefs

Cf. A000929, A001511, A005187, A006697, A007814, A046699, A055938, A094913, A163988.

Cf. A035263, A066519.

Sequence in context: A286937 A080813 A100672 * A175480 A285568 A014577

Adjacent sequences: A079556 A079557 A079558 * A079560 A079561 A079562

Keyword

nonn,nice

Author

Vladeta Jovovic, Jan 25 2003

Extensions

Edited by M. F. Hasler, Jan 03 2008

Status

approved