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A079559 Number of partitions of n into distinct parts of the form 2^j-1, j=1,2,.... 46
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 (list; graph; refs; listen; history; text; internal format)
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

R. Zumkeller, Table of n, a(n) for n = 0..1000 - Reinhard Zumkeller, Mar 18 2009

Joerg Arndt, Matters Computational (The Fxtbook), section 1.26.4, p. 73

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]

F. Ruskey, 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.: prod( n>=1, 1+x^(2^n-1) ).

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

a(n) = p(n,1) with p(n,k) = if k<=n then p(n-k,2*k+1)+p(n,2*k+1) else 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 xrange(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.

Sequence in context: A145006 A080813 A100672 * A175480 A229062 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

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Last modified November 19 08:51 EST 2017. Contains 294923 sequences.