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A079559
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Number of partitions of n into distinct parts of the form 2^j-1, j=1,2,....
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10
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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; internal format)
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OFFSET
| 0,1
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COMMENTS
| Differences of the Meta-Fibonacci sequence for s=0. - Frank Ruskey (http://www.cs.uvic.ca/~ruskey/) and Chris Deugau (deugaucj(AT)uvic.ca)
Fixed point of morphism 0-->0, 1-->110 - Joerg Arndt, Jun 07 2007
Also, the infinite word generated by 1 -> 110, 0 -> 0. A006697(k) gives number of distinct subwords of length k, conjectured to be equal to A094913(k)+1. - M. F. Hasler, Dec 19 2007
Contribution from Reinhard Zumkeller, Mar 18 2009: (Start)
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. (End)
It appears that this is a triangle (see example). [From Omar E. Pol, Nov 30 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]
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REFERENCES
| B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
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LINKS
| R. Zumkeller, Table of n, a(n) for n = 0..1000 [From Reinhard Zumkeller, Mar 18 2009]
Joerg Arndt, Fxtbook, section 1.26.4, p. 73
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences, J. Integer Seq., Vol. 12. [This is a later version than that in the GenMetaFib.html link]
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
Index entries for characteristic functions [From Reinhard Zumkeller, Mar 18 2009]
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FORMULA
| G.f.: Product_{n>0} 1+x^(2^n-1).
a(n) = if n=0 then 1 else A043545(n+1)*a(n-A053644(n)). - 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. [From 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 6 2011]
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EXAMPLE
| a(11)=1 because we have [7,3,1].
1 + x + x^3 + x^4 + x^7 + x^8 + x^10 + x^11 + x^15 + x^16 + x^18 + ...
Contribution from Omar E. Pol, Nov 30 2009: (Start)
Triangle 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,... (End)
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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 (http://www.cs.uvic.ca/~ruskey/) and Chris Deugau (deugaucj(AT)uvic.ca)
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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
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CROSSREFS
| Cf. A005187, A055938, A000929, A046699, A006697, A094913.
Cf. A000079, A163988. [From Omar E. Pol, Nov 30 2009]
Cf. A007814, A001511.
Sequence in context: A145006 A080813 A100672 * A175480 A014577 A157926
Adjacent sequences: A079556 A079557 A079558 * A079560 A079561 A079562
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KEYWORD
| nonn,nice
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 25 2003
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EXTENSIONS
| Edited by M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jan 03 2008
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