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A023434
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Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).
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8
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0, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 113, 150, 199, 264, 350, 464, 615, 815, 1080, 1431, 1896, 2512, 3328, 4409, 5841, 7738, 10251, 13580, 17990, 23832, 31571, 41823, 55404, 73395, 97228
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| lim n->inf a(n)/a(n-1) = positive root of 1+x-x^3 (smallest Pisot-Vijayaraghavan number, A060006) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 19 2004
a(n) = number of distinct even run-types taken over nonempty subsets of [n+1]. The run-type of a set of positive integers is the sequence of lengths when the set is decomposed into maximal runs of consecutive integers and it is even if all its entries are even. For example, the set {2,3,5,6,9,10,11} has run-type (2,2,3) and a(6)=6 counts (2),(4),(6),(2,2),(2,4),(4,2). - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
Partial sums of the sequence obtained by deleting the first 2 terms of A000931. Example: 0+1+0+1+1 = 3 = a(4). - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
One less than the sequence obtained by deleting the first 7 terms of A000931. - Ira Gessel (gessel(AT)brandeis.edu), May 02 2007
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REFERENCES
| J. H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.
V. E. Hoggatt, Jr. and D. A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1070
Index to sequences with linear recurrences with constant coefficients, signature (1,1,0,-1).
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FORMULA
| a(0)=0, a(1)=1, a(2)=1 then for n>2 a(n)=ceiling(r*a(n-1)) where r is the positive root of x^3-x-1=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 19 2004
G.f.: x/((1-x)(1-x^2-x^3)). - Jon Perry (perry(AT)globalnet.co.uk), Jul 04 2004
For n>2 a(n) = floor{sqrt(a(n-3)*a(n-2) + a(n-2)*a(n-1) + a(n-1)*a(n-3))} + 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 19 2004
a(n) = Sum_{1<=k<=(n+2)/3} binomial[Floor[(n+2-k)/2],k]. This formula counts even run-types by length. - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
a(n) = a(n-2) + a(n-3) + 1 [From M. Dols (markdols(AT)yahoo.com), Feb 01 2010]
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MATHEMATICA
| a=b=c=0; d=1; lst={c, d}; Do[e=c+d-a; AppendTo[lst, e]; a=b; b=c; c=d; d=e, {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 30 2009]
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CROSSREFS
| Cf. A060006.
a(n) = A000931(n+7)-1.
Sequence in context: A035990 A036001 A027336 * A087192 A188917 A046935
Adjacent sequences: A023431 A023432 A023433 * A023435 A023436 A023437
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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