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A006894
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Number of planted 3-trees of height < n.
(Formerly M1254)
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12
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1, 2, 4, 11, 67, 2279, 2598061, 3374961778892, 5695183504492614029263279, 16217557574922386301420536972254869595782763547561, 131504586847961235687181874578063117114329409897615188504091716162522225834932122128288032336298142
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Representation requires n triangular numbers with greedy algorithm.
Comment from Marc LeBrun (mlb(AT)well.com): Maximum possible number of distinct values after applying a commuting operation from 0 to N times to a single initial value.
Divide the natural numbers in sets of consecutive numbers, starting with {1}, each set with number of elements equal to the sum of elements of the preceding set. The greatest element of the n-th set gives a(n). The sets begin {1}, {2}, {3,4}, {5,6,7,8,9,10,11}, ... - Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 16 2002
a(n+1) = (a(n)) th triangular numbers + 1 = A000217(a(n)) + 1. a(n) = A072638(n-1) + 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 11 2009]
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REFERENCES
| F. Harary et al., Counting free binary trees..., J. Combin. Inform. System Sciences, 17 (1992), 175-181.
E. Lemoine, ``Note sur deux nouvelles d\'{e}compositions des nombres entiers,'' Assoc. fran\c{c}aise pour l'avancement des sciences. Vol. 29, pp. 72-74, 1900.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| David Wasserman, Table of n, a(n) for n = 1..14
Index entries for "core" sequences
Index entries for sequences related to rooted trees
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FORMULA
| Partial sums of A002658; a(n+1) = a(n)(a(n)+1)/2 + 1 (from Marc LeBrun).
Sequence arises from a self-recursive process: a[1]=1, a[n]=a[n-1]*(a[n-1]+1)/2+1. E.g. a(1)=1, a(2)=1*2/2+1=2, a(3)=2*3/2+1=4, a(4)=4*5/2+1=11, a(5)=11*12/2+1=67... - Miklos Kristof (kristmikl(AT)freemail.hu), Dec 11 2007
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MAPLE
| A006894 := proc(n) option remember; if n=1 then 1 else A006894(n-1)*(A006894(n-1)+1)/2+1 fi end; [ seq(A006894(i), i=1..11) ];
a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2, 2) od: seq(a[n]+1, n=-1..9); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 08 2007
a[1]:=1:for n from 2 to 10 do a[n]:=a[n-1]*(a[n-1]+1)/2+1 od: seq(a[n], n=1..10); - Miklos Kristof (kristmikl(AT)freemail.hu), Dec 11 2007
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MATHEMATICA
| NestList[(#(#+1))/2+1&, 1, 12] (* From Harvey P. Dale, May 24 2011 *)
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PROG
| (PARI) v=vector(15); v[1]=1; for(i=2, #v, v[i]=binomial(v[i-1]+1, 2)+1); v \\ Charles R Greathouse IV, Feb 11 2011
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CROSSREFS
| Row sums of A036602.
Sequence in context: A173312 A156434 A007903 * A193565 A198864 A038093
Adjacent sequences: A006891 A006892 A006893 * A006895 A006896 A006897
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KEYWORD
| nonn,easy,core,nice
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AUTHOR
| Jeffrey Shallit, N. J. A. Sloane (njas(AT)research.att.com).
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