

A002658


a(0) = a(1) = 1; for n > 0, a(n+1) = a(n)*(a(0)+...+a(n1)) + a(n)*(a(n)+1)/2.
(Formerly M1814 N0718)


12




OFFSET

0,3


COMMENTS

Number of planted trees in which every node has degree <=3 and of height n; or products of height n when multiplication is commutative but nonassociative.
Also called planted 3trees or planted unarybinary trees.
The next term (which was incorrectly given) is in fact too large to include. See the bfile.
Comment from Marc LeBrun: Maximum possible number of distinct new values after applying a commuting operation 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 number of elements in the nth (n>0) set gives a(n). The sets begin {1}, {2}, {3,4}, {5,6,7,8,9,10,11}, ...  Floor van Lamoen, Jan 16 2002
Consider the free algebraic system with one binary commutative (x+y) operator and one generator A. The number of elements of height n is a(n) where the height of A is zero and the height of (x+y) is one more than the maximum height of x and y.  Michael Somos, Mar 06 2012
Sergey Zimnitskiy, May 08 2013, provided an illustration for A006894 and A002658 in terms of packing circles inside circles. The following description of the figure was supplied by Allan Wilks. Label a blank page "1" and draw a black circle labeled "2". Subsequent circles are labeled "3", "4", ... . In the black circle put two red circles (numbered "3" and "4"); two because the label of the black circle is "2". Then in each of the red circles put blue circles in number equal to the labels of the red circles. So these get labeled "5", ..., "11". Then in each of the blue circles, starting with circle "5", place a set of green (say) circles, equal in number to the label of the enclosing blue circle. When all of the green circles have been drawn, they will be labeled "12", ..., "67". If you take the maximum circle label at each colored level, you get 1,2,4,11,67,2279,..., which is A006894, which itself is the partial sums of A002658. The picture is a visualization of Floor van Lamoen's comment above.
See A067338 for a variant of the integer partitioning construction, starting with {1,2}, {3,4,5}, ...  M. F. Hasler, Jan 21 2015


REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

David Wasserman, Table of n, a(n) for n = 0..13
I. M. H. Etherington, Nonassociate powers and a functional equation, Math. Gaz. 21 (1937), 3639; addendum 21 (1937), 153.
I. M. H. Etherington, On nonassociative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 193839), 153162. [Annotated scanned copy]
I. M. H. Etherington, Some problems of nonassociative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. ivi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages viixiv of the same issue.
A. Erdelyi and I. M. H. Etherington, Some problems of nonassociative combinations (II), Edinburgh Math. Notes, 32 (1940), pp. viixiv.
F. Harary, et al., Counting free binary trees admitting a given height, J. Combin. Inform. System Sci. 17 (1992), no. 12, 175181. MR1216977 (94c:05039)
Harary, Frank; Palmer, Edgar M.; Robinson, Robert W., Counting free binary trees admitting a given height, J. Combin. Inform. System Sci. 17 (1992), no. 12, 175181. (Annotated scanned copy)
Z. A. Melzak, A note on homogeneous dendrites, Canad. Math. Bull., 11 (1968), 8593.
Sergey Zimnitskiy, Illustration of initial terms of A006894 and A002658
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for "core" sequences


FORMULA

a(n + 1) = a(n) * (a(n) / a(n1) + (a(n) + a(n1)) / 2) [equation (5) on page 87 of Melzak 1968 with a() instead of his f()].
a(n) ~ 2 * c^(2^n), where c = 1.2460208329836625089431529441999359284665241772983812581752523573774108242448... .  Vaclav Kotesovec, May 21 2015


MAPLE

s := proc(n) local i, j, ans; ans := [ 1 ]; for i to n do ans := [ op(ans), ans[ i ]*(add(j, j=ans)ans[ i ])+ans[ i ]*(ans[ i ]+1)/2 ] od; RETURN(ans); end; t1 := s(10); A002658 := n>t1[n];


MATHEMATICA

Clear[a, b]; a[0] = a[1] = 1; b[0] = b[1] = 1; b[n_] := b[n] = b[n1] + a[n1]; a[n_] := a[n] = (a[n1]+1)*a[n1]/2 + a[n1]*b[n1]; Table[a[n], {n, 0, 9}] (* JeanFrançois Alcover, Jan 31 2013, after Frank Harary *)
RecurrenceTable[{a[n] == a[n1]*(a[n1]/a[n2]+(a[n1]+a[n2])/2), a[0]==1, a[1]==1}, a, {n, 0, 10}] (* Vaclav Kotesovec, May 21 2015 *)


PROG

(PARI) {a(n) = local(a1, a2); if( n<2, n>=0, a2 = a(n1); a1 = a(n2); a2 * (a2 / a1 + (a1 + a2) / 2))} /* Michael Somos, Mar 06 2012 */
(Haskell)
a002658 n = a002658_list !! n
a002658_list = 1 : 1 : f [1, 1] where
f (x:xs) = y : f (y:x:xs') where y = x * sum xs + x * (x + 1) `div` 2
 Reinhard Zumkeller, Apr 10 2012
(PARI) print1(s=a=1); for(i=1, 9, print1(", "a*=(1a)/2+s); s+=a) \\ M. F. Hasler, Jan 21 2015


CROSSREFS

Cf. A006894, A005588. First differences of A072638.
Sequence in context: A182055 A211209 A271440 * A175818 A034939 A048898
Adjacent sequences: A002655 A002656 A002657 * A002659 A002660 A002661


KEYWORD

nonn,easy,core,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Corrected by David Wasserman, Nov 20 2006


STATUS

approved



