A002658 a(0) = a(1) = 1; for n > 0, a(n+1) = a(n)*(a(0) + ... + a(n-1)) + a(n)*(a(n) + 1)/2. (Formerly M1814 N0718)

1, 1, 2, 7, 56, 2212, 2595782, 3374959180831, 5695183504489239067484387, 16217557574922386301420531277071365103168734284282

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 non-associative.

Also called planted 3-trees or planted unary-binary trees.

The next term (which was incorrectly given) is in fact too large to include. See the b-file.

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 n-th (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

Mayfawny Bergmann, Efficiency of Lossless Compression of a Binary Tree via its Minimal Directed Acyclic Graph Representation. Rose-Hulman Undergraduate Mathematics Journal: Vol. 15 : Iss. 2, Article 1. (2014).

I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.

I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162. [Annotated scanned copy]

I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.

A. Erdelyi and I. M. H. Etherington, Some problems of non-associative combinations (II), Edinburgh Math. Notes, 32 (1940), pp. vii-xiv.

F. Harary, et al., Counting free binary trees admitting a given height, J. Combin. Inform. System Sci. 17 (1992), no. 1-2, 175--181. 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. 1-2, 175-181. (Annotated scanned copy)

Z. A. Melzak, A note on homogeneous dendrites, Canad. Math. Bull., 11 (1968), 85-93.

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(n-1) + (a(n) + a(n-1)) / 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[n-1] + a[n-1]; a[n_] := a[n] = (a[n-1]+1)*a[n-1]/2 + a[n-1]*b[n-1]; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Jan 31 2013, after Frank Harary *)
RecurrenceTable[{a[n] == a[n-1]*(a[n-1]/a[n-2]+(a[n-1]+a[n-2])/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(n-1); a1 = a(n-2); a2 * (a2 / a1 + (a1 + a2) / 2))} /* Michael Somos, Mar 06 2012 */
(PARI) print1(s=a=1); for(i=1, 9, print1(", "a*=(1-a)/2+s); s+=a) \\ M. F. Hasler, Jan 21 2015
(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
(Python)
from itertools import islice
def agen():
    yield 1
    an = s = 1
    while True:
        yield an
        an1 = an*s + an*(an+1)//2
        an, s = an1, s+an
print(list(islice(agen(), 10))) # Michael S. Branicky, Nov 14 2022

Crossrefs

Cf. A006894, A005588. First differences of A072638.

Sequence in context: A363205 A271440 A304984 * 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