A007501 a(0) = 2; for n >= 0, a(n+1) = a(n)*(a(n)+1)/2.

2, 3, 6, 21, 231, 26796, 359026206, 64449908476890321, 2076895351339769460477611370186681, 2156747150208372213435450937462082366919951682912789656986079991221

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• easy

Comments

Number of nonisomorphic complete binary trees with leaves colored using two colors. - Brendan McKay, Feb 01 2001

With a(0) = 2, a(n+1) is the number of possible distinct sums between any number of elements in {1,...,a(n)}. - Derek Orr, Dec 13 2014

Examples

			Example for depth 2 (the nonisomorphic possibilities are AAAA, AAAB, AABB, ABAB, ABBB, BBBB):
         o
        / \
       /   \
      o     o
     / \   / \
    /   \ /   \
    A   B B   B
		

References

• W. H. Cutler, Subdividing a Box into Completely Incongruent Boxes, J. Rec. Math., 12 (1979), 104-111.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..12
• G. L. Honaker, Jr., 41041 (another Prime Pages' Curiosity)
• J. C. Kieffer, Hierarchical Type Classes and Their Entropy Functions, in 2011 First International Conference on Data Compression, Communications and Processing, pp. 246-254; Digital Object Identifier: 10.1109/CCP.2011.36.
• J. V. Post, Math Pages [wayback copy]
• J.S. Seneschal, Iteration of Complete Graphs
• Stephan Wagner, Enumeration of highly balanced trees

Crossrefs

Cf. A000217, A006893.

Cf. A117872 (parity), A275342 (2-adic valuation).

Cf. A129440.

Cf. A013589 (start=4), A050542 (start=5), A050548 (start=7), A050536 (start=8), A050909 (start=9).

Cf. A108225, A145272.

Programs

Haskell

a007501 n = a007501_list !! n
a007501_list = iterate a000217 2  -- Reinhard Zumkeller, Aug 15 2013

Mathematica

f[n_Integer] := n(n + 1)/2; NestList[f, 2, 10]

PARI

a(n)=if(n<1,2,a(n-1)*(1+a(n-1))/2)

Formula

a(n) = A006893(n+1) + 1.

a(n+1) = A000217(a(n)). - Reinhard Zumkeller, Aug 15 2013

a(n) ~ 2 * c^(2^n), where c = 1.34576817070125852633753712522207761954658547520962441996... . - Vaclav Kotesovec, Dec 17 2014

a(n) = A145272(n) + a(n-1). - J.S. Seneschal, Jul 17 2025