

A114852


The number of closed lambda calculus terms of size n, where size(lambda x.M)=2+size(M), size(M N)=2+size(M)+size(N), and size(V)=1+i for a variable V bound by the ith enclosing lambda (corresponding to a binary encoding).


6



0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 6, 5, 13, 14, 37, 44, 101, 134, 298, 431, 883, 1361, 2736, 4405, 8574, 14334, 27465, 47146, 89270, 156360, 293840, 522913, 978447, 1761907, 3288605, 5977863, 11148652, 20414058, 38071898, 70125402, 130880047
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OFFSET

0,9


REFERENCES

K. Grygiel, P. Lescanne, Counting terms in the binary lambda calculus, arXiv preprint arXiv:1401.0379, 2014


LINKS

Table of n, a(n) for n=0..40.
John Tromp, John's Lambda Calculus and Combinatory Logic Playground
John Tromp, Binary Lambda Calculus and Combinatory Logic
John Tromp, TITLE FOR LINK


FORMULA

a(n) = N(0,n) with
N(k,0) = N(k,1) = 0
N(k,n+2) = (if k>n then 1 else 0) +
N(k+1,n) +
Sum_{i=0..n} N(k,i) * N(k,ni)


EXAMPLE

a(8) = 2 because lambda x.lambda y.lambda z.z and lambda x.(x x) are the only two closed lambda terms of size 8.


PROG

a114852 = closed 0 where
closed k n = if n<2 then 0 else
(if n2<k then 1 else 0) +
closed (k+1) (n2) +
sum [closed k i * closed k (n2i)  i < [0..n2]]
 See link for a more efficient version.


CROSSREFS

Cf. A114851, A195691.
Sequence in context: A156993 A030770 A188652 * A188048 A191529 A095132
Adjacent sequences: A114849 A114850 A114851 * A114853 A114854 A114855


KEYWORD

nonn


AUTHOR

John Tromp, Feb 20 2006


STATUS

approved



