login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A195691 The number of closed normal form 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 i-th enclosing lambda (corresponding to a binary encoding). 4
0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 4, 8, 7, 18, 23, 42, 50, 105, 153, 271, 385, 721, 1135, 1992, 3112, 5535, 9105, 15916, 26219, 45815, 77334, 135029, 229189, 399498, 685710, 1198828, 2070207, 3619677, 6286268, 11024475, 19241836, 33795365, 59197968, 104234931 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Table of n, a(n) for n=0..44.

Wikipedia, Binary lambda calculus

John Tromp, A195691.hs

FORMULA

a(n) = N(1,0,n) with

  N(q,k,0) = N(q,k,1) = 0

  N(q,k,n+2) = (if k>n then 1 else 0) +

               q * N(1,k+1,n) +

               Sum_{i=0..n} N(0,k,i) * N(1,k,n-i)

EXAMPLE

This sequence first differs from A114852 at n=10, where it excludes the two reducible terms (lambda x.x)(lambda x.x) and lambda x.(lambda x.x)x, so normal 10 = (closed 10)-2 = 6-2 = 4.

MATHEMATICA

A[_, _, 0] = A[_, _, 1] = 0; A[q_, k_, n_] := A[q, k, n] = Boole[k > n-2] + q*A[1, k+1, n-2] + Sum[A[0, k, i]*A[1, k, n-i-2], {i, 0, n-2}];

a[n_] := A[1, 0, n];

Table[a[n], {n, 0, 44}] (* Jean-Fran├žois Alcover, May 23 2017 *)

PROG

(Haskell)

a195691 = normal True 0 where

  normal qLam k n = if n<2 then 0 else

    (if n-2<k then 1 else 0) +

    (if qLam then normal True (k+1) (n-2) else 0) +

    sum [normal False k i * normal True k (n-2-i) | i <- [0..n-2]]

-- See link for a more efficient version.

CROSSREFS

Cf. A114851, A114852.

Cf. A224345 for another way of counting normal forms in lambda-calculus.

Sequence in context: A008312 A060723 A300622 * A074763 A099932 A175000

Adjacent sequences:  A195688 A195689 A195690 * A195692 A195693 A195694

KEYWORD

nonn

AUTHOR

John Tromp, Sep 22 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)