A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways). (Formerly M1139 N0435)

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 851, 1928, 4396, 10087, 23273, 53948, 125608, 293543, 688366, 1619087, 3818818, 9029719, 21400706, 50828664, 120963298, 288405081, 688821573, 1647853491, 3948189131, 9473431479

Offset

1,4

Comments

a(n) <= A002955(n). - Max Alekseyev, Sep 23 2009

References

J. Q. Longyear, personal communication.

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

Table of n, a(n) for n=1..31.

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy, with permission)

Sean A. Irvine, Java program (github).

J. Longyear, T. Trotter, and N. J. A. Sloane, Correspondence.

MathOverflow, MathOverflow discussion of related questions.

Vladimir Reshetnikov, Computes terms of OEIS sequence A002845 (C# library).

Jon E. Schoenfield, The 851 values for n=12.

Index entries for sequences related to parenthesizing

Example

From M. F. Hasler, Apr 17 2024: (Start)
The table with explicit lists of values starts as follows:
   n | distinct values of 2^...^2 with all possible parenthesizations
-----+---------------------------------------------------------------
   1 | 2
   2 | 2^2 = 4
   3 | (2^2)^2 = 2^(2^2) = 16
   4 | (2^2^2)^2 = 2^8 = 256, (2^2)^(2^2) = 2^(2^2^2) = 2^16 (= 65536)
   5 | 256^2 = 2^16, (2^16)^2 = 2^32, 2^256, 2^2^16 (~ 2*10^19728)
   6 | (2^16)^2 = 2^32, 2^64, 2^512, 2^2^16, 2^2^17, 2^2^32, 2^2^256, 2^2^2^16
   7 | 2^64, 2^128, 2^256, 2^1024, 2^2^17, 2^2^18, 2^2^32, 2^2^33, 2^2^64, 2^2^257,
     | 2^2^512, 2^2^2^16, 2^2^65537, 2^2^2^17, 2^2^2^32, 2^2^2^256, 2^2^2^2^16
  ...| ...
(When parentheses are omitted above, we use that ^ is right associative.) (End)

Prog

(PARI) /* Define operators for numbers represented (recursively) as list of positions of bits 1. Illustration using the commands below: T = 3.bits; T.int */
n.bits = vector(hammingweight(n), v,  n -= 1 << v= valuation(n, 2); v.bits)
ONE = 1.bits; m.int = sum(i=1, #m, 1<<m[i].int) /* Convert back. (Not needed.) */
{POW(m, n)= if(#m==1, [MUL(m[1], n)], my(p=ONE); until(!#n || !#m=MUL(m, m), #n[1] || p=MUL(p, m); n=RSHIFT(n)); p)}\\ binary exponentiation unless only 1 bit set
{MUL(m, n)= my(S=[]); #n && foreach(m, b, S=ADD(S, [ADD(c, b)| c<-n])); S}
{RSHIFT(m, n=ONE)= if(!#m|| !#n|| !(#m[1]|| #m=m[^1]), m, [SUB(b, n)| b<-m, CMP(b, n)>=0])}
{ADD(m, n, a=#m, b=#n)= if(!a, n, !b, m, a=b=1; until(a>#m|| b>#n, if(m[a]==n[b], until(a>=#m|| m[a]!=m[a+1]|| !#m=m[^a], m[a]=ADD(m[a], ONE)); b++, CMP(m[a], n[b])<0, a++, m=concat([m[1..a-1], [n[b]], m[a..#m]]); b++)); b>#n|| m=concat(m, n[b..#n]); m)}
{CMP(m, n, a=#m, b=#n, c=0)= if(!b, a, !a, -1, while(!(c=CMP(m[a], n[b]))&& a--&& b--, ); if(c, c, 1-b))}
{SUB(m, n, a=#n)= if(!a, m, my(b=a=1, c, i); while(a<=#m && b<=#n, if(0>c=CMP(m[a], n[b]), a++, c, i=[c=n[b]]; b++; while(m[a]!=c=ADD(c, ONE), if(b<=#n && c==n[b], b++, i=concat(i, [c]))); m=concat([m[1..a-1], i, m[a+1..#m]]); a += #i, m=m[^a]; b++)); m)}
A2845 = List([[2.bits]]) /* List of values for each n */
{A002845(n)= while(#A2845<n, my(S=[], n=#A2845); for(k=1, n, foreach(A2845[n-k+1], b, S=setunion(S, Set([POW(a, b)| a<-A2845[k]])))); listput(A2845, S)); #A2845[n]}
\\ Unoptimized code, for illustration. Slow for n >= 15. - M. F. Hasler, Apr 28 2024

Crossrefs

Cf. A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

Sequence in context: A251691 A157904 A182901 * A072925 A002955 A202844

Adjacent sequences: A002842 A002843 A002844 * A002846 A002847 A002848

Keyword

nonn,nice,more,changed

Author

N. J. A. Sloane

Extensions

a(12)-a(13) corrected and a(14)-a(27) added by Jon E. Schoenfield, Oct 11 2008

a(28)-a(29) computed by Kirill Osenkov, added by Vladimir Reshetnikov, Feb 07 2019

a(30)-a(31) added by Sean A. Irvine, Feb 18 2019

Status

approved