A165420 -id:A165420 - cp's OEIS frontend

A001146 a(n) = 2^(2^n).

2, 4, 16, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 0

N. J. A. Sloane

Or, write previous term in base 2, read in base 4.
a(1) = 2, a(n) = smallest power of 2 which does not divide the product of all previous terms.
Number of truth tables generated by Boolean expressions of n variables. - C. Bradford Barber (bradb(AT)shore.net), Dec 27 2005
From Ross Drewe, Feb 13 2008: (Start)
Or, number of distinct n-ary operators in a binary logic. The total number of n-ary operators in a k-valued logic is T = k^(k^n), i.e., if S is a set of k elements, there are T ways of mapping an ordered subset of n elements from S to an element of S. Some operators are "degenerate": the operator has arity p, if only p of the n input values influence the output. Therefore the set of operators can be partitioned into n+1 disjoint subsets representing arities from 0 to n.
For n = 2, k = 2 gives the familiar Boolean operators or functions, C = F(A,B). There are 2^2^2 = 16 operators, composed of: arity 0: 2 operators (C = 0 or 1), arity 1: 4 operators (C = A, B, not(A), not(B)), arity 2: 10 operators (including well-known pairs AND/NAND, OR/NOR, XOR/EQ). (End)
From José María Grau Ribas, Jan 19 2012: (Start)
  Or, numbers that can be formed using the number 2, the power operator (^), and parenthesis. (End) [The paper by Guy and Selfridge (see also A003018) shows that this is the same as the current sequence. - N. J. A. Sloane, Jan 21 2012]
a(n) is the highest value k such that A173419(k) = n+1. - Charles R Greathouse IV, Oct 03 2012
Let b(0) = 8 and b(n+1) = the smallest number not in the sequence such that b(n+1) - Product_{i=0..n} b(i) divides b(n+1)*Product_{i=0..n} b(i). Then b(n) = a(n) for n > 0. - Derek Orr, Jan 15 2015
Twice the number of distinct minimal toss sequences of a coin to obtain all sequences of length n, which is 2^(2^n-1). This derives from the 2^n ways to cut each of the de Bruijn sequences B(2,n). - Maurizio De Leo, Feb 28 2015
I conjecture that { a(n) ; n>1 } are the numbers such that n^4-1 divides 2^n-1, intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015
Erdős has shown that it is an irrationality sequence (see Guy reference). - Stefano Spezia, Oct 13 2024

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E24.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..11
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Jan Brandts and Apo Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, Special Matrices, Vol. 5, No. 1 (2017), pp. 158-201, arXiv preprint, arXiv:1512.03044 [math.CO], 2015.
John H. Conway, Sphere packings, lattices, codes and greed, pp. 45-55 of Proc. Intern. Congr. Math., Vol. 2, 1994, alternative link.
Jose María Grau and A. M. Oller-Marcén On the last digit and the last non-zero digit of n^n in base b, arXiv:1203.4066 [math.NT], 2012.
Richard K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Rudolf Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969), 456.
Rudolf Ondrejka, Letter to N. J. A. Sloane, May 15 1976
Eric Weisstein's World of Mathematics, Irrationality Sequence, Quadratic Recurrence Equation, Coin Tossing.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000079, A000215, A007404, A026477, A062090, A062091, A112535, A155538, A215016.

Cf. also A003018, A051179, A173419, A165420, A247165, A247219.

a001146 = (2 ^) . (2 ^)
a001146_list = iterate (^ 2) 2  -- Reinhard Zumkeller, Jun 04 2012

[2^(2^n): n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

A001146:=n->2^(2^n): seq(A001146(n), n=0..9); # Wesley Ivan Hurt, Sep 19 2014

2^2^Range[0,10] (* Harvey P. Dale, Jul 20 2011 *)

a(n)=1<<2^n \\ Charles R Greathouse IV, Jul 25 2011

a(n)=2^2^n \\ Charles R Greathouse IV, Oct 03 2012

def A001146(n): return 1<<(1<Chai Wah Wu, Mar 14 2023

a(n+1) = (a(n))^2.
1 = Sum_{n>=0} a(n)/A051179(n+1) = 2/3 + 4/15 + 16/255 + 256/65535, ..., with partial sums: 2/3, 14/15, 254/255, 65534/65535, ... - Gary W. Adamson, Jun 15 2003
a(n) = A000079(A000079(n)). - Robert Israel, Jan 15 2015
Sum_{n>=0} 1/a(n) = A007404. - Amiram Eldar, Oct 14 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = 2.
Product_{n>=0} (1 - 1/a(n)) = A215016. (End)

A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 4, 14, 16, 8, 1, 16, 112, 324, 508, 474, 268, 88, 16, 1, 256, 3584, 22912, 88832, 233936, 443936, 628064, 675456, 557492, 353740, 171644, 62878, 17000, 3264, 416, 32, 1, 65536, 1835008, 24576000, 209715200, 1281482752, 5974786048, 22114709504, 66752724992

Offset: 1

Vladimir Shevelev, Dec 11 2010

Length of the first row is 2; for i>=2, length of the i-th row is 2^{i-2}+1.

			Triangle begins:
   1,   1;
   2,   1;
   2,   4,   1;
   4,  14,  16,   8,   1;
  16, 112, 324, 508, 474, 268, 88, 16, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..1035
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A067686, A110360, A000027, A005408, A056220, A177888.

First column gives: A165420.

p:= proc(n) option remember;
       z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
deg:= n-> `if`(n=0, 1, 2^(n-1)):
T:= (n,k)-> coeff(p(n)(z), z, deg(n)-k):
seq(seq(T(n,k), k=0..deg(n)), n=0..6); # Alois P. Heinz, Dec 13 2010

P[0][z_] := z + 1;
P[n_][z_] := P[n][z] = z + Product[P[k][z], {k, 0, n-1}];
row[n_] := CoefficientList[P[n][z], z] // Reverse;
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

Another recursion is: P_n(z)=z+P_(n-1)(z)(P_(n-1)(z)-z).

Private values: P_n(0)=1; P_n(-1)=delta_(n,0)-1; {P_n(1)}=A000058; {P_n(2)}=A000215; {P_n(3)}={A000289(n+1)}; {P_n(4)}={A000324(n+1)}; {P_n(5)}={A001543(n+1)}; {P_n(6)}={A001544(n+1)}; {P_n(7)}={A067686(n)}; {P_n(8)}={A110360(n)}; {P_0(n)}={A000027(n+1)}; {P_1(n)}={A005408(n)}; {P_2(n)}={A056220(n+1)}.

More terms from Alois P. Heinz, Dec 13 2010

A320603 a(0) = 1; if n is odd, a(n) = Product_{i=0..n-1} a(i); if n is even, a(n) = Sum_{i=0..n-1} a(i).

Original entry on oeis.org

1, 1, 2, 2, 6, 24, 36, 20736, 20808, 8947059130368, 8947059171984, 716210897494804754044764041567551881216, 716210897494804754044764059461670225184

Offset: 0

Iain Fox, Oct 17 2018

nonn

Next term is too large to include.

Odd terms are the product of previous terms and even terms are the sum of previous terms.

			5 is odd, so a(5) = 1 * 1 * 2 * 2 * 6 = 24.
6 is even, so a(6) = 1 + 1 + 2 + 2 + 6 + 24 = 36.

Iain Fox, Table of n, a(n) for n = 0..18
Iain Fox, Table of n, a(n) for n = 0..32

Cf. A039941, A077753, A122961.
Sum of previous terms: A011782.
Product of previous terms: A165420.

a[0]:= 1; a[n_]:= If[OddQ[n], Product[a[j], {j,0,n-1}], Sum[a[j], {j,0,n -1}]]; Table[a[n], {n, 0, 15}] (* G. C. Greubel, Oct 19 2018 *)

first(n) = my(res = vector(n, i, 1)); for(x=3, n, res[x]=if(x%2, sum(i=1, x-1, res[i]), prod(i=1, x-1, res[i]))); res

first(n) = my(res = vector(n, i, 1)); res[3]++; for(x=4, n, res[x]=if(x%2, res[x-1]+2*res[x-2], res[x-1]*res[x-2]^2)); res

a(n) = a(n-1) + 2*a(n-2), for even n > 2.

a(n) = a(n-1) * a(n-2)^2, for odd n > 1.

A171163 Number of children at height n in a doubly logarithmic tree. Leaves are at height 0.

Original entry on oeis.org

0, 2, 2, 4, 16, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 0

Chad Brewbaker, Dec 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..13
Omer Berkman, Baruch Schieber and Uzi Vishkin, Optimal doubly logarithmic parallel algorithms based on finding all nearest smaller values, Journal of Algorithms, 14(1993)(3): 344-370.

Equals A001146 with the prefix 0, 2.

Essentially the same as A165420.

Join[{0, 2}, Table[2^2^(n-2), {n, 2, 9}]] (* Harvey P. Dale, Feb 01 2014 *)

def doubly_log_tree_children(n):
    if n==0:
        return 0
    if n==1:
        return 2
    return 2**(2**(n-2))

a(0)=0, a(1)=2; for n>1, a(n) = 2^(2^(n-2)).

More terms from Harvey P. Dale, Feb 01 2014

cp's OEIS Frontend

A001146 a(n) = 2^(2^n).

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A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

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A320603 a(0) = 1; if n is odd, a(n) = Product_{i=0..n-1} a(i); if n is even, a(n) = Sum_{i=0..n-1} a(i).

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A171163 Number of children at height n in a doubly logarithmic tree. Leaves are at height 0.

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