A006894 -id:A006894 - cp's OEIS frontend

A064065 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=0, a(1)=1.

0, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 2, 3, 3, 3, 4, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 5, 4, 5, 5, 5, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 6, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 4, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 5, 5, 3, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 5, 5

Offset: 0

Henry Bottomley, Aug 31 2001

nonn

Each positive number appears an infinite number of times: e.g., a(k)=1 whenever k-1 is in A006894.

			Start with (0,1). So after initial step have (0, *1*, 0+1 = 1), then (0, 1, *1*, 0+1 = 1, 1+1 = 2), then (0, 1, 1, *1*, 2, 0+1 = 1, 1+1 = 2, 1+1 = 2), then (0, 1, 1, 1, *2*, 1, 2, 2, 0+2 = 2, 1+2 = 3, 1+2 = 3, 1+2 = 3), etc.

Cf. A064064, A064066, A064067.

A356082 Matula-Goebel number of the complete binary tree of n levels.

Original entry on oeis.org

1, 4, 49, 51529, 400034745289, 135016053798647886015597889

Offset: 1

Kevin Ryde, Jul 26 2022

An estimate for a(7) is 7.304058*10^55. - Hugo Pfoertner, Jul 26 2022

			For n=3, the complete binary tree of 3 levels is
        49
      /    \     a(3) = prime(4)^2
    4       4         = 49
   / \     / \
  1   1   1   1

Cf. A006894 (Colijn-Plazzotta), A084107 (balanced binary).

Cf. A356083 (ternary), A356084 (quaternary).

a(n) = my(ret=1); for(i=2,n, ret=prime(ret)^2); ret;

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

a(6) from Rémy Sigrist, Jul 26 2022

A067339 Divide the natural numbers in sets of consecutive numbers, starting with {1,2}, each set with number of elements equal to the sum of elements of the preceding set. The final element of the n-th set gives a(n).

Original entry on oeis.org

2, 5, 17, 155, 12092, 73114280, 2672849006516342, 3572060905817699556013859788655, 6379809557435582128907282471160505774257452233828787563248842

Offset: 1

Floor van Lamoen, Jan 16 2002

The sets begin {1, 2}, {3, 4, 5}, {6, 7, 8, ..., 17}, ...

Cf. A006894, A002658. Partial sums of A067338.

RecurrenceTable[{a[n] == a[n-1]*(a[n-1]+1)/2 + 2, a[1]==2}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 09 2015 *)
NestList[(#(#+1))/2+2&,2,10] (* Harvey P. Dale, Jun 17 2017 *)

a(n) = if(n>1,a(n-1)*(a(n-1)+1)/2)+2 \\ Edited by M. F. Hasler, Jan 23 2015

vector(10,i,if(i>1,n=n*(a+a-n+1)/2;a+=n,n=a=2)) \\ M. F. Hasler, Jan 23 2015

a(n)=a(n-1)*(a(n-1)+1)/2 + 2
a(n)=a(n-1)+A067338(n). - M. F. Hasler, Jan 23 2015
a(n) ~ 2 * c^(2^n), where c = 1.312718001584962838462131787518361199185077166417566246117... . - Vaclav Kotesovec, Dec 09 2015

More terms from Jason Earls, Jan 16 2002

A283793 Number of elements formable in <= n steps, starting with 4 elements, combining 2 elements into a new element at each step.

Original entry on oeis.org

4, 14, 109, 5999, 17997004, 161946085486514, 13113267302202731189080679359, 85978889669509647874887802052390686151982448025024665124

Offset: 0

Michael Turniansky, Mar 16 2017

nonn

In a game such as Doodle God (see links), you start with Earth, Air, Fire and Water, and combine them two at a time (including combining an element with itself) into new elements. A(n) is the hypothetical maximum number of different possible elements you could reach from clicking at most n times.

This list is akin to A006894, which is the sequence if we started with 1 element instead of 4.

			Starting with {A, B, C, D}, we can make {AA, AB, AC, AD, BB, BC, BD, CC, CD, and DD}.  The union of these two sets has cardinality 14 = a(1).

Michael Turniansky, Table of n, a(n) for n = 0..9
Anton Rybakov as Joybits Ltd., Doodle God game homepage
Cary Kaiming Huang, Elements 3. An online version which is not bounded by predefined elements, so a(n) could theoretically be reached. (No longer works.)

Cf. A006894.

a[0]=4; a[n_] := a[n] = 4 + a[n-1] (a[n-1] + 1)/2; a /@ Range[0, 7] (* Giovanni Resta, Mar 16 2017 *)

a(n) = if(n<1, 4, 4 + a(n - 1) * (a(n - 1) + 1) / 2);
for(n=0, 7, print1(a(n),", ")) \\ Indranil Ghosh, Mar 16 2017

a(n) = 4 + T(a(n-1)) where T(m) is the m-th triangular number.

A363257 a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 31, 257, 16642, 69247363, 1198799355237125, 359279973529237254190922184970, 32270524844792355518177347536627638351478874995525184567711

Offset: 0

Harry Richman, May 23 2023

nonn

Iterated application of A033638, with a shift.

Andrew Howroyd, Table of n, a(n) for n = 0..16
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. A003095, A006894, A014980, A033638.

a(n) = if(n < 1, 0, floor( ((a(n-1) + 1) / 2)^2 ) + 1) \\ Andrew Howroyd, Jan 01 2024

a(n) = A033638(a(n-1)+1) for n > 0.

log a(n) ~ C * 2^n for some constant C.

A173498 Partial sums of A005588.

Original entry on oeis.org

2, 9, 61, 2194, 2592601, 3374954133663, 5695183504482491594510172, 16217557574922386301420519886707289378131172220652, 131504586847961235687181874578063117114329409897566535831366955410641808739121788386036154689297602

Offset: 1

Jonathan Vos Post, Feb 19 2010

nonn

Partial sums of number of free binary rooted trees of height n. The subsequence of primes in this partial sum begins: 2, 61, no more through a(12).

			a(9) = 2 + 7 + 52 + 2133 + 2590407 + 3374951541062 + 5695183504479116640376509 + 16217557574922386301420514191523784895639577710480 + 131504586847961235687181874578063117114329409897550318273792033024340388219235081096658023517076950.

Cf. A005588, A002658, A006894.

a(n) = Sum_{i=1..n} A005588(i).

cp's OEIS Frontend

A064065 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=0, a(1)=1.

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A356082 Matula-Goebel number of the complete binary tree of n levels.

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A067339 Divide the natural numbers in sets of consecutive numbers, starting with {1,2}, each set with number of elements equal to the sum of elements of the preceding set. The final element of the n-th set gives a(n).

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A283793 Number of elements formable in <= n steps, starting with 4 elements, combining 2 elements into a new element at each step.

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A363257 a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.

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A173498 Partial sums of A005588.

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