A113880 Variation on Recamán's sequence utilizing the four basic operations (/,-,+,*) in that order.

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 432, 407, 381, 354, 326, 297, 267, 236, 204, 171, 137, 102, 66, 29, 67, 28, 68, 27, 69, 26, 70, 115, 161, 114, 162, 113, 163, 112, 60, 3180, 3126, 3071, 3015, 2958, 51, 110, 50, 111, 49

Offset: 0

Author

Sergio Pimentel, Jan 27 2006

Keywords

• nonn

Comments

More precisely:

a(n) = a(n-1)/n if a(n-1)/n is an integer and not already in the sequence. Else:

a(n) = a(n-1)-n if a(n-1)-n is positive and not already in the sequence. Else:

a(n) = a(n-1)+n if a(n-1)+n is not already in the sequence. Else:

a(n) = a(n-1)*n if a(n-1)*n is not already in the sequence. Else STOP.

In other words, divide if you can, else subtract, else add, else multiply.

By a(1000) there are 3 division steps, 928 subtraction steps, 59 addition steps and 10 multiplication steps. It is unlikely that every number belongs to the sequence since there are many "holes". It is an open question if there are any repetitions after a multiplication step. Can anybody expand the series?

At a(2500000000) = 2285684529311288243, there have been 44 divisions, 2499821613 subtractions, 178253 additions, and 90 multiplications. The largest value seen was a(1926305697) = 3555357710450807490. No multiplication step has produced a duplicate term. - Benjamin Chaffin, Sep 22 2016

Examples

			a(24) = 432 because:
- a(23) = 18,
- 18/24 is not an integer,
- 18 - 24 is negative,
- 18 + 24 = 42 is already in the sequence,
- 18 * 24 = 432 is not already in the sequence.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to Recamán's sequence

Crossrefs

Cf. A005132.

Programs

Mathematica

f[s_List] := Block[{l = s[[-1]], n = Length@ s}, If[ IntegerQ[l/n] && !MemberQ[s, l/n], Append[s, l/n], If[l > n && !MemberQ[s, l - n], Append[s, l - n], If[ !MemberQ[s, l + n], Append[s, l + n], Append[s, l*n]]]]]; Nest[f, {0}, 62] (* Robert G. Wilson v, Sep 10 2016 *)

Python

from itertools import count, islice
def A113880(): # generator of terms
    aset, an = {0, 1}, 1
    yield from [0, 1]
    for n in count(2):
        q, r = divmod(an, n)
        if r == 0 and q not in aset: an = q
        elif (d:=an-n) > 0 and d not in aset: an = d
        elif (s:=an+n) not in aset: an = s
        elif (m:=an*n) not in aset: an = m
        else: return
        yield an
        aset.add(an)
print(list(islice(A113880(), 63))) # Michael S. Branicky, May 26 2023

Extensions

a(0) = 0 prepended by Robert G. Wilson v, Sep 10 2016