A053461 -id:A053461 - cp's OEIS frontend

A076042 a(0) = 0; thereafter a(n) = a(n-1) + n^2 if a(n-1) < n^2, otherwise a(n) = a(n-1) - n^2.

0, 1, 5, 14, 30, 5, 41, 90, 26, 107, 7, 128, 272, 103, 299, 74, 330, 41, 365, 4, 404, 845, 361, 890, 314, 939, 263, 992, 208, 1049, 149, 1110, 86, 1175, 19, 1244, 2540, 1171, 2615, 1094, 2694, 1013, 2777, 928, 2864, 839, 2955, 746, 3050, 649, 3149

Offset: 0

Amarnath Murthy, Oct 29 2002

Does not return to zero within first 2^25000 =~ 10^7525 terms.  Define an epoch as an addition followed by a sequence of (addition, subtraction) pairs.  The first epoch has length 1 (+), the second 3 (++-), the third 5 (++-+-), and so forth (cf. A324792).  The epoch lengths increase geometrically by about the square root of 3, and the value at the end of each epoch is the low value in the epoch.  These observations lead to the Python program given. - Tomas Rokicki, Aug 31 2019
Using the Maple program in A324791, I confirmed that a(n) != 0 for 0 < n < 10^2394. See the a- and b-files in A325056 and A324791. - N. J. A. Sloane, Oct 03 2019
'Easy Recamán transform' of the squares. - Daniel Forgues, Oct 25 2019

Tomas Rokicki, Table of n, a(n) for n = 0..20000

Cf. A076039, A003462, A076041, A046901.
See also A325056, A324791, A324792.
Cf. A053461 ('Recamán transform' of the squares).
Cf. A000290, A008344, A008345.

a:= proc(n) option remember; `if`(n<0, 0,
      ((s, t)-> s+`if`(sAlois P. Heinz, Jan 11 2020

a[0] = 0;
a[n_] := a[n] = a[n-1] + If[a[n-1] < n^2, n^2, -n^2];
a /@ Range[0, 50] (* Jean-François Alcover, Apr 11 2020 *)

v=vector(50); v[1]=1; for(n=2,50,if(v[n-1]

More terms from Ralf Stephan, Mar 20 2003
a(0)=0 prepended, at the suggestion of Allan C. Wechsler, by N. J. A. Sloane, Aug 31 2019
Offset set to 0, to cohere with previous action of N. J. A. Sloane, by Allan C. Wechsler, Sep 08 2019

A064365 a(0) = 0; thereafter a(n) = a(n-1)-prime(n) if positive and new, otherwise a(n) = a(n-1)+prime(n), where prime(n) is the n-th prime.

Original entry on oeis.org

0, 2, 5, 10, 3, 14, 1, 18, 37, 60, 31, 62, 25, 66, 23, 70, 17, 76, 15, 82, 11, 84, 163, 80, 169, 72, 173, 276, 383, 274, 161, 34, 165, 28, 167, 316, 467, 310, 147, 314, 141, 320, 139, 330, 137, 334, 135, 346, 123, 350, 121, 354, 115, 356, 105, 362, 99, 368, 97, 374, 93

Offset: 0

Neil Fernandez, Sep 25 2001

'Recamán transform' (see A005132) of the prime sequence. Note that the definition permits repeated terms [though only by addition] (and there are many repeated terms, just as there are in A005132).
Does every positive integer appear in the sequence? This seems unlikely, since 4 has not appeared in 70000 terms.
Note: this is similar to Clark Kimberling's A022831, except in the latter sequence the words 'and new' have been omitted.
The smallest numbers not occurring in the first million terms: 4, 6, 7, 12, 13, 16, 19, 20, 21, 22, 24, 26, 27, 29, 30, 32, 36, 39, 41, 42. - Reinhard Zumkeller, Apr 26 2012

			To find a(9) we try subtracting the 9th prime, which is 23, from a(8), which is 37. 37 - 23 = 14, but 14 is already in the sequence (it is a(5)), so we must add. a(9) = 37 + 23 = 60.

N. J. A. Sloane, First 70000 terms
Index entries for sequences related to Recamán's sequence

Cf. A005132, A022831, A117128.

import Data.Set (singleton, notMember, insert)
a064365 n = a064365_list !! n
a064365_list = 0 : f 0 a000040_list (singleton 0) where
   f x (p:ps) s | x' > 0 && x' `notMember` s = x' : f x' ps (insert x' s)
                | otherwise                  = xp : f xp ps (insert xp s)
                where x' = x - p; xp = x + p
-- Reinhard Zumkeller, Apr 26 2012

a = {0}; Do[ If[ a[ [ -1 ] ] - Prime[ n ] > 0 && Position[ a, a[ [ -1 ] ] - Prime[ n ] ] == {}, a = Append[ a, a[ [ -1 ] ] - Prime[ n ] ], a = Append[ a, a[ [ -1 ] ] + Prime[ n ] ] ], {n, 1, 70} ]; a (* Modified by Ivan N. Ianakiev, Aug 05 2019, to accommodate the new initial term of a(0). *)

A064365(N,s/*=1 to print all terms*/)={ my(a=0,u=0); N & forprime(p=1,prime(N), s & print1(a","); u=bitor(u,2^a+=if(a<=p || bittest(u,a-p),p,-p)));a}  \\ M. F. Hasler, Mar 07 2012

from sympy import primerange, prime
def aupton(terms):
  alst = [0]
  for n, pn in enumerate(primerange(1, prime(terms)+1), start=1):
    x = alst[-1] - pn
    alst += [x if x > 0 and x not in alst else alst[-1] + pn]
  return alst
print(aupton(60)) # Michael S. Branicky, May 30 2021

a(n) = A117128(n) - 1. - Thomas Ordowski, Dec 05 2016

More terms from Robert G. Wilson v, Sep 26 2001
Further terms from N. J. A. Sloane, Feb 10 2002
Added initial term a(0)=0, in analogy with A128204, A005132, A053461, A117073/A078783. - M. F. Hasler, Mar 07 2012

A128204 a(0) = 0; a(n) = a(n-1) - (2n-1) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1) + (2n-1).

Original entry on oeis.org

0, 1, 4, 9, 2, 11, 22, 35, 20, 3, 22, 43, 66, 41, 14, 43, 12, 45, 10, 47, 8, 49, 6, 51, 98, 147, 96, 149, 94, 37, 96, 157, 220, 155, 88, 19, 90, 17, 92, 15, 94, 13, 96, 181, 268, 179, 270, 177, 82, 179, 80, 181, 78, 183, 76, 185, 74, 187, 72, 189, 70, 191, 68, 193, 320

Offset: 0

Nick Hobson, Feb 19 2007

'Recamán transform' (see A005132) of the odd numbers.

			Consider n=7. We have a(6)=22 and try to subtract 13, the 7th odd number. The result, 9, is certainly positive, but we cannot use it because 9 is already in the sequence. So we must add 13 instead, getting a(7) = 22 + 13 = 35.

Cf. A005132, A053461, A064365, A123483.

A128204(N,s/*=1 to print all terms*/)={my(a=0,u=0);  for( n=1,N, s&print1(a","); u=bitor(u,2^a+=if(a<2*n || bittest(u,a+1-2*n), 2*n-1,1-2*n)));a} \\ M. F. Hasler, Mar 07 2012

cp's OEIS Frontend

A076042 a(0) = 0; thereafter a(n) = a(n-1) + n^2 if a(n-1) < n^2, otherwise a(n) = a(n-1) - n^2.

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A064365 a(0) = 0; thereafter a(n) = a(n-1)-prime(n) if positive and new, otherwise a(n) = a(n-1)+prime(n), where prime(n) is the n-th prime.

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A128204 a(0) = 0; a(n) = a(n-1) - (2n-1) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1) + (2n-1).

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