A033627 0-additive sequence: not the sum of any previous pair.

1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175

Offset

1,2

Comments

Conjecture: a(n+1) is the number of distinct numbers of steps required for the last n digits of integers to repeat themselves by iterating the map m -> m^2 + 1. - Ya-Ping Lu, Oct 19 2021

References

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section C4, pp. 166-167.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Igor Dolinka, James East, and Robert D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015-2016 (A sequence in Table 5 appears to match this. - N. J. A. Sloane, Sep 17 2016)

Eric Weisstein's World of Mathematics, Stöhr Sequence.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

2 together with numbers of form 3k+1 (A016777).

From Gary W. Adamson, May 10 2008: (Start)

Equals binomial transform of [1, 1, 1, 0, -1, 2, -3, 4, -5, 6, -7, ...].

Equals sum of antidiagonal terms of the following arithmetic array: 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, ... 1, 3, 5, 7, 9, ... . (End)

From Colin Barker, Sep 19 2012: (Start)

a(n) = 3*n - 5, for n > 2.

a(n) = 2*a(n-1) - a(n-2), for n > 4;

G.f.: x*(1+x^2+x^3)/(1-x)^2. (End)

E.g.f.: 5 + 3*x + x^2/2 + exp(x)*(3*x - 5). - Stefano Spezia, Apr 15 2023

Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - Pi/(3*sqrt(3)) - log(2)/3. - Amiram Eldar, Dec 27 2025

a(n) = A016789(n-2)-1 for all n > 2. - Ali Sada and M. F. Hasler, Jun 03 2026

Mathematica

Join[{1, 2}, Range[4, 200, 3]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
(* Alternative: *)
f[s_List] := Block[{k = s[[-1]] + 1, ss = Union[ Plus @@@ Subsets[s, {2}]]}, While[ MemberQ[ ss, k], k++]; Append[ s, k]]; Nest[f, {1}, 70] (* Robert G. Wilson v, Jun 23 2014 *)
(* Alternative: *)
CoefficientList[Series[x(1+x^2+x^3)/(1-x)^2 , {x, 0, 70}], x] (* Stefano Spezia, Oct 04 2018 *)

Prog

(Haskell)
import Data.List ((\\))
a033627 n = a033627_list !! (n-1)
a033627_list = f [1..] [] where
   f (x:xs) ys = x : f (xs \\ (map (+ x) ys)) (x:ys)
-- Reinhard Zumkeller, Jan 11 2012
(PARI) a(n)=if(n>2, 3*n-5, n) \\ Charles R Greathouse IV, Sep 01 2016
(Python)
def a(n): return 3*n-5 if n > 2 else n
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Jun 09 2025

Crossrefs

Cf. A002858, A010672, A016777, A016789.

See A244151 for another version.

Sequence in context: A190279 A186325 A272059 * A066512 A304116 A226596

Adjacent sequences: A033624 A033625 A033626 * A033628 A033629 A033630

Keyword

nonn,easy

Author

Jud McCranie

Status

approved