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A224839 a(1) = 1 and a(n) is the smallest number greater than a(n-1) with no square difference with any preceding term.

%I A224839 #36 Mar 05 2025 20:38:45
%S A224839 1,3,6,8,11,13,16,18,21,23,35,40,45,53,58,63,66,68,73,86,96,110,120,
%T A224839 125,128,131,133,138,143,148,151,171,178,181,183,188,193,198,205,211,
%U A224839 216,239,244,250,256,258,261,263,268,273
%N A224839 a(1) = 1 and a(n) is the smallest number greater than a(n-1) with no square difference with any preceding term.
%C A224839 The first 11 terms are identical to those of A210570.
%C A224839 This sequence represents the losing positions in the misère version of a one-heap nim game, where players remove a square number of objects (> 0) on their turn. - _Kiran Ananthpur Bacche_, Feb 23 2025
%H A224839 Jean-François Alcover, <a href="/A224839/b224839.txt">Table of n, a(n) for n = 1..1000</a>
%F A224839 a(n) >= A210570(n) by definition of the latter. - _Charles R Greathouse IV_, Nov 28 2024
%F A224839 a(n) = A030193(n-1) + 1. - _Kiran Ananthpur Bacche_, Feb 23 2025
%p A224839 N:= 1000: # to get all terms <= N
%p A224839 V:= Vector(N):
%p A224839 Res:= NULL:
%p A224839 for m from 1 to N do
%p A224839   if V[m] = 0 then
%p A224839     V[m]:= 1;
%p A224839     Res:= Res, m;
%p A224839     for k from 1 to floor(sqrt(N-m)) do V[m+k^2]:= 1 od:
%p A224839   fi
%p A224839 od:
%p A224839 Res; # _Robert Israel_, Nov 30 2016
%t A224839 a[1] = 1; a[n_] := a[n] = For[k = a[n-1]+1, True, k++, If[FreeQ[k - Array[a, n-1], d_ /; IntegerQ[Sqrt[d]]], Return[k]]]; Table[a[n], {n, 1, 40}]
%o A224839 (Haskell)
%o A224839 a224839 n = a224839_list !! (n-1)
%o A224839 a224839_list = f [1..] [] where
%o A224839    f (x:xs) ys = if all ((== 0) . a010052) $ map (x -) ys
%o A224839                     then x : f xs (x:ys) else f xs ys
%o A224839 -- _Reinhard Zumkeller_, May 02 2014
%Y A224839 Cf. A030193, A210570, A010052.
%K A224839 nonn
%O A224839 1,2
%A A224839 _Jean-François Alcover_, Sep 18 2013