|
%I #27 Sep 10 2023 01:42:29
%S 3,0,5,2,7,4,9,6,11,8,13,10,15,12,17,14,19,16,21,18,23,20,25,22,27,24,
%T 29,26,31,28,33,30,35,32,37,34,39,36,41,38,43,40,45,42,47,44,49,46,51,
%U 48,53,50,55,52,57,54,59,56,61,58,63,60,65,62,67,64,69,66,71,68,73,70
%N a(n) = nimsum(n+(n+1)+(n+2)).
%C Start with 3. Then repeat the cycle: subtract 3, add 5. The odd-indexed terms give the odd numbers, beginning with 3. The even-indexed terms give the even numbers, beginning with 0. In the infinite sequence, every positive integer except 1 is listed.
%H Colin Barker, <a href="/A165754/b165754.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = 2*n-a(n-1)+1 (with a(0)=3). - _Vincenzo Librandi_, Dec 02 2010
%F a(n) = 1 + n + 2*(-1)^n. - _R. J. Mathar_, Dec 02 2010
%F From _Colin Barker_, Nov 05 2015: (Start)
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.
%F G.f.: (2*x^2-3*x+3) / ((x-1)^2*(x+1)). (End)
%F Sum_{n>=2} (-1)^(n+1)/a(n) = 4/3 - log(2). - _Amiram Eldar_, Sep 10 2023
%e For n = 3, Nimsum(3 + 4 + 5) = 2, as shown: 011 XOR 100 XOR 101 010.
%p read("transforms") ; A165754 := proc(n) nimsum(nimsum(n,n+1),n+2) ; end: seq(A165754(n),n=0..120) ; # _R. J. Mathar_, Sep 28 2009
%t Flatten[NestList[{Last[#]+5,Last[#]+2}&,{3,0},40]] (* _Harvey P. Dale_, Dec 04 2011 *)
%o (Python)
%o n = 0
%o while n < 100:
%o print(n^(n+1)^(n+2), end=',')
%o n += 1
%o (PARI) Vec((2*x^2-3*x+3)/((x-1)^2*(x+1)) + O(x^100)) \\ _Colin Barker_, Nov 05 2015
%Y Cf. A065168. - _R. J. Mathar_, Sep 28 2009
%K nonn,easy
%O 0,1
%A Mick Purcell (mickpurcell(AT)gmail.com), Sep 26 2009
%E Extended by _R. J. Mathar_, Sep 28 2009
|
