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A131717 Natural numbers A000027 with 6n+4 and 6n+5 terms swapped.

%I A131717 #44 Jan 01 2024 23:49:44
%S A131717 1,2,3,5,4,6,7,8,9,11,10,12,13,14,15,17,16,18,19,20,21,23,22,24,25,26,
%T A131717 27,29,28,30,31,32,33,35,34,36,37,38,39,41,40,42,43,44,45,47,46,48,49,
%U A131717 50,51,53,52,54,55,56,57,59,58,60,61,62,63,65,64,66,67,68,69,71,70,72
%N A131717 Natural numbers A000027 with 6n+4 and 6n+5 terms swapped.
%C A131717 Hexaperiodic differences: 1, 1, 2, -1, 2, 1; 0, 1, -3, 3, -1, 0 (even palindromic signed); 1,-4, 6, -4, 1, 0.
%H A131717 Frieder Mittmann, <a href="/A131717/b131717.txt">Table of n, a(n) for n = 1..1002</a>
%H A131717 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).
%F A131717 a(n) = A008585(n/3) if n is congruent to 0 mod 3. - _Frieder Mittmann_, Nov 11 2014
%F A131717 a(n) = A007310((n-1)/3) if n is congruent to 1 mod 3. - _Frieder Mittmann_, Nov 11 2014
%F A131717 a(n) = A047235((n-2)/3) if n is congruent to 2 mod 3. - _Frieder Mittmann_, Nov 11 2014
%F A131717 G.f.: x*(2*x^5-x^4+2*x^3+x^2+x+1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - _Colin Barker_, Nov 11 2014
%F A131717 a(n) = (24*floor(n/6)-3*(n^2-3*n-2)-9*floor(n/3)*(3*floor(n/3)-2*n+3)+(-1)^floor(n/3)*(3*n^2-5*n-6+3*floor(n/3)*(9*floor(n/3)-6*n+5)))/4. - _Luce ETIENNE_, Apr 18 2017
%p A131717 seq(seq(6*i+s,s=[1,2,3,5,4,6]),i=0..100); # _Robert Israel_, Nov 11 2014
%t A131717 Drop[CoefficientList[Series[x (2x^5 - x^4 + 2x^3 + x^2 + x + 1)/((x - 1)^2 (x + 1) (x^2 - x + 1) (x^2 + x + 1)), {x, 0, 100}], x], 1] (* _Indranil Ghosh_, Apr 18 2017 *)
%t A131717 Table[Sum[(7 #1 - 13 #2 + 17 #3 - 3 #4 + 2 #5 + 2 #6)/30 & @@ Mod[k + Range[0, 5], 6], {k, 0, n}], {n, 0, 71}] (* _Michael De Vlieger_, Apr 22 2017 *)
%o A131717 (PARI) Vec(x*(2*x^5-x^4+2*x^3+x^2+x+1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Nov 11 2014
%Y A131717 Cf. A131042.
%K A131717 nonn,easy
%O A131717 1,2
%A A131717 _Paul Curtz_, Sep 15 2007