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A098180 Odd numbers with twice the odd numbers repeated in order between them.

%I A098180 #24 Sep 08 2022 08:45:14
%S A098180 1,2,2,3,5,6,6,7,9,10,10,11,13,14,14,15,17,18,18,19,21,22,22,23,25,26,
%T A098180 26,27,29,30,30,31,33,34,34,35,37,38,38,39,41,42,42,43,45,46,46,47,49,
%U A098180 50,50,51,53,54,54,55,57,58,58,59,61,62,62,63,65,66,66,67,69,70,70,71
%N A098180 Odd numbers with twice the odd numbers repeated in order between them.
%C A098180 Partial sums of A098178.
%C A098180 Also A042968 with the even terms repeated. - _Michel Marcus_, Apr 14 2015
%C A098180 Fixed points are [2,3,6,7,10,11,..] = A042964. - _Wesley Ivan Hurt_, Oct 13 2015
%H A098180 Iain Fox, <a href="/A098180/b098180.txt">Table of n, a(n) for n = 0..10000</a>
%H A098180 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A098180 G.f.: (1+x)(1-x+x^2)/((1-x)^2(1+x^2)).
%F A098180 a(n) = sqrt(2)*sin(Pi*n/2+Pi/4)/2+n+1/2.
%F A098180 a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4), n>3.
%F A098180 From _Wesley Ivan Hurt_, Apr 12 2015, Oct 13 2015: (Start)
%F A098180 a(n) = (2n+1-(-1)^((n+1)*(n+2)/2))/2.
%F A098180 a(n) = n + A002265(n) - A002265(n-2). (End)
%F A098180 E.g.f: (exp(-i*x)*((1+i) + (1-i)*exp(2*i*x) + exp((1+i)*x)*(2+4*x)))/4, where i = sqrt(-1). - _Iain Fox_, Oct 17 2018
%p A098180 A098180:=n->(2*n+1-(-1)^((n+1)*(n+2)/2))/2: seq(A098180(n), n=0..100); # _Wesley Ivan Hurt_, Apr 12 2015
%t A098180 Table[(2 n + 1 - (-1)^((n + 1) (n + 2)/2))/2, {n, 0, 40}] (* _Wesley Ivan Hurt_, Apr 12 2015 *)
%o A098180 (Magma) [Floor((2*n+1-(-1)^((n+1)*(n+2)/2))/2): n in [0..80]]; // _Vincenzo Librandi_, Apr 13 2015
%o A098180 (PARI) first(n) = Vec((1+x)*(1-x+x^2)/((1-x)^2*(1+x^2)) + O(x^n)) \\ _Iain Fox_, Oct 17 2018
%o A098180 (PARI) a(n) = (2*n+1-(-1)^((n+1)*(n+2)/2))/2 \\ _Iain Fox_, Oct 17 2018
%Y A098180 Cf. A002265, A042964, A042968, A098178.
%K A098180 easy,nonn
%O A098180 0,2
%A A098180 _Paul Barry_, Aug 30 2004