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A291681 First differences of A067046.

%I A291681 #29 Feb 26 2018 16:57:50
%S A291681 1,8,0,25,-7,56,-24,105,-55,176,-104,273,-175,400,-272,561,-399,760,
%T A291681 -560,1001,-759,1288,-1000,1625,-1287,2016,-1624,2465,-2015,2976,
%U A291681 -2464,3553,-2975,4200,-3552,4921,-4199,5720,-4920,6601,-5719,7568,-6600,8625,-7567,9776,-8624
%N A291681 First differences of A067046.
%C A291681 a(2n) > 0 and a(2n+1) < 0 for all n >= 2.
%H A291681 Colin Barker, <a href="/A291681/b291681.txt">Table of n, a(n) for n = 1..1000</a>
%H A291681 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,3,-3,-3,1,1).
%F A291681 a(n) = A067046(n+1) - A067046(n).
%F A291681 G.f.: -x*(x^6+x^5-3*x^4-2*x^3+5*x^2+9*x+1)/((x-1)^3*(x+1)^4). - _Alois P. Heinz_, Sep 04 2017
%F A291681 From _Colin Barker_, Sep 29 2017: (Start)
%F A291681 a(n) = (n^3 + 9*n^2 + 20*n + 12) / 12 for n even.
%F A291681 a(n) = (-n^3 + 7*n + 6) / 12 for n odd.
%F A291681 (End)
%p A291681 a:= n-> `if`(irem(n-1, 2, 'r')=0, -(r-1)*(2*r+3)*(r+1)/3
%p A291681                                 , (2*r+3)*(r+4)*(r+2)/3):
%p A291681 seq(a(n), n=1..100);  # _Alois P. Heinz_, Sep 04 2017
%o A291681 (PARI) Vec(x*(1 + 9*x + 5*x^2 - 2*x^3 - 3*x^4 + x^5 + x^6) / ((1 - x)^3*(1 + x)^4) + O(x^60)) \\ _Colin Barker_, Sep 29 2017
%Y A291681 Cf. A067046.
%K A291681 sign,easy
%O A291681 1,2
%A A291681 _Enrique Navarrete_, Sep 04 2017