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%I #48 Sep 08 2022 08:46:13
%S 1,2,3,4,6,5,8,7,10,9,13,10,15,12,17,14,20,15,22,17,24,19,27,20,29,22,
%T 31,24,34,25,36,27,38,29,41,30,43,32,45,34,48,35,50,37,52,39,55,40,57,
%U 42,59,44,62,45,64,47,66,49,69,50,71,52,73,54,76,55,78
%N a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.
%C A260708 difference table rows have the same nine-step recurrence:
%C 0, 1, 3, 6, 10, 16, 21, 29, 36, 46, 55, 65, 78, 93, ...
%C 1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, ... = a(n)
%C 1, 1, 1, 2, -1, 3, -1, 3, -1, 4, -3, 5, -3, 5, ... = b(n)
%C 0, 0, 1, -3, 4, -4, 4, -4, 5, -7, 8, -8, 8, -8, ... (see A042965(n)).
%C (b(2n) + b(2n+1) = A052901(n+2).)
%H G. C. Greubel, <a href="/A260307/b260307.txt">Table of n, a(n) for n = 0..1500</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,0,0,1,0,-1).
%F a(2n) = A047282(n). a(2n+1) = A047212(n+1).
%F a(n) = A260708(n+1) - A260708(n).
%F a(n+6) = a(n) + period of length 2: repeat 7, 5.
%F a(2n) + a(2n+1) = 3 + 4*n.
%F a(n) = n + 1 + (-1)^n*A152467(n+2).
%F From _Colin Barker_, Nov 22 2015: (Start)
%F a(n) = a(n-2) + a(n-6) - a(n-8) for n>7.
%F G.f.: (x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)).
%F (End)
%t RecurrenceTable[{a[n] == a[n-2] + a[n-6] - a[n-8], a[0]=1, a[1]=2, a[2]=3, a[3]=4, a[4]=6, a[5]=5, a[6]=8, a[7]=7}, a, {n,0,100}] (* _G. C. Greubel_, Nov 23 2015 *)
%o (PARI) Vec((x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Nov 22 2015
%o (PARI) vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ _Altug Alkan_, Nov 24 2015
%o (Magma) I:=[1,2,3,4,6,5,8,7];[n le 8 select I[n] else Self(n-2) + Self(n-6) - Self(n-8): n in [1..70]]; // _Vincenzo Librandi_, Dec 26 2015
%Y Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Nov 22 2015
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