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A260307 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.

%I A260307 #48 Sep 08 2022 08:46:13
%S A260307 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 A260307 31,24,34,25,36,27,38,29,41,30,43,32,45,34,48,35,50,37,52,39,55,40,57,
%U A260307 42,59,44,62,45,64,47,66,49,69,50,71,52,73,54,76,55,78
%N A260307 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 A260307 A260708 difference table rows have the same nine-step recurrence:
%C A260307 0, 1, 3,  6, 10, 16, 21, 29, 36, 46, 55, 65, 78,  93, ...
%C A260307 1, 2, 3,  4,  6,  5,  8,  7, 10,  9, 13, 10, 15,  12, ...     = a(n)
%C A260307 1, 1, 1,  2, -1,  3, -1,  3, -1,  4, -3,  5, -3,   5, ...     = b(n)
%C A260307 0, 0, 1, -3,  4, -4,  4, -4,  5, -7,  8, -8,  8,  -8, ... (see A042965(n)).
%C A260307 (b(2n) + b(2n+1) = A052901(n+2).)
%H A260307 G. C. Greubel, <a href="/A260307/b260307.txt">Table of n, a(n) for n = 0..1500</a>
%H A260307 <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 A260307 a(2n) = A047282(n). a(2n+1) = A047212(n+1).
%F A260307 a(n) = A260708(n+1) - A260708(n).
%F A260307 a(n+6) = a(n) + period of length 2: repeat 7, 5.
%F A260307 a(2n) + a(2n+1) = 3 + 4*n.
%F A260307 a(n) = n + 1 + (-1)^n*A152467(n+2).
%F A260307 From _Colin Barker_, Nov 22 2015: (Start)
%F A260307 a(n) = a(n-2) + a(n-6) - a(n-8) for n>7.
%F A260307 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 A260307 (End)
%t A260307 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 A260307 (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 A260307 (PARI) vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ _Altug Alkan_, Nov 24 2015
%o A260307 (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 A260307 Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.
%K A260307 nonn,easy
%O A260307 0,2
%A A260307 _Paul Curtz_, Nov 22 2015