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%I #38 Jul 26 2022 22:02:31
%S 1,0,0,0,2,-2,2,0,0,0,3,-3,3,0,0,0,4,-4,4,0,0,0,5,-5,5,0,0,0,6,-6,6,0,
%T 0,0,7,-7,7,0,0,0,8,-8,8,0,0,0,9,-9,9,0,0,0,10,-10,10,0,0,0,11,-11,11,
%U 0,0,0,12,-12,12,0,0,0,13,-13,13,0,0,0,14
%N Expansion of tri-digit zeros interlaced with an arithmetic progression of positive and negative numbers.
%C This sequence is observed as the second difference in the expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^6)) (A029000), a sequence noted for its interlaced and structural coordination numbers.
%H Colin Barker, <a href="/A258133/b258133.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1,1,0,1,1,0,-1,-1).
%F a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-6) + a(n-7) - a(n-9) - a(n-10). a(6*k-2) = -a(6*k-1) = a(6*k) = k+1 for k >= 1.
%F G.f.: (x^9-x^7-x^6+x^4-x^3+x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2). - _Colin Barker_, May 24 2015
%F a(n) = -a(-14-n) for all n in Z. - _Michael Somos_, Jun 07 2015
%e G.f. = 1 + 2*x^4 - 2*x^5 + 2*x^6 + 3*x^10 - 3*x^11 + 3*x^12 + 4*x^16 + ...
%t a[ n_] := With[ {m=n-1}, If[ OddQ[ Quotient[ m, 3]], Quotient[ m+9, 6] (-1)^Mod[m, 3], 0]]; (* _Michael Somos_, Jun 07 2015 *)
%o (PARI) Vec((x^9-x^7-x^6+x^4-x^3+x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2) + O(x^100)) \\ _Colin Barker_, May 24 2015
%o (PARI) {a(n) = n--; if(n\3%2, (n+9)\6 * (-1)^(n%3), 0)}; /* _Michael Somos_, Jun 07 2015 */
%K sign,easy
%O 0,5
%A _Avi Friedlich_, May 21 2015
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