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%I #24 Jan 12 2023 01:50:01
%S 1,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,
%T 4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,
%U 2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2,3,4,3,2
%N Expansion of (1 - x^2)^2 * (1 - x^3) / ((1 - x)^3 * (1 - x^4)) in powers of x.
%C Coordination sequence for a chain of hexagons joined by single edges. - _N. J. A. Sloane_, Nov 21 2019
%H G. C. Greubel, <a href="/A164358/b164358.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1).
%F a(n) = 3*b(n) unless n=0 where b(n) is multiplicative with b(2) = 4/3, b(2^e) = 2/3 if e>1, b(p^e) = 1 if p>2.
%F Euler transform of length 4 sequence [3, -2, -1, 1].
%F Moebius transform is length 4 sequence [3, 1, 0, -2].
%F a(n) = a(-n) for all n in Z. a(n+4) = a(n) unless n=0 or n=-4. a(4*n) == 2 unless n=0. a(2*n + 1) = 3. a(4*n + 2) = 4.
%F G.f.: -1 + 3 / (1 - x) - 1 / (1 + x^2).
%F G.f.: (1+x)*(1+x+x^2)/((1-x)*(1+x^2)). - _N. J. A. Sloane_, Nov 21 2019
%e G.f. = 1 + 3*x + 4*x^2 + 3*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 3*x^7 + 2*x^8 + ...
%t a[ n_] := - Boole[n == 0] + 3 - If[ EvenQ[n], (-1)^(n/2), 0];
%t CoefficientList[Series[(1+3*x+4*x^2+3*x^3+x^4)/(1-x^4), {x, 0, 150}], x] (* _G. C. Greubel_, Sep 26 2018 *)
%o (PARI) {a(n) = -(n==0) + 3 - if( n%2 == 0, (-1)^(n/2), 0)};
%o (PARI) x='x+O('x^150); Vec((1+3*x+4*x^2+3*x^3+x^4)/(1-x^4)) \\ _G. C. Greubel_, Sep 26 2018
%o (Magma) m:=150; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x+4*x^2+3*x^3+x^4)/(1-x^4))); // _G. C. Greubel_, Sep 26 2018
%K nonn,easy
%O 0,2
%A _Michael Somos_, Aug 13 2009
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