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A195417 a(n) = sum of the absolute values of the coefficients in (1+x-x^3)^n.

%I A195417 #11 Jan 05 2023 04:41:38
%S A195417 1,3,9,25,63,171,471,1193,3209,8841,22537,60393,165489,424451,1134849,
%T A195417 3093893,7987289,21288933,57973543,150409931,399788279,1085870421,
%U A195417 2831439537,7501994181,20330726077,53274740933,140615421393,380366331549,1002078031627,2634444052559
%N A195417 a(n) = sum of the absolute values of the coefficients in (1+x-x^3)^n.
%C A195417 What is the limit a(n)^(1/n)?  The limit appears to be near 2.657...
%C A195417 The limit a(n+1)/a(n) does not exist, but seems to cycle between attractors.
%H A195417 Vaclav Kotesovec, <a href="/A195417/b195417.txt">Table of n, a(n) for n = 0..1000</a>
%e A195417 Coefficients in (1+x-x^3)^n begin:
%e A195417 n=0: [1];
%e A195417 n=1: [1,1,0,-1];
%e A195417 n=2: [1,2,1,-2,-2,0,1];
%e A195417 n=3: [1,3,3,-2,-6,-3,3,3,0,-1];
%e A195417 n=4: [1,4,6,0,-11,-12,2,12,6,-4,-4,0,1];
%e A195417 n=5: [1,5,10,5,-15,-29,-10,25,30,0,-20,-10,5,5,0,-1];
%e A195417 n=6: [1,6,15,14,-15,-54,-44,30,84,40,-45,-60,-5,30,15,-6,-6,0,1];
%e A195417 n=7: [1,7,21,28,-7,-84,-112,1,168,168,-35,-189,-105,70,105,14,-42,-21,7,7,0,-1];
%e A195417 n=8: [1,8,28,48,14,-112,-224,-104,253,448,132,-392,-462,0,364,224,-98,-168,-28,56,28,-8,-8,0,1]; ...
%e A195417 where the sum of the absolute values of the coefficients generate this sequence.
%e A195417 The limit a(n)^(1/n) seems to exist:
%e A195417 a(1000)^(1/1000) = 2.65862772...
%e A195417 a(2000)^(1/2000) = 2.65827933...
%e A195417 a(3000)^(1/3000) = 2.65816317...
%e A195417 a(4000)^(1/4000) = 2.65810624...
%e A195417 a(5000)^(1/5000) = 2.65807183...
%t A195417 Table[Total[Abs[CoefficientList[Expand[(1+x-x^3)^n],x]]],{n,0,30}] (* _Harvey P. Dale_, Nov 26 2011 *)
%o A195417 (PARI) {a(n)=local(V=Vec((1+x-x^3+O(x^(3*n+1)))^n));abs(V)*vector(#V,i,1)~}
%Y A195417 Cf. A084611.
%K A195417 nonn
%O A195417 0,2
%A A195417 _Paul D. Hanna_, Sep 17 2011