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

%I #11 Jan 05 2023 04:41:38

%S 1,3,9,25,63,171,471,1193,3209,8841,22537,60393,165489,424451,1134849,

%T 3093893,7987289,21288933,57973543,150409931,399788279,1085870421,

%U 2831439537,7501994181,20330726077,53274740933,140615421393,380366331549,1002078031627,2634444052559

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

%C What is the limit a(n)^(1/n)? The limit appears to be near 2.657...

%C The limit a(n+1)/a(n) does not exist, but seems to cycle between attractors.

%H Vaclav Kotesovec, <a href="/A195417/b195417.txt">Table of n, a(n) for n = 0..1000</a>

%e Coefficients in (1+x-x^3)^n begin:

%e n=0: [1];

%e n=1: [1,1,0,-1];

%e n=2: [1,2,1,-2,-2,0,1];

%e n=3: [1,3,3,-2,-6,-3,3,3,0,-1];

%e n=4: [1,4,6,0,-11,-12,2,12,6,-4,-4,0,1];

%e n=5: [1,5,10,5,-15,-29,-10,25,30,0,-20,-10,5,5,0,-1];

%e n=6: [1,6,15,14,-15,-54,-44,30,84,40,-45,-60,-5,30,15,-6,-6,0,1];

%e 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 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 where the sum of the absolute values of the coefficients generate this sequence.

%e The limit a(n)^(1/n) seems to exist:

%e a(1000)^(1/1000) = 2.65862772...

%e a(2000)^(1/2000) = 2.65827933...

%e a(3000)^(1/3000) = 2.65816317...

%e a(4000)^(1/4000) = 2.65810624...

%e a(5000)^(1/5000) = 2.65807183...

%t Table[Total[Abs[CoefficientList[Expand[(1+x-x^3)^n],x]]],{n,0,30}] (* _Harvey P. Dale_, Nov 26 2011 *)

%o (PARI) {a(n)=local(V=Vec((1+x-x^3+O(x^(3*n+1)))^n));abs(V)*vector(#V,i,1)~}

%Y Cf. A084611.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 17 2011