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A093951 Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1,k-1,..,1. 1

%I

%S 1,2,4,8,17,36,80,176,403,910,2128,4896,11628,27132,65208,153824,

%T 373175,888030,2170740,5202600,12797265,30853680,76292736,184863168,

%U 459162452,1117370696,2786017120,6804995008,17024247304,41717833740,104673837384

%N Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1,k-1,..,1.

%C Substitutions 1->{2}, 2->{3,1}, 3->{4,2}, 4->{5,3,1}, 5->{6,4,2}, 6->{7,5,3,1}, 7->{8,6,4,2}, etc. The function f[n] gives Det[IdentityMatrix[n]-x*A[n]] with A[n]=Table[If[j > i+1, 0, Mod[i+j,2]], {i,n}, {j,n}] and can be written in terms of Dickson polynomials as : g(w)= x D_(w-1)(1+x, x*(1+x)) +(1-2*x)*E_(w-1)(1+x, x*(1+x)) Francisco Salinas (franciscodesalinas(AT)hotmail.com), Apr 13 2004; Count of integers is A047749. Sum of integers with substitution starting from 0 is A084081.

%F GF up to n-th term = GF[n]= g[n]/f[n] with f[1]=1; f[2]=1-x^2; f[3]=1-2x^2; f[n]=f[n-1]-x^2 f[n-3] and g[1]=1; g[2]=1+2x; g[3]=1+2x+2x^2; g[n]=g[n-1]-x^2 g[n-3]+2 x^(n-1)

%F a(2n) = 4*binomial(3n,n-1)/(n+1) = 2*A006629(n-1); a(2n+1) = 6*binomial(3n+2,n)/(2n+3) - binomial(3n+1,n)/(n+1) = A056096(n+3). - _Paul D. Hanna_, Apr 24 2006

%e GF[12]=(1 +2*x -7*x^2 -14*x^3 +9*x^4 +20*x^5 +2*x^6 -2*x^7 +2*x^11)/(1 -11*x^2 +36*x^4 -35*x^6 +5*x^8) produces a[1] to a[12].

%e a(4)=8 since 4-1= 3 substitutions on 1 produce 1-> 2-> 3+1-> 4+2+2 =8.

%t Plus@@@Flatten/@NestList[ #/.k_Integer:>Range[k+1, 1, -2]&, {1}, 8];(*or for n>16 *); f[1]=1; f[2]=1-x^2; f[3]=1-2x^2; f[n_]:=f[n]=Expand[f[n-1]-x^2 f[n-3]]; g[1]=1; g[2]=1+2x; g[3]=1+2x+2x^2; g[n_]:=g[n]=Expand[g[n-1] -x^2 g[n-3]+2 x^(n-1)]; GF[n_]:=g[n]/f[n]; CoefficientList[Series[GF[36], {x, 0, 36}], x]

%o (PARI) {a(n)=if(n%2==0,4*binomial(3*n/2,n/2-1)/(n/2+1), 6*binomial(3*(n\2)+2,n\2)/(2*(n\2)+3) - binomial(3*(n\2)+1,n\2)/(n\2+1))} - _Paul D. Hanna_, Apr 24 2006

%Y Cf. A084081, A047749.

%Y Cf. A006629, A056096.

%K nonn

%O 1,2

%A _Wouter Meeussen_, Apr 18 2004

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Last modified October 15 05:43 EDT 2019. Contains 328026 sequences. (Running on oeis4.)