1,2

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.

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)

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 (pauldhanna(AT)juno.com), Apr 24 2006

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].

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

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]

(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 (pauldhanna(AT)juno.com), Apr 24 2006

Cf. A084081, A047749.

Cf. A006629, A056096.

Sequence in context: A002845 A072925 A002955 this_sequence A137255 A076892 A106462

Adjacent sequences: A093948 A093949 A093950 this_sequence A093952 A093953 A093954

nonn

Wouter Meeussen (wouter.meeussen(AT)pandora.be), Apr 18 2004