This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A036675 #26 Jul 08 2025 21:53:53 %S A036675 1,1,2,6,18,59,198,690,2450,8878,32632,121518,457262,1736526,6646340, %T A036675 25613086,99298674,387021728,1515594560,5960406102,23530528512, %U A036675 93216984177,370450977206,1476458287082,5900150928510,23635544130948 %N A036675 G.f. satisfies A(x) = 1 + x*A(x)^2*A(x^2). %H A036675 Vaclav Kotesovec, <a href="/A036675/b036675.txt">Table of n, a(n) for n = 0..1000</a> %F A036675 G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x^2/(1-x^2)) (continued fraction); more generally g.f. C(x/(1-x^2/(1-x^2))) where C(x) is the g.f. for the Catalan numbers (A000108). [_Joerg Arndt_, Mar 18 2011] %F A036675 a(n) ~ c * d^n / n^(3/2), where d = 4.250770453055989899189676464071962617426..., c = 0.600960911911396921862654605015399962... . - _Vaclav Kotesovec_, Aug 10 2014 %F A036675 a(n) = T(2*n+1,1), where T(n,m) = sum(i=1..n-m, (m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i)), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 18 2015 %p A036675 A := 1; f := proc(n) global A; coeff(series( 1+x*(A*subs(x=x^2,A)), x, n+1), x,n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A; %t A036675 terms = 26; A[_] = 0; Do[A[x_] = 1 + x*A[x]^2*A[x^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 15 2018 *) %o A036675 (PARI) a(n)=local(A,m); if(n<0,0,m=2; A=1+O(x); while(m<=n+1,m*=2; A=2/(1+sqrt(1-4*x*subst(A,x,x^2)))); polcoeff(A,n)) %o A036675 (Maxima) %o A036675 T(n,m):=if m=n then 1 else sum((m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i),i,1,n-m); %o A036675 makelist(T(2*n+1,1),n,0,30); /* _Vladimir Kruchinin_, Mar 18 2015 */ %Y A036675 Cf. A000621, A101913 %K A036675 nonn,easy %O A036675 0,3 %A A036675 _N. J. A. Sloane_