A046126 Denominators q[ n ] of convergents to Stern's non-simple continued fraction for Pi/2.
1, 3, -3, -15, 45, 315, -1575, -14175, 99225, 1091475, -9823275, -127702575, 1404728325, 21070924875, -273922023375, -4656674397375, 69850115960625, 1327152203251875, -22561587455281875, -473793336560919375
Offset: 0
Links
- Eric Weisstein's World of Mathematics, Pi.
- Eric Weisstein's World of Mathematics, Pi Continued Fraction.
- Index entries for sequences related to Stern's sequences
Programs
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Mathematica
b[ n_ ] := 2-(-1)^n; a[ 1 ] := -1; a[ n_Integer?EvenQ ] := -n(n+1); a[ n_Integer?OddQ ] := -(n-2)(n-1); then use the standard algorithm to get p[ n ]/q[ n ]. a[n_] := Product[If[OddQ[k], k+2, 1-k], {k, 1, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 06 2012, after 1st Pari program *) -
PARI
a(n)=if(n<0,0,prod(k=1,n,if(k%2,k+2,1-k)))
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PARI
{a(n)=local(A); if(n<0, 0, A=matrix(2,n+1); for(k=0, n, A[2, k+1]=if(k%2, 3, 1); A[1, k+1]=if(k<2, (-1)^k, if(k%2, -(k-2)*(k-1), -k*(k+1)))); contfracpnqn(A)[2,1])} /* Michael Somos, Jul 15 2003 */
Formula
E.g.f.: exp(asinh(x))((1+x)/(1+x^2)+(2-x+x^2)/(1+x^2)^(3/2))-2. - Michael Somos, Mar 11 2004
E.g.f.: (1+3*x+2*x^3)/(1+x^2)^(3/2). - Vaclav Kotesovec, Oct 05 2013
a(n) ~ 2*(cos(Pi*n/2)+sin(Pi*n/2)) * n^(n+1) / exp(n). - Vaclav Kotesovec, Oct 05 2013