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A007759 Knopfmacher expansion of sqrt(2): a(2n) = 2*(a(2n-1) + 1)^2 - 1, a(2n+1) = 2*(a(2n)^2 - 1).

%I A007759 #19 Sep 08 2022 08:44:35
%S A007759 2,17,576,665857,886731088896,1572584048032918633353217,
%T A007759 4946041176255201878775086487573351061418968498176,
%U A007759 48926646634423881954586808839856694558492182258668537145547700898547222910968507268117381704646657
%N A007759 Knopfmacher expansion of sqrt(2): a(2n) = 2*(a(2n-1) + 1)^2 - 1, a(2n+1) = 2*(a(2n)^2 - 1).
%H A007759 G. C. Greubel, <a href="/A007759/b007759.txt">Table of n, a(n) for n = 1..11</a>
%H A007759 A. Knopfmacher and J. Knopfmacher, <a href="https://doi.org/10.1007/978-94-009-1910-5_24">An alternating product representation for real numbers</a>, in Applications of Fibonacci numbers, Vol. 3 (Kluwer 1990), pp. 209-216.
%p A007759 a:= proc(n) option remember;
%p A007759 if n=1 then 2
%p A007759 elif `mod`(n,2) = 0 then 2*(a(n-1) +1)^2 -1
%p A007759 else 2*(a(n-1)^2 -1)
%p A007759 end if; end proc;
%p A007759 seq(a(n), n = 1..9); # _G. C. Greubel_, Mar 04 2020
%t A007759 a[n_]:= a[n]= If[n==1, 2, If[EvenQ[n], 2*(a[n-1] +1)^2 -1, 2*a[n-1]^2 -2]]; Table[a[n], {n,9}] (* _G. C. Greubel_, Mar 04 2020 *)
%o A007759 (PARI) a(n) = if (n==1, 2, if (n % 2, 2*a(n-1)^2 - 2, 2*(a(n-1)+1)^2 - 1)); \\ _Michel Marcus_, Feb 20 2019
%o A007759 (Magma)
%o A007759 function a(n)
%o A007759   if n eq 1 then return 2;
%o A007759   elif n mod 2 eq 0 then return 2*(a(n-1) +1)^2 -1;
%o A007759   else return 2*(a(n-1)^2 -1);
%o A007759   end if; return a; end function;
%o A007759 [a(n): n in [1..9]]; // _G. C. Greubel_, Mar 04 2020
%o A007759 (Sage)
%o A007759 @CachedFunction
%o A007759 def a(n):
%o A007759     if (n==1): return 2
%o A007759     elif (n%2==0): return 2*(a(n-1) +1)^2 -1
%o A007759     else: return 2*(a(n-1)^2 -1)
%o A007759 [a(n) for n in (1..9)] # _G. C. Greubel_, Mar 04 2020
%Y A007759 Cf. A002193 (sqrt(2)), A001601.
%K A007759 nonn
%O A007759 1,1
%A A007759 _Arnold Knopfmacher_
%E A007759 More terms from _Christian G. Bower_, Jan 06 2006