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%I #30 Sep 08 2022 08:45:21
%S 3,1,5,1,11,3,21,5,43,11,85,21,171,43,341,85,683,171,1365,341,2731,
%T 683,5461,1365,10923,2731,21845,5461,43691,10923,87381,21845,174763,
%U 43691,349525,87381,699051,174763,1398101,349525,2796203,699051,5592405
%N Numerator of 3/4 + 1/4 - 3/8 - 1/8 + 3/16 + 1/16 - 3/32 - 1/32 + 3/64 + ....
%C Numerator of partial sums of A112030(n)/A016116(n+4), denominators = A112032;
%C a(n)/A112032(n) - 2/3 = (-1)^floor(n/2) / A112033(n);
%C lim_{n->infinity} a(n)/A112032(n) = 2/3.
%D G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 4, Sect. 1, Problem 148.
%H Vincenzo Librandi, <a href="/A112031/b112031.txt">Table of n, a(n) for n = 0..3000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,2).
%F a(n) = (2^(floor(n/2) + 2 + (-1)^n) + (-1)^floor(n/2)) / 3.
%F From _Colin Barker_, Apr 05 2013: (Start)
%F a(n) = a(n-2) + 2*a(n-4);
%F g.f.: (2*x^2+x+3) / ((1+x^2)*(1-2*x^2)). (End)
%t LinearRecurrence[{0,1,0,2},{3,1,5,1},50] (* _Harvey P. Dale_, Dec 31 2017 *)
%o (Magma) [(2^(Floor(n/2) + 2 + (-1)^n) + (-1)^Floor(n/2)) / 3: n in [0..50]]; // _Vincenzo Librandi_, Aug 17 2011
%o (PARI) m=50; v=concat([3,1,5,1], vector(m-4)); for(n=5, m, v[n]=v[n-2] +2*v[n-4]); v \\ _G. C. Greubel_, Nov 08 2018
%Y Cf. A016116, A112030, A112032, A112033, A001045 (bisections).
%K nonn,frac
%O 0,1
%A _Reinhard Zumkeller_, Aug 27 2005
%E a(22) corrected by _Vincenzo Librandi_, Aug 17 2011
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