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 A158780 #25 May 14 2023 02:10:21
%S A158780 0,1,1,1,2,2,4,4,8,8,16,16,32,32,64,64,128,128,256,256,512,512,1024,
%T A158780 1024,2048,2048,4096,4096,8192,8192,16384,16384,32768,32768,65536,
%U A158780 65536,131072,131072,262144,262144,524288,524288,1048576,1048576,2097152,2097152,4194304
%N A158780 a(2n) = A131577(n), a(2n+1) = A011782(n).
%C A158780 This construction combines the 2 basic sequences which equal their first differences in the same way as A138635 does for sequences which equal their 3rd differences and A137171 does for sequences which equal their fourth differences.
%C A158780 Essentially the same as A016116, A060546, and A131572. - _R. J. Mathar_, Apr 08 2009
%C A158780 Dropping a(0), this is the inverse binomial transform of A024537. - _R. J. Mathar_, Apr 08 2009
%H A158780 G. C. Greubel, <a href="/A158780/b158780.txt">Table of n, a(n) for n = 0..1000</a>
%H A158780 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2).
%F A158780 a(2n) + a(2n+1) = A000079(n).
%F A158780 G.f.: x*(1+x-x^2)/(1-2*x^2). - _R. J. Mathar_, Apr 08 2009
%F A158780 a(n) = (1/2)*(2^floor(n/2) + [n=1] - [n=0]). - _G. C. Greubel_, Apr 19 2023
%F A158780 E.g.f.: (2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) + 2*x - 2)/4. - _Stefano Spezia_, May 13 2023
%t A158780 Table[(2^Floor[n/2] +Boole[n==1] -Boole[n==0])/2, {n,0,50}] (* or *) LinearRecurrence[{0,2}, {0,1,1,1}, 51] (* _G. C. Greubel_, Apr 19 2023 *)
%o A158780 (PARI) a(n)=if(n>3,([0,1; 2,0]^n*[1;1])[1,1]/2,n>0) \\ _Charles R Greathouse IV_, Oct 18 2022
%o A158780 (Magma) [0,1] cat [2^Floor((n-2)/2): n in [2..50]]; // _G. C. Greubel_, Apr 19 2023
%o A158780 (SageMath)
%o A158780 def A158780(n): return (2^(n//2) + int(n==1) - int(n==0))/2
%o A158780 [A158780(n) for n in range(51)] # _G. C. Greubel_, Apr 19 2023
%Y A158780 The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - _N. J. A. Sloane_, Jul 14 2022
%K A158780 nonn,easy
%O A158780 0,5
%A A158780 _Paul Curtz_, Mar 26 2009
%E A158780 Edited by _R. J. Mathar_, Apr 08 2009