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A354788 a(2*k) = 3*2^k - 3, a(2*k+1) = 2^(k+2) - 3.

%I A354788 #22 Oct 17 2023 05:03:44
%S A354788 0,1,3,5,9,13,21,29,45,61,93,125,189,253,381,509,765,1021,1533,2045,
%T A354788 3069,4093,6141,8189,12285,16381,24573,32765,49149,65533,98301,131069,
%U A354788 196605,262141,393213,524285,786429,1048573,1572861,2097149,3145725,4194301,6291453,8388605,12582909,16777213,25165821,33554429,50331645
%N A354788 a(2*k) = 3*2^k - 3, a(2*k+1) = 2^(k+2) - 3.
%H A354788 Paolo Xausa, <a href="/A354788/b354788.txt">Table of n, a(n) for n = 0..1000</a>
%H A354788 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).
%F A354788 a(n) = A136252(n-1). - _R. J. Mathar_, Jul 14 2022
%F A354788 G.f.: x*(1 + 2*x)/((x - 1)*(2*x^2 - 1)). - _R. J. Mathar_, Jul 14 2022
%F A354788 E.g.f.: 3*(cosh(sqrt(2)*x) - cosh(x)) - 3*sinh(x) + 2*sqrt(2)*sinh(sqrt(2)*x). - _Stefano Spezia_, Feb 04 2023
%p A354788 f1:=proc(n) if (n mod 2) = 1 then 2^((n+3)/2)-3 else 3*2^(n/2)-3; fi; end;
%p A354788 [seq(f1(n),n=0..45)];
%t A354788 LinearRecurrence[{1,2,-2},{0,1,3},100] (* _Paolo Xausa_, Oct 17 2023 *)
%Y A354788 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 A354788 nonn,easy
%O A354788 0,3
%A A354788 _N. J. A. Sloane_, Jul 13 2022