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A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).

%I A343177 #21 Jul 26 2024 12:33:20
%S A343177 4,6,7,9,11,15,19,27,35,51,67,99,131,195,259,387,515,771,1027,1539,
%T A343177 2051,3075,4099,6147,8195,12291,16387,24579,32771,49155,65539,98307,
%U A343177 131075,196611,262147,393219,524291,786435,1048579,1572867,2097155,3145731,4194307,6291459
%N A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).
%C A343177 Number of edges along the boundary of the graph G(n) described in A342759.
%H A343177 Paolo Xausa, <a href="/A343177/b343177.txt">Table of n, a(n) for n = 0..1000</a>
%H A343177 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).
%F A343177 G.f.: (4 + 2*x - 7*x^2 - 2*x^3)/((1 - x)*(1 - 2*x^2)). - _Stefano Spezia_, Feb 04 2023
%F A343177 E.g.f.: 3*cosh(x) + 2*cosh(sqrt(2)*x) + 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2) - 1. - _Stefano Spezia_, Jul 25 2024
%p A343177 f:=n->if n = 0 then 4 elif (n mod 2) = 0 then 2^(n/2+1)+3 else 3*(2^((n-1)/2)+1); fi;
%p A343177 [seq(f(n),n=0..40)];
%t A343177 LinearRecurrence[{1, 2, -2}, {4, 6, 7, 9}, 50] (* or *)
%t A343177 A343177[n_] := Which[n == 0, 4, OddQ[n], 3*(2^((n-1)/2)+1), True, 2^(n/2+1)+3];
%t A343177 Array[A343177, 50, 0] (* _Paolo Xausa_, Feb 02 2024 *)
%Y A343177 Cf. A342759.
%Y A343177 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 A343177 nonn,easy
%O A343177 0,1
%A A343177 _N. J. A. Sloane_, Apr 26 2021