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 A279312 #24 Mar 30 2025 11:18:42
%S A279312 1,2,4,8,12,24,40,80,128,256,416,832,1344,2688,4352,8704,14080,28160,
%T A279312 45568,91136,147456,294912,477184,954368,1544192,3088384,4997120,
%U A279312 9994240,16171008,32342016,52330496
%N A279312 Number of subsets of {1, 2, 3, ..., n} that include no consecutive even integers.
%C A279312 Let b(n) be the number of subsets of [n] that include no consecutive odd integers then b(n) satisfies the recurrence b(0)=1, b(1)=2, b(2)=4, b(3)=6; for n > 3, b(n) = 2b(n-2) + 4b(n-4). For that sequence see A279245.
%C A279312 Let a(n) be the number of subsets of [n] that include no consecutive even integers. If n is an even integer then, a(n) = b(n). Since in the set S = {1, 2, 3, ..., n} where n is even, the number of odd integers is equal to the number of even integers. For example, let S ={1, 2, 3, 4} In this set there are 2 odd and 2 even integers. So the number of subsets of S contain no consecutive odd integers is equal to the number of subsets of S contain no consecutive even integers. In the other case if n is an odd integer then, a(n) = 2b(n-1). Since in the set S = {1, 2, 3, ..., n} where n is odd; Let K = {1, 2, 3, ..., n-1}, n-1 is an even integer so there are b(n-1) subsets containing no consecutive even integers in the set K. And prepending the last element 'n' to each of those gives us another b(n-1) subsets, for a total of 2b(n-1) subsets. Hence if n is even then, a(n) = b(n). If n is odd then, a(n) = 2b(n-1). For k = 0, 1, 2, 3, ... ; a(2k+1) = 2a(2k).
%H A279312 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,4).
%F A279312 a(n) = A279245(n) if n is even; a(n) = 2*A279245(n-1) if n is odd.
%F A279312 G.f.: (2*x^3+4*x^2+2*x+2)/(4*x^4+2*x^2-1).
%F A279312 a(n) = 2a(n-2) + 4a(n-4). - _Charles R Greathouse IV_, Dec 13 2016
%e A279312 For n=4, a(n)=12. The number of subsets of {1, 2, 3, 4} that include no consecutive even integers are: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {3,4}, {1,2,3}, {1,3,4}.
%t A279312 LinearRecurrence[{0,2,0,4},{1,2,4,8},40] (* _Harvey P. Dale_, Mar 30 2025 *)
%o A279312 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 4,0,2,0]^n*[1;2;4;8])[1,1] \\ _Charles R Greathouse IV_, Dec 13 2016
%Y A279312 Cf. A279245
%K A279312 nonn,easy
%O A279312 0,2
%A A279312 _Baris Arslan_, Dec 09 2016
%E A279312 More terms from _Charles R Greathouse IV_, Dec 13 2016
%E A279312 Edited by _Michel Marcus_, Jul 04 2017