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 A049673 #26 Sep 21 2022 17:21:17
%S A049673 0,1,3,12,49,205,864,3653,15463,65484,277365,1174889,4976832,21082073,
%T A049673 89304891,378301260,1602509321,6788337557,28755857952,121811766781,
%U A049673 516002920895,2185823443596,9259296684333,39223010163217,166151337308544,703828359351025
%N A049673 a(n) = (F(3n) + F(n))/3, where F = A000045 (the Fibonacci sequence).
%C A049673 This is an odd divisibility sequence, that is, if n divides m and n/m is odd then a(n) divides a(m). More generally, if r and s are positive integers such that r = s (mod 2) then the sequence Fibonacci(r*n) + Fibonacci(s*n) is an odd divisibility sequence. In the particular case that r is even and s = r + 2 then Fibonacci(r*n) + Fibonacci(s*n) is, in fact, a divisibility sequence. See for example A215466 and A273624. - _Peter Bala_, May 29 2016
%H A049673 Colin Barker, <a href="/A049673/b049673.txt">Table of n, a(n) for n = 0..1000</a>
%H A049673 P. Bala, <a href="/A273622/a273622.pdf">Lucas sequences and divisibility sequences</a>
%H A049673 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2,-5,-1).
%F A049673 G.f.: x*(1-2*x-x^2) / ((x^2+4*x-1)*(x^2+x-1)). - _R. J. Mathar_, Oct 26 2015
%F A049673 a(n) = 5*a(n-1) - 2*a(n-2) - 5*a(n-3) - a(n-4) for n>3. - _Wesley Ivan Hurt_, Jun 01 2016
%F A049673 a(n) = ((-(1/2*(1-sqrt(5)))^n-(2-sqrt(5))^n+(1/2*(1+sqrt(5)))^n+(2+sqrt(5))^n))/(3*sqrt(5)). - _Colin Barker_, Jun 02 2016
%F A049673 G.f.: G(F(t)), where G(t) is g.f. of A001045 and F(t) is g.f. of A000129. - _Oboifeng Dira_, Dec 07 2016
%p A049673 with(combinat): A049673:=n->(fibonacci(3*n)+fibonacci(n))/3: seq(A049673(n), n=0..30); # _Wesley Ivan Hurt_, Jun 01 2016
%t A049673 Table[(Fibonacci[3 n] + Fibonacci[n])/3, {n, 0, 30}] (* _Wesley Ivan Hurt_, Jun 01 2016 *)
%t A049673 LinearRecurrence[{5,-2,-5,-1},{0,1,3,12},30] (* _Harvey P. Dale_, Sep 21 2022 *)
%o A049673 (PARI) concat(0, Vec(x*(1-2*x-x^2)/((x^2+4*x-1)*(x^2+x-1)) + O(x^30))) \\ _Colin Barker_, Jun 02 2016
%o A049673 (Magma) [(Fibonacci(3*n)+Fibonacci(n))/3: n in [0..30]]; // _Vincenzo Librandi_, Jun 04 2016
%Y A049673 Cf. A000045, A000129, A001045, A215466, A273624.
%K A049673 nonn,easy
%O A049673 0,3
%A A049673 _Clark Kimberling_