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 A078836 #45 Feb 16 2025 08:32:48
%S A078836 6,14,32,72,160,352,768,1664,3584,7680,16384,34816,73728,155648,
%T A078836 327680,688128,1441792,3014656,6291456,13107200,27262976,56623104,
%U A078836 117440512,243269632,503316480,1040187392,2147483648,4429185024,9126805504,18790481920,38654705664
%N A078836 a(n) = n*2^(n-6).
%C A078836 a(n) is the number of occurrences of 5s in the palindromic compositions of 2n-1 = the number of occurrences of 6s in the palindromic compositions of 2n.
%C A078836 This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A079859, A079861 - A079863. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.
%C A078836 Also the number of independent vertex sets and vertex covers in the (n-4)-sun graph. - _Eric W. Weisstein_, Sep 27 2017
%H A078836 Vincenzo Librandi, <a href="/A078836/b078836.txt">Table of n, a(n) for n = 6..3000</a>
%H A078836 Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, <a href="https://www.calstatela.edu/sites/default/files/users/u1231/Papers/freqs.pdf">The frequency of summands of a particular size in Palindromic Compositions</a>, Ars Combin., Vol. 69 (2003), pp. 65-78.
%H A078836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>.
%H A078836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>.
%H A078836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>.
%H A078836 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F A078836 From _Colin Barker_, Sep 29 2015: (Start)
%F A078836 a(n) = 2*A045891(n-4).
%F A078836 a(n) = 4*a(n-1) - 4*a(n-2) for n > 7.
%F A078836 G.f.: -2*x^6*(5*x-3) / (2*x-1)^2.
%F A078836 (End)
%F A078836 From _Amiram Eldar_, Jan 12 2021: (Start)
%F A078836 Sum_{n>=6} 1/a(n) = 64*log(2) - 661/15.
%F A078836 Sum_{n>=6} (-1)^n/a(n) = 391/15 - 64*log(3/2). (End)
%e A078836 a(6) = 6 since the palindromic compositions of 11 that contain a 5 are 3+5+3, 1+2+5+2+1, 2+1+5+1+2, 1+1+1+5+1+1+1 and 5+1+5, for a total of 6 5s. The palindromic compositions of 12 that contain a 6 are 3+6+3, 1+2+6+2+1, 2+1+6+1+2, 1+1+1+6+1+1+1 and 6+6.
%t A078836 Table[n 2^(n - 6), {m, 6, 50}]
%t A078836 LinearRecurrence[{4, -4}, {6, 14}, 20] (* _Eric W. Weisstein_, Sep 27 2017 *)
%t A078836 CoefficientList[Series[-2 (-3 + 5 x)/(-1 + 2 x)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 27 2017 *)
%o A078836 (PARI) a(n)=n<<(n-6) \\ _Charles R Greathouse IV_, Oct 03 2011
%o A078836 (Magma) [n*2^(n-6): n in [6..40]]; // _Vincenzo Librandi_, Oct 04 2011
%o A078836 (PARI) Vec(-2*x^6*(5*x-3)/(2*x-1)^2 + O(x^100)) \\ _Colin Barker_, Sep 29 2015
%o A078836 (Python)
%o A078836 def a(n): return n << (n-6)
%o A078836 print([a(n) for n in range(6, 37)]) # _Michael S. Branicky_, Jun 14 2021
%Y A078836 Cf. A057711, A001792, A079859, A079861, A079862, A079863.
%K A078836 easy,nonn
%O A078836 6,1
%A A078836 Silvia Heubach (sheubac(AT)calstatela.edu), Jan 17 2003