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 A078836 a(n) = n*2^(n-6). 9
 6, 14, 32, 72, 160, 352, 768, 1664, 3584, 7680, 16384, 34816, 73728, 155648, 327680, 688128, 1441792, 3014656, 6291456, 13107200, 27262976, 56623104, 117440512, 243269632, 503316480, 1040187392, 2147483648, 4429185024, 9126805504, 18790481920, 38654705664 (list; graph; refs; listen; history; text; internal format)
 OFFSET 6,1 COMMENTS 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. 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. Also the number of independent vertex sets and vertex covers in the (n-4)-sun graph. - Eric W. Weisstein, Sep 27 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 6..3000 Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78. Eric Weisstein's World of Mathematics, Independent Vertex Set. Eric Weisstein's World of Mathematics, Sun Graph. Eric Weisstein's World of Mathematics, Vertex Cover. Index entries for linear recurrences with constant coefficients, signature (4,-4). FORMULA From Colin Barker, Sep 29 2015: (Start) a(n) = 2*A045891(n-4). a(n) = 4*a(n-1) - 4*a(n-2) for n > 7. G.f.: -2*x^6*(5*x-3) / (2*x-1)^2. (End) From Amiram Eldar, Jan 12 2021: (Start) Sum_{n>=6} 1/a(n) = 64*log(2) - 661/15. Sum_{n>=6} (-1)^n/a(n) = 391/15 - 64*log(3/2). (End) EXAMPLE 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. MATHEMATICA Table[n 2^(n - 6), {m, 6, 50}] LinearRecurrence[{4, -4}, {6, 14}, 20] (* Eric W. Weisstein, Sep 27 2017 *) CoefficientList[Series[-2 (-3 + 5 x)/(-1 + 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *) PROG (PARI) a(n)=n<<(n-6) \\ Charles R Greathouse IV, Oct 03 2011 (Magma) [n*2^(n-6): n in [6..40]]; // Vincenzo Librandi, Oct 04 2011 (PARI) Vec(-2*x^6*(5*x-3)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Sep 29 2015 (Python) def a(n): return n << (n-6) print([a(n) for n in range(6, 37)]) # Michael S. Branicky, Jun 14 2021 CROSSREFS Cf. A057711, A001792, A079859, A079861, A079862, A079863. Sequence in context: A199705 A225972 A332724 * A340735 A142875 A074981 Adjacent sequences: A078833 A078834 A078835 * A078837 A078838 A078839 KEYWORD easy,nonn AUTHOR Silvia Heubach (sheubac(AT)calstatela.edu), Jan 17 2003 STATUS approved

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Last modified September 23 02:06 EDT 2023. Contains 365532 sequences. (Running on oeis4.)