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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) = 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

P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size ..., Ars Combin. 69 (2003), 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)

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

CROSSREFS

Cf. A057711, A001792, A079859, A079861, A079862, A079863.

Sequence in context: A199705 A225972 A332724 * A142875 A074981 A066510

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 August 10 01:31 EDT 2020. Contains 336361 sequences. (Running on oeis4.)