A078836 a(n) = n*2^(n-6).

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

Offset: 6

Author

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 17 2003

Keywords

• easy
• nonn

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

Examples

			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.
		

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).

Crossrefs

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

Programs

Magma

[n*2^(n-6): n in [6..40]]; // Vincenzo Librandi, Oct 04 2011

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 *)

PARI

a(n)=n<<(n-6) \\ Charles R Greathouse IV, Oct 03 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

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)