A079863 a(n) is the number of occurrences of 11s in the palindromic compositions of m=2*n-1 = the number of occurrences of 12s in the palindromic compositions of m=2*n.

34, 70, 144, 296, 608, 1248, 2560, 5248, 10752, 22016, 45056, 92160, 188416, 385024, 786432, 1605632, 3276800, 6684672, 13631488, 27787264, 56623104, 115343360, 234881024, 478150656, 973078528, 1979711488, 4026531840, 8187281408, 16642998272, 33822867456

Offset

12,1

Comments

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, A078836, A079861, A079862. 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.

Links

Colin Barker, Table of n, a(n) for n = 12..1000

P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 65-78.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Formula

a(n) = (n+22)*2^(n-12).

From Colin Barker, Sep 29 2015: (Start)

a(n) = 4*a(n-1) - 4*a(n-2) for n>13.

G.f.: -2*x^12*(33*x-17) / (2*x-1)^2.

(End)

Example

a(12) = 34 since the palindromic compositions of 23 that contain a 11 are 11+1+11 and the 32 compositions of the form c+11+(reverse of c), where c represents a composition of 6.

Mathematica

Table[(22 + i)*2^(i - 12), {i, 12, 50}]
LinearRecurrence[{4, -4}, {34, 70}, 30] (* Harvey P. Dale, Jan 30 2017 *)

Prog

(PARI) Vec(-2*x^12*(33*x-17)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Sep 29 2015
(PARI) a(n)=(n+22)<<(n-12) \\ Charles R Greathouse IV, Sep 29 2015

Crossrefs

Cf. A057711, A001792, A079859, A078836, A079861, A079862.

Sequence in context: A044136 A044517 A063333 * A154095 A184066 A277449

Adjacent sequences: A079860 A079861 A079862 * A079864 A079865 A079866

Keyword

easy,nonn

Author

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

Status

approved