A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2*n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2*n.

10, 22, 48, 104, 224, 480, 1024, 2176, 4608, 9728, 20480, 43008, 90112, 188416, 393216, 819200, 1703936, 3538944, 7340032, 15204352, 31457280, 65011712, 134217728, 276824064, 570425344, 1174405120, 2415919104, 4966055936

Offset: 8

Author

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

Keywords

• easy
• nonn

Comments

This sequence is part of a family of sequences, namely R(n,k), the number of k's 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 >= 2*k.

Examples

			a(8)=10 since the palindromic compositions of 15 that contain a 7 are 7+1+7, 4+7+4, 1+3+7+3+1, 3+1+7+1+3, 2+2+7+2+2, 1+1+1+1+7+1+1+1+1, 1+1+2+7+2+1+1, 1+2+1+7+1+2+1 and 2+1+1+7+1+1+2, for a total of 10 7's.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 8..1000
• 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.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Cf. A057711, A001792, A079859, A061256, A079862, A079863, A111297.

Programs

Magma

[(2+n)*2^(n-8) : n in [8..40]]; // Vincenzo Librandi, Sep 22 2011

Mathematica

Table[(2 + i)*2^(i - 8), {i, 8, 50}]
LinearRecurrence[{4,-4},{10,22},50] (* Harvey P. Dale, Jun 04 2025 *)

PARI

Vec(-2*x^8*(9*x-5)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Dec 16 2014

Formula

a(n) = (2+n)*2^(n-8).

a(n) = 2*A111297(n-6). - Colin Barker, Dec 16 2014

a(n) = 4*a(n-1) - 4*a(n-2). - Colin Barker, Dec 16 2014

G.f.: -2*x^8*(9*x-5) / (2*x-1)^2. - Colin Barker, Dec 16 2014

From Amiram Eldar, Jan 13 2021: (Start)

Sum_{n>=8} 1/a(n) = 1024*log(2) - 447047/630.

Sum_{n>=8} (-1)^n/a(n) = 261617/630 - 1024*log(3/2). (End)