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%I #36 Sep 08 2022 08:45:08
%S 10,22,48,104,224,480,1024,2176,4608,9728,20480,43008,90112,188416,
%T 393216,819200,1703936,3538944,7340032,15204352,31457280,65011712,
%U 134217728,276824064,570425344,1174405120,2415919104,4966055936
%N 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.
%C 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.
%H Vincenzo Librandi, <a href="/A079861/b079861.txt">Table of n, a(n) for n = 8..1000</a>
%H Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, <a href="http://www.calstatela.edu/sites/default/files/users/u1231/Papers/freqs.pdf">The Frequency of Summands of a Particular Size in Palindromic Compositions</a>, Ars Combin., Vol. 69 (2003), pp. 65-78.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F a(n) = (2+n)*2^(n-8).
%F a(n) = 2*A111297(n-6). - _Colin Barker_, Dec 16 2014
%F a(n) = 4*a(n-1) - 4*a(n-2). - _Colin Barker_, Dec 16 2014
%F G.f.: -2*x^8*(9*x-5) / (2*x-1)^2. - _Colin Barker_, Dec 16 2014
%F From _Amiram Eldar_, Jan 13 2021: (Start)
%F Sum_{n>=8} 1/a(n) = 1024*log(2) - 447047/630.
%F Sum_{n>=8} (-1)^n/a(n) = 261617/630 - 1024*log(3/2). (End)
%e 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.
%t Table[(2 + i)*2^(i - 8), {i, 8, 50}]
%o (Magma) [(2+n)*2^(n-8) : n in [8..40]]; // _Vincenzo Librandi_, Sep 22 2011
%o (PARI) Vec(-2*x^8*(9*x-5)/(2*x-1)^2 + O(x^100)) \\ _Colin Barker_, Dec 16 2014
%Y Cf. A057711, A001792, A079859, A061256, A079862, A079863, A111297.
%K easy,nonn
%O 8,1
%A Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003
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