A079859 - OEIS

%I #48 Sep 08 2022 08:45:08

%S 4,10,24,56,128,288,640,1408,3072,6656,14336,30720,65536,139264,

%T 294912,622592,1310720,2752512,5767168,12058624,25165824,52428800,

%U 109051904,226492416,469762048,973078528,2013265920,4160749568,8589934592,17716740096,36507222016

%N a(n) = n*2^(n-4).

%C a(n) = the number of occurrences of 3s in the palindromic compositions of m = 2*n-1 = the number of occurrences of 4s in the palindromic compositions of k = 2*n.

%C 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, 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.

%C Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in same relative order as those in the triple (x,y,z). - _Sergey Kitaev_, Nov 11 2004

%C a(n) appears to be the coefficient of Pi^n in the closed-form expression for the expected value of X^n, where X is the area of a spherical triangle formed by three random points on the unit sphere. (The n*2^(n-4) formula applies when n=2,3 as well, and produces fractional coefficients.) - _Drake Thomas_, Jan 24 2021

%H Vincenzo Librandi, <a href="/A079859/b079859.txt">Table of n, a(n) for n = 4..1000</a>

%H Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, <a href="https://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 Sergey Kitaev, <a href="http://www.emis.de/journals/INTEGERS/papers/e21/e21.Abstract.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), A21, 20pp.

%H Sergey Kitaev, <a href="https://web.archive.org/web/20130625171839/http://www.ms.uky.edu/~math/MAreport/4-ser.ps">On multi-avoidance of right angled numbered polyomino patterns</a>, University of Kentucky Research Reports (2004).

%H Math StackExchange, <a href="https://math.stackexchange.com/a/3993104">The distribution of areas of a random triangle on the sphere</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).

%F O.g.f.: 2*x^4*(2-3*x)/(1-2*x)^2. a(n) = 2*A045623(n-3). - _R. J. Mathar_, Jun 13 2008

%F a(n) = 4*a(n-1) - 4*a(n-2) for n>5. - _Colin Barker_, Sep 29 2015

%F From _Amiram Eldar_, Jan 12 2021: (Start)

%F Sum_{n>=4} 1/a(n) = 16*log(2) - 32/3.

%F Sum_{n>=4} (-1)^n/a(n) = 20/3 - 16*log(3/2). (End)

%F E.g.f.: x*(exp(2*x) - 1 - 2*x - 2*x^2)/8. - _Stefano Spezia_, Apr 06 2021

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

%t Table[i*2^(i - 4), {i, 4, 50}]

%o (Magma) [n*2^(n-4) : n in [4..40]]; // _Vincenzo Librandi_, Sep 22 2011

%o (PARI) Vec(-2*x^4*(3*x-2)/(2*x-1)^2 + O(x^50)) \\ _Colin Barker_, Sep 29 2015

%o (PARI) a(n) = n*2^(n-4);

%o vector(40, n, a(n+3)) \\ _Altug Alkan_, Sep 29 2015

%Y Cf. A057711, A001792, A078836, A079861, A079862.

%Y Main diagonal of A049089.

%K easy,nonn

%O 4,1

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