A119749 - OEIS

%I #36 Aug 28 2020 02:02:48

%S 1,1,4,7,15,32,65,137,284,591,1231,2560,5329,11089,23076,48023,99935,

%T 207968,432785,900633,1874236,3900319,8116639,16890880,35150241,

%U 73148321,152223044,316779047,659223215,1371856032,2854858465

%N Number of compositions of n into odd blocks with one element in each block distinguished.

%C The sequence is the INVERT transform of the aerated odd integers. - _Gary W. Adamson_, Feb 02 2014

%C Number of compositions of n into odd parts where there is 1 sort of part 1, 3 sorts of part 3, 5 sorts of part 5, ... , 2*k-1 sorts of part 2*k-1. - _Joerg Arndt_, Aug 04 2014

%H R. X. F. Chen and L. W. Shapiro, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL10/Chen/chen509.html">On Sequences G(n) satisfying G(n)=(d+2)*G(n-1)-G(n-2)</a>, J. Int. Seq. 10 (2007) #07.8.1, Theorem 16.

%H Y-h. Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Guo/guo4.html">Some n-Color Compositions</a>, J. Int. Seq. 15 (2012) 12.1.2, eq. (6).

%H Y.-h. Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Guo/guo9.html">n-Color Odd Self-Inverse Compositions</a>, J. Int. Seq. 17 (2014) # 14.10.5, eq. (2).

%H B. Hopkins, <a href="https://web.archive.org/web/20171111231553/http://www.westga.edu/~integers/a6intproc11/a6intproc11.pdf">Spotted tilings and n-color compositions</a>, INTEGERS 12B (2012/2013), #A6.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,1,-1).

%F G.f.: (x+x^3)/(x^4 - x^3 -2x^2 -x +1).

%F a(n) = A092886(n)+A092886(n-2). - _R. J. Mathar_, Mar 08 2018

%F Sum_{k=0..n} a(k) = (3*a(n) + 2*a(n-1) - a(n-3))/2 - 1. - _Xilin Wang_ and _Greg Dresden_, Aug 27 2020

%e a(3) = 4 since Abc, aBc, abC come from one block of size 3 and A/B/C comes from having three blocks. The capital letters are the distinguished elements.

%t Rest@ CoefficientList[ Series[x(1 + x^2)/(x^4 - x^3 - 2x^2 - x + 1), {x, 0, 50}], x] (* _Robert G. Wilson v_ *)

%Y Cf. A105309, A052530, A000045, A030267. Row sums of A292835.

%K easy,nonn

%O 1,3

%A _Louis Shapiro_, Jul 30 2006