A119749 Number of compositions of n into odd blocks with one element in each block distinguished.

1, 1, 4, 7, 15, 32, 65, 137, 284, 591, 1231, 2560, 5329, 11089, 23076, 48023, 99935, 207968, 432785, 900633, 1874236, 3900319, 8116639, 16890880, 35150241, 73148321, 152223044, 316779047, 659223215, 1371856032, 2854858465

Offset: 1

Author

Louis Shapiro, Jul 30 2006

Keywords

• easy
• nonn

Comments

The sequence is the INVERT transform of the aerated odd integers. - Gary W. Adamson, Feb 02 2014

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

Examples

			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.
		

Links

• R. X. F. Chen and L. W. Shapiro, On Sequences G(n) satisfying G(n)=(d+2)*G(n-1)-G(n-2), J. Int. Seq. 10 (2007) #07.8.1, Theorem 16.
• Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (6).
• Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, eq. (2).
• B. Hopkins, Spotted tilings and n-color compositions, INTEGERS 12B (2012/2013), #A6.
• Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).

Crossrefs

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

Programs

Mathematica

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

Formula

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

a(n) = A092886(n)+A092886(n-2). - R. J. Mathar, Mar 08 2018

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