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

1,3

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

Links

Table of n, a(n) for n=1..31.

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).

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A131090 A178615 A131935 * A201498 A145970 A232048

Adjacent sequences: A119746 A119747 A119748 * A119750 A119751 A119752

Keyword

easy,nonn

Author

Louis Shapiro, Jul 30 2006

Status

approved