0,2

From M. F. Hasler, May 01 2009: (Start)

Also, number of 5-element subsets of {1,...,n+5} whose elements sum to an odd integer, i.e. column 5 of A159916.

A linear recurrent sequence with constant coefficients and characteristic polynomial x^9 - 3*x^8 + 8*x^6 - 6*x^5 - 6*x^4 + 8*x^3 - 3*x + 1. (End)

Equals (1/2)*((1, 6, 21, 56, 126, 252, ...) + (1, 0, 3, 0, 6, 0, 10, ...)), see A000389 and A000217.

Equals row sums of triangle A160770.

F(1,5,n) is the number of bracelets with 1 blue, 5 identical red and n identical black beads. If F(1,5,1) = 3 and F(1,5,2) = 12 taken as a base, F(1,5,n) = n(n+1)(n+2)(n+3)/24 + F(1,3,n) + F(1,5,n-2). [F(1,3,n) is the number of bracelets with 1 blue, 3 identical red and n identical black beads. If F(1,3,1) = 2 and F(1,3,2) = 6 taken as a base F(1,3,n) = n(n+1)/2 + [|n/2|] + 1 + F(1,3,n-2)], where [|x|]: if a is an integer and a<=x<a+1, [|x|]=a. - Ata A. Uslu and Hamdi G. Ozmenekse, Mar 16 2012

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Winston C. Yang (paper in preparation).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

N. J. A. Sloane, Classic Sequences

L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250.

Ata A. Uslu and Hamdi G. Ozmenekse, F(1,3,n)

Ata A. Uslu and Hamdi G. Ozmenekse, F(1,5,n)

Index entries for linear recurrences with constant coefficients, signature (3, 0, -8, 6, 6, -8, 0, 3, -1).

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

a(2n-1) = n(n+1)(n+2)(2n+1)(2n+3)/30, a(2n) = (n+1)(n+2)(n+3)(4n^2+6n+5)/ 30. [M. F. Hasler, May 01 2009]

a(n) = (1/240)*(n+1)*(n+2)*(n+3)*(n+4)*(n+5) + (1/16)*(n+2)*(n+4)*(1/2)*(1+(-1)^n). [Yosu Yurramendi, Jun 20 2013]

a(n) = A038163(n)+3*A038163(n-2). - R. J. Mathar, Mar 29 2018

a:= n-> (Matrix([[1, 0$6, -3, -9]]). Matrix(9, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -8, 6, 6, -8, 0, 3, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Jul 31 2008

a[n_?OddQ] := 1/240*(n+1)*(n+2)*(n+3)*(n+4)*(n+5); a[n_?EvenQ] := 1/240*(n+2)*(n+4)*(n+6)*(n^2+3*n+5); Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Mar 17 2014, after M. F. Hasler *)

LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1}, {1, 3, 12, 28, 66, 126, 236, 396, 651}, 40] (* Ray Chandler, Sep 23 2015 *)

(PARI) a(k)= if(k%2, (k+1)*(k+3)*(k+5), (k+6)*(k^2+3*k+5))*(k+2)*(k+4)/240 \\ M. F. Hasler, May 01 2009

Cf. A160770, A053132 (bisection), A271870 (bisection), A018210 (partial sums).

Sequence in context: A294418 A308669 A115549 * A351643 A034503 A382648

Adjacent sequences: A005992 A005993 A005994 * A005996 A005997 A005998

nonn,easy

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

approved