OFFSET
0,3
COMMENTS
For n >= i, i = 6,7, a(n - i) is the number of incongruent two-color bracelets of n beads, i of which are black (cf. A005513, A032280), having a diameter of symmetry. The latter means the following: if we imagine (0,1)-beads as points (with the corresponding labels) dividing a circumference of a bracelet into n identical parts, then a diameter of symmetry is a diameter (connecting two beads or not) such that a 180-degree turn of one of two sets of points around it (obtained by splitting the circumference by this diameter) leads to the coincidence of the two sets (including their labels). - Vladimir Shevelev, May 03 2011
From Johannes W. Meijer, May 20 2011: (Start)
The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Fi1(n) = a(n-1) + 5*a(n-2) + a(n-3) + 5*a(n-4).
The Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above. (End)
The number of quadruples of integers [x, u, v, w] that satisfy x > u > v > w >= 0, n + 5 = x + u. - Michael Somos, Feb 09 2015
Also, this sequence is the fourth column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - Mohammad K. Azarian, Jul 18 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..880
Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., Vol. 10, No. 8 (1979), 964-999.
Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
FORMULA
a(n) = A006918(n+1) - a(n-1).
a(2*n) = a(2*n+1) = A000292(n) = (n+1)*(n+2)*(n+3)/6.
a(n) = (2*n^3 + 21*n^2 + 67*n + 63)/96 + (n^2 + 7*n + 11)(-1)^n/32. - Paul Barry, Aug 19 2003
a(n) = A108299(n-3,n)*(-1)^floor(n/2) for n > 2. - Reinhard Zumkeller, Jun 01 2005
Euler transform of finite sequence [1, 3]. - Michael Somos, Jun 07 2005
G.f.: 1 / ((1 - x) * (1 - x^2)^3) = 1 / ((1 + x)^3 * (1 - x)^4). a(n) = -a(-7-n) for all n in Z.
a(n) = binomial(floor(n/2) + 3, 3). - Vladimir Shevelev, May 03 2011
Sum_{n>=0} 1/a(n) = 3. - Amiram Eldar, Aug 18 2022
MAPLE
A058187:= proc(n) option remember; A058187(n):= binomial(floor(n/2)+3, 3) end: seq(A058187(n), n=0..51); # Johannes W. Meijer, May 20 2011
MATHEMATICA
a[n_]:= Length @ FindInstance[{x>u, u>v, v>w, w>=0, x+u==n+5}, {x, u, v, w}, Integers, 10^9]; (* Michael Somos, Feb 09 2015 *)
With[{tetra=Binomial[Range[30]+2, 3]}, Riffle[tetra, tetra]] (* Harvey P. Dale, Mar 22 2015 *)
PROG
(PARI) {a(n) = binomial(n\2+3, 3)}; /* Michael Somos, Jun 07 2005 */
(Haskell)
a058187 n = a058187_list !! n
a058187_list = 1 : f 1 1 [1] where
f x y zs = z : f (x + y) (1 - y) (z:zs) where
z = sum $ zipWith (*) [1..x] [x, x-1..1]
-- Reinhard Zumkeller, Dec 21 2011
(Sage) [binomial((n//2)+3, 3) for n in (0..60)] # G. C. Greubel, Feb 18 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, Nov 20 2000
STATUS
approved