

A058187


Expansion of (1+x)/(1x^2)^4: duplicated tetrahedral numbers.


32



1, 1, 4, 4, 10, 10, 20, 20, 35, 35, 56, 56, 84, 84, 120, 120, 165, 165, 220, 220, 286, 286, 364, 364, 455, 455, 560, 560, 680, 680, 816, 816, 969, 969, 1140, 1140, 1330, 1330, 1540, 1540, 1771, 1771, 2024, 2024, 2300, 2300, 2600, 2600, 2925, 2925, 3276, 3276
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OFFSET

0,3


COMMENTS

For n >= i, i = 6,7, a(n  i) is the number of incongruent twocolor 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 180degree 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
The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums, see A180662 for their definitions, of the ConnellPol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Fi1(n) = a(n1) + 5*a(n2) + a(n3) + 5*a(n4).
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



FORMULA

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
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(7n) for all n in Z.


MAPLE



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, x1..1]
(Sage) [binomial((n//2)+3, 3) for n in (0..60)] # G. C. Greubel, Feb 18 2022


CROSSREFS

Cf. A057884. Sum of 2 consecutive terms gives A006918, whose sum of 2 consecutive terms gives A002623, whose sum of 2 consecutive terms gives A000292, which is this sequence without the duplication. Continuing to sum 2 consecutive terms gives A000330, A005900, A001845, A008412 successively.


KEYWORD

easy,nonn


AUTHOR



STATUS

approved



