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A058187
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Expansion of (1+x)/(1-x^2)^4: duplicated tetrahedral numbers.
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17
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = A108299(n-3,n)*(-1)^floor(n/2) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
For n>=i,i=6,7, a(n-i) is the number of incongruent two-color bracelets of n beads, i from them are black (Cf. A005513, A032280), having a diameter of symmetry. The latter means the following: if to imagine (0,1)-beads as points (with the corresponding labels) dividing a circumference of a bracelet in n the same parts, then a diameter of symmetry is a diameter (connecting or not two beads) such that a 180 degree turn around it of one of two sets of points (obtained by splitting circumference by this diameter) leads to the coincidence of the two sets (including their labels). - Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 3 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)
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REFERENCES
| H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..880
V. Shevelev, A problem of enumeration of two-color bracelets with several variations
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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 (pbarry(AT)wit.ie), Aug 19 2003
Euler transform of finite sequence [1, 3]. - Michael Somos Jun 07 2005
G.f.: 1/((1-x)*(1-x^2)^3). a(n)=-a(-7-n).
a(n)=C(floor(n/2)+3, 3). - Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 3 2011
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MAPLE
| A058187:= proc(n) option remember; A058187(n):= binomial(floor(n/2)+3, 3) end: seq(A058187(n), n=0..51); [From Johannes W. Meijer, May 20 2011]
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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
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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.
Cf. A096338, A005513, A032280
Cf. A000292, A190717, A190718. [From Johannes W. Meijer, May 20 2011]
Sequence in context: A168326 A101256 A116569 * A188271 A006477 A182699
Adjacent sequences: A058184 A058185 A058186 * A058188 A058189 A058190
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KEYWORD
| easy,nonn
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Nov 20 2000
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