%I A005900 M4128
%S A005900 0,1,6,19,44,85,146,231,344,489,670,891,1156,1469,1834,2255,2736,3281,
%T A005900 3894,4579,5340,6181,7106,8119,9224,10425,11726,13131,14644,16269,
%U A005900 18010,19871,21856,23969,26214,28595,31116,33781,36594,39559,42680
%N A005900 Octahedral numbers: (2n^3 + n)/3.
%C A005900 a(n) = 1^2 + 2^2 + ... + (n-1)^2 + n^2 + (n-1)^2 + ... + 2^2 + 1^2. -
Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 28 2001
%C A005900 Series reversion of g.f. A(x) is Sum_{n>0} -A066357(n)(-x)^n.
%C A005900 Also as a(n)=(1/6)*(4*n^3+2*n), n>0: structured tetragonal diamond numbers
(vertex structure 5) (Cf. A000447 - structured diamonds); and structured
trigonal anti-prism numbers (vertex structure 5) (Cf. A100185 - structured
anti-prisms). Cf. A100145 for more on structured polyhedral numbers.
- James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.
%C A005900 Schlaefli symbol for this polyhedron: {3,4}
%C A005900 If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal
to the number of 5-subests of X intersecting both Y and Z. - Milan
R. Janjic (agnus(AT)blic.net), Aug 26 2007
%C A005900 Starting with "1" = binomial transform of [1, 5, 8, 4, 0, 0, 0,...] where
(1, 5, 8, 4) = row 3 of the Chebyshev triangle A081277. - Gary W.
Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008
%C A005900 a(n) = largest coefficient of (1+...+x^(n-1)) ^ 4 [From Ron Hardin (rhhardin(AT)att.net),
Jul 23 2009]
%C A005900 Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 +
7x^3 + ...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27
2009]
%C A005900 Starting with offset 1 = the triangular series convolved with [1, 3,
4, 4, 4,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 28
2009]
%D A005900 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A005900 H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J.
Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical
and political essays in honor of Dirk J. Struik, Reidel, Dordrecht,
1974.
%D A005900 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq.
(5).
%D A005900 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral
clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A005900 T. D. Noe, <a href="b005900.txt">Table of n, a(n) for n=0..1000</a>
%H A005900 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A005900 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A005900 X. Acloque, <a href="http://members.fortunecity.fr/polynexus/index.html">
Polynexus Numbers and other mathematical wonders</a>.
%H A005900 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A005900 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
On Regular Polytope Numbers</a>
%H A005900 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A005900 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005900 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
OctahedralNumber.html">Link to a section of The World of Mathematics.</
a>
%F A005900 Partial sums of centered square numbers A001844. - Paul Barry (pbarry(AT)wit.ie),
Jun 26 2003
%F A005900 G.f.: x(1+x)^2/(1-x)^4. a(n)=-a(-n)=(2n^3+n)/3.
%F A005900 a(n)=((n^5-(n-1)^5)-((n-1)^5-(n-2)^5))/30 - Xavier Acloque Oct 17 2003
%F A005900 a(n) is the sum of the products pq, where p and q are both positive and
odd and p+q=2n, e.g. a(4) = 7*1 + 5*3 + 3*5 + 1*7 = 44 - Jon Perry
(perry(AT)globalnet.co.uk), May 17 2005
%F A005900 a(n) = 4C(n,3) + 2C(n,2) + n^2 - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu),
Jul 06 2006
%F A005900 a(n) = C(n+2,3) + 2 C(n+1,3) + C(n,3)
%F A005900 G.f.: sage: taylor( mul( x*(x^2+2*x+1)/(x-1)^4 for i in xrange(1,2)),
x,0,40)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun
03 2009]
%F A005900 sum(1/((2*n^3+n)*(1/3)),n=1..infinity)=3*gamma+3*Psi((I*(1/2))*sqrt(2))-(1/
2)*(3*I)*Pi*coth((1/2)*Pi*sqrt(2))-(1/2)*(3*I)*sqrt(2)=1.27818515909094617954039094836757133842390...
[From Stephen Crowley (crow(AT)crowlogic.net), Jul 14 2009]
%p A005900 al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,
r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(3,n),n=0..100)];
%p A005900 A005900:=(z+1)**2/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
%p A005900 with (combinat):seq(fibonacci(4,2*n)/12, n=0..40); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Apr 21 2008
%o A005900 (PARI) a(n)=(2*n^3+n)/3
%Y A005900 Sums of 2 consecutive terms give A001845. Cf. A001844.
%Y A005900 1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003,
A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467,
A007588, A062025, A063521, A063522, A063523.
%Y A005900 Cf. A022521.
%Y A005900 Cf. A081277.
%Y A005900 Sequence in context: A096957 A035495 A061293 this_sequence A138357 A005712
A070893
%Y A005900 Adjacent sequences: A005897 A005898 A005899 this_sequence A005901 A005902
A005903
%K A005900 nonn,easy
%O A005900 0,3
%A A005900 N. J. A. Sloane (njas(AT)research.att.com).
%E A005900 More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001
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