|
| |
|
|
A005900
|
|
Octahedral numbers: (2*n^3 + n)/3.
(Formerly M4128)
|
|
77
|
|
|
|
0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, 19871, 21856, 23969, 26214, 28595, 31116, 33781, 36594, 39559, 42680
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
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
Series reversion of g.f. A(x) is Sum_{n>0} -A066357(n)(-x)^n.
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.
Schlaefli symbol for this polyhedron: {3,4}
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-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
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, Jul 19 2008
a(n) = largest coefficient of (1+...+x^(n-1)) ^ 4 [From R. H. Hardin, Jul 23 2009]
Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 + 7x^3 + ...) = A005408(x). [From Gary W. Adamson, Jul 27 2009]
Starting with offset 1 = the triangular series convolved with [1, 3, 4, 4, 4,...]. [From Gary W. Adamson, Jul 28 2009]
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). [From Daniel Forgues, May 14 2010]
Let b be any product of four different primes. Then the divisor lattice of b^n is of width a(n+1). [From Jean Drabbe (jdrabbe(AT)skynet.be), Oct 13 2010]
Arises in Bezdek's proof on contact numbers for congruent sphere packings (see preprint). - Jonathan Vos Post, Feb 08 2011
A005900(n+1) is the number of 2x2 matrices with all terms in {0,1,...,n} and (sum of terms)=2n. [Clark Kimberling, Mar 19 2012]
a(n) is the number of semistandard Young tableaux over all partitions of 3 with maximal element <= n. - Alois P. Heinz, Mar 22 2012
Self convolution of odd numbers. [Reinhard Zumkeller, Apr 04 2012]
a(n) is the number of (w,x,y,z) with all terms in {1,...,n} and w+x=y+z; also the number of (w,x,y,z) with all terms in {0,...,n} and |w-x|<=y. [Clark Kimberling, Jun 02 2012]
|
|
|
REFERENCES
|
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.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (5).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Karoly Bezdek, Contact numbers for congruent sphere packings, arXiv:1102.1198
Milan Janjic, Two Enumerative Functions
Hyun Kwang Kim, On Regular Polytope Numbers
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Octahedral Number.
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-6,4,-1)
|
|
|
FORMULA
|
Partial sums of centered square numbers A001844. - Paul Barry, Jun 26 2003
Euler transform of length 2 sequence [6, -2]. - Michael Somos Mar 27 2011
G.f.: x * (1 + x)^2 / (1 - x)^4. a(n) = -a(-n) = (2*n^3 + n) / 3.
a(n)=( ((n+1)^5-n^5) -(n^5-(n-1)^5)) )/30 - Xavier Acloque Oct 17 2003
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, May 17 2005
a(n) = 4C(n,3) + 4C(n,2) + C(n,1) - Mitch Harris, Jul 06 2006
a(n) = C(n+2,3) + 2 C(n+1,3) + C(n,3). (this pair generalizes - see A014820 the 4-cross polytope numbers)
sum(1/a(n),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) = A175577. [From Stephen Crowley (crow(AT)crowlogic.net), Jul 14 2009]
a(n) = A035597(n)/2. - J. M. Bergot, Jun 11 2012
|
|
|
EXAMPLE
|
x + 6*x^2 + 19*x^3 + 44*x^4 + 85*x^5 + 146*x^6 + 231*x^7 + 344*x^8 + ...
|
|
|
MAPLE
|
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)];
A005900:=(z+1)**2/(z-1)**4; [Simon Plouffe in his 1992 dissertation.]
with (combinat):seq(fibonacci(4, 2*n)/12, n=0..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2008
|
|
|
MATHEMATICA
|
f[n_]:=(n+1)^2+n^2; s=0; lst={}; Do[AppendTo[lst, s+=f[n]], {n, 0, 4!}]; lst..and/or.. f[n_]:=(2*n^3+n)/3; lst={}; Do[AppendTo[lst, f[n]], {n, 0, 4!}]; lst [From Vladimir Joseph Stephan Orlovsky, Feb 17 2010]
|
|
|
PROG
|
(PARI) {a(n) = (2*n^3 + n) / 3}
(Haskell)
a005900 n = sum $ zipWith (*) odds $ reverse odds where odds = take n a005408_list
-- Reinhard Zumkeller, Apr 04 2012
(Maxima) A005900(n):=(2*n^3 + n)/3$ makelist(A005900(n), n, 0, 20); /* Martin Ettl, Jan 07 2013 */
|
|
|
CROSSREFS
|
Sums of 2 consecutive terms give A001845. Cf. A001844.
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.
Cf. A022521.
Cf. A081277.
Row n=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A005408.
Sequence in context: A212684 A035495 A061293 * A138357 A183763 A209403
Adjacent sequences: A005897 A005898 A005899 * A005901 A005902 A005903
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
EXTENSIONS
|
More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001
|
|
|
STATUS
|
approved
|
| |
|
|
