OFFSET
0,2
COMMENTS
Binomial transform of (1,4,6,4,0,0,0,...). - Paul Barry, Jul 01 2003
If X is an n-set and Y a fixed 4-subset of X then a(n-4) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).
FORMULA
a(n) = (2*n + 1)*(n^2 + n + 3)/3.
G.f.: (1+x)*(1+x^2)/(1-x)^4.
a(n) = C(n, 0) + 4*C(n, 1) + 6*C(n, 2) + 4*C(n, 3). - Paul Barry, Jul 01 2003
a(n) is the sum of 4 consecutive tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)*(n+2)*(n+3)/6 = A000292(n). a(n) = A000292(n-3) + A000292(n-2) + A000292(n-1) + A000292(n). - Alexander Adamchuk, May 20 2006
a(n) = binomial(n+3,n) + binomial(n+2,n-1) + binomial(n+1,n-2) + binomial(n,n-3). (modified by G. C. Greubel, Nov 30 2017)
a(n) = a(n-1) + 2*n^2 + 2, n>=1 (first differences A005893). - Vincenzo Librandi, Mar 27 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=5, a(2)=15, a(3)=35. - Harvey P. Dale, Nov 03 2011
E.g.f.: (3 + 12*x + 9*x^2 + 2*x^3)*exp(x)/3. - G. C. Greubel, Nov 30 2017
MAPLE
A005894:=(z+1)*(1+z**2)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(2n+1)(n^2+n+3)/3, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 5, 15, 35}, 40] (* Harvey P. Dale, Nov 03 2011 *)
PROG
(PARI) a(n)=(2*n+1)*(n^2+n+3)/3 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [(2*n+1)*(n^2+n+3)/3: n in [0..30]]; // G. C. Greubel, Nov 30 2017
CROSSREFS
(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Cf. A000292.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved