0,2

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

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 127.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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).

a(n) = (2*n + 1)*(n^2 + n + 3)/3.

G.f.: (1+x)*(1+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation

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: Sum_{i=-2..1} A000292(n+i). - 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

a(n) = A006527(n)+A006527(n+1) = A000330(n-1)+A000330(n+1). - R. J. Mathar, Jun 05 2025

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 *)

(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

(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.

Sequence in context: A063382 A393110 A069983 * A374711 A015622 A346761

Adjacent sequences: A005891 A005892 A005893 * A005895 A005896 A005897

nonn,easy,nice

approved