login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005894 Centered tetrahedral numbers.
(Formerly M3850)
19
1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, 18971, 20895, 22945, 25125, 27439, 29891, 32485, 35225, 38115 (list; graph; refs; listen; history; text; internal format)
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

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).

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

Milan Janjic, Two Enumerative Functions

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Index to sequences with 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)+4C(n, 1)+6C(n, 2)+4C(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

binomial(n+6,n+3)+binomial(n+5,n+2)+binomial(n+4,n+1)+binomial(n+3,n).

a(n) = a(n-1) +2*n^2+2, n>=1 (first differences A005893). - Vincenzo Librandi, Mar 27 2011

a(0)=1, a(1)=5, a(2)=15, a(3)=35, a(n)=4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4) [From Harvey P. Dale, Nov 03 2011]

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 *)

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.

Sequence in context: A061829 A063382 A069983 * A015622 A000750 A008487

Adjacent sequences:  A005891 A005892 A005893 * A005895 A005896 A005897

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 2 05:40 EDT 2014. Contains 247537 sequences.