|
Search: id:A005894
|
|
|
| A005894 |
|
Centered tetrahedral numbers.
(Formerly M3850)
|
|
+0
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; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Binomial transform of (1,4,6,4,0,0,0,.......) - Paul Barry (pbarry(AT)wit.ie), 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 R. Janjic (agnus(AT)blic.net), Jul 30 2007
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. 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.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
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
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
|
|
FORMULA
|
a(n)=(1/3)*(2*n+1)*(n^2+n+3). G.f.: (1-x^4)/(1-x)^5.
a(n)=C(n, 0)+4C(n, 1)+6C(n, 2)+4C(n, 3) - Paul Barry (pbarry(AT)wit.ie), 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 (alex(AT)kolmogorov.com), May 20 2006
binomial(n+6,n+3)+binomial(n+5,n+2)+binomial(n+4,n+1)+binomial(n+3,n).
|
|
MAPLE
|
A005894:=(z+1)*(1+z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
|
|
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 this_sequence A015622 A000750 A008487
Adjacent sequences: A005891 A005892 A005893 this_sequence A005895 A005896 A005897
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
Search completed in 0.002 seconds
|
