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A005894
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Centered tetrahedral numbers.
(Formerly M3850)
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19
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
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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.
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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.
Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1)
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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 (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).
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]
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MAPLE
| A005894:=(z+1)*(1+z**2)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[(2n+1)(n^2+n+3)/3, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 5, 15, 35}, 40] (* From Harvey P. Dale, Nov 03 2011 *)
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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
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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