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A005917
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Rhombic dodecahedral numbers: n^4 - (n-1)^4.
(Formerly M4968)
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27
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1, 15, 65, 175, 369, 671, 1105, 1695, 2465, 3439, 4641, 6095, 7825, 9855, 12209, 14911, 17985, 21455, 25345, 29679, 34481, 39775, 45585, 51935, 58849, 66351, 74465, 83215, 92625, 102719, 113521, 125055, 137345, 150415, 164289, 178991
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Final digits of a(n), Mod[a(n),10], are repeated periodically with period of length 5 {1,5,5,5,9}. There is a symmetry in this list since the sum of two numbers equally distant from the ends is equal to 10 = 1 + 9 = 5 + 5 = 2*5. Last two digits of a(n), Mod[a(n),100], are repeated periodically with period of length 50. - Alexander Adamchuk, Aug 11 2006
a(n) = VarScheme(n,2) in the scheme displayed in A128195. - Peter Luschny, Feb 26 2007
If Y is a 3-subset of a 2n-set X then, for n>=2, a(n-2) is the number of 4-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Nov 18 2007
The numbers are the constant number found in magic squares of order n, where n is an odd number, see the comment in A006003. A Magic Square of side 1 is 1; 3 is 15; 5 is 65 and so on. [From David Quentin Dauthier (d_dauthier(AT)yahoo.com), Nov 07 2008]
a(n) = A045975(2*n-1,n) = A204558(2*n-1)/(2*n-1). [Reinhard Zumkeller, Jan 18 2012]
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REFERENCES
| J. H. Conway and R. K. Guy, The Book of Numbers, p. 53.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (9).
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.
D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..10000
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.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number, MathWorld.
Eric Weisstein's World of Mathematics, Nexus Number
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
| a(n) = (2*n-1)*(2*n^2-2*n +1).
First differences of A000583.
G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^4. More generally, g.f. for n^m - (n-1)^m is Euler(m, x)/(1-x)^m, where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292). E.g.f.: x*(exp(y/(1-x))-exp(x*y/(1-x)))/(exp(x*y/(1-x))-x*exp(y/(1-x))). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 08 2002
Sum{a(i), i=1..n} = n^4. - Mario Catalani (mario.catalani(AT)unito.it), Jun 20 2003
a(n) = sum of the next (2*n-1) odd numbers; i.e. group the odd numbers so that the n-th group contains (2*n-1) elements like this (1), (3, 5, 7), (9, 11, 13, 15, 17), (19, 21, 23, 25, 27, 29, 31), ... E.g. a(3)=65 because 9+11+13+15+17=65 - Xavier Acloque Oct 11 2003
a(n)= 2*n-1+12*sum_{i=1..n} (i-1)^2. - Xavier Acloque Oct 16 2003
a(n) = (4*binomial(n, 2)+1)*sqrt(8*binomial(n, 2)+1). - Paul Barry, Mar 14 2004
Binomial transform of [1, 14, 36, 24, 0, 0, 0,...], if the offset is 0. - Gary W. Adamson, Dec 20 2007
Sum(a(i)+a(i+1), i=1..n-1) = 8*sum(i^3+i, i=1..n) = 16*A002817(n-1) for n>1. - Bruno Berselli, Mar 04 2011
a(n+1) = a(n) + 2*(6*n^2+1) =a(n)+ A005914(n). - Vincenzo Librandi, Mar 16 2011
a(n) = -a(-n+1). a(n) = (1/6)*(A181475(n)-A181475(n-2)). - Bruno Berselli, Sep 26 2011
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MAPLE
| A005917:=(z+1)*(z**2+10*z+1)/(z-1)**4; [S. Plouffe in his 1992 dissertation, for offset 0.]
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MATHEMATICA
| Table[n^4-(n-1)^4, {n, 40}] (* From Harvey P. Dale, Apr 01 2011 *)
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PROG
| (PARI) a(n)=n^4-(n-1)^4 \\ Charles R Greathouse IV, Jul 31 2011
(MAGMA) [n^4 - (n-1)^4: n in [1..50]]; // Vincenzo Librandi, Aug 01 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.
A row of A047969.
Cf. A128195.
Sequence in context: A096905 A147857 A147858 * A027455 A152729 A055268
Adjacent sequences: A005914 A005915 A005916 * A005918 A005919 A005920
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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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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000
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