Search: id:A005917
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%I A005917 M4968
%S A005917 1,15,65,175,369,671,1105,1695,2465,3439,4641,6095,7825,9855,12209,
%T A005917 14911,17985,21455,25345,29679,34481,39775,45585,51935,58849,66351,
%U A005917 74465,83215,92625,102719,113521,125055,137345,150415,164289,178991
%N A005917 Rhombic dodecahedral numbers: n^4 - (n-1)^4.
%C A005917 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 
               (alex(AT)kolmogorov.com), Aug 11 2006
%C A005917 a(n) = VarScheme(n,2) in the scheme displayed in A128195. - Peter Luschny 
               (peter(AT)luschny.de), Feb 26 2007
%C A005917 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
%C A005917 It appears that this sequence of numbers is the constant number found 
               in magic squares of order n, where n is an odd number. 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]
%D A005917 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005917 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.
%D A005917 J. H. Conway and R. K. Guy, The Book of Numbers, p. 53.
%D A005917 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (9).
%D A005917 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral 
               clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%D A005917 D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 
               819-822.
%H A005917 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A005917 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A005917 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005917 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005917 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RhombicDodecahedralNumber.html">Link to a section of The World of 
               Mathematics.</a>
%H A005917 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NexusNumber.html">Nexus Number</a>
%F A005917 a(n) = (2n-1)(2n^2 - 2n +1).
%F A005917 G.f.: x*(1+11*x+11*x^2+x^3)/(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
%F A005917 Sum a(i), {i=0, .., n)=(n+1)^4 - Mario Catalani (mario.catalani(AT)unito.it), 
               Jun 20 2003
%F A005917 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
%F A005917 a(n)=12*(sum(n^2))+(2*n+1) - Xavier Acloque Oct 16 2003
%F A005917 a(n)=(4*binomial(n+1, 2)+1)sqrt(8*binomial(n+1, 2)+1). - Paul Barry (pbarry(AT)wit.ie), 
               Mar 14 2004
%F A005917 Binomial transform of [1, 14, 36, 24, 0, 0, 0,...]. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Dec 20 2007
%p A005917 A005917:=(z+1)*(z**2+10*z+1)/(z-1)**4; [Conjectured by S. Plouffe in 
               his 1992 dissertation.]
%t A005917 lst={};Do[a=n^2;b=(n+1)^2;s=(a+b)*(b-a);AppendTo[lst,s],{n,0,5!}];lst 
               [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 23 2009]
%Y A005917 (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.
%Y A005917 Cf. A000538, A000583. First differences of A000583.
%Y A005917 A row of A047969.
%Y A005917 Cf. A128195.
%Y A005917 Sequence in context: A096905 A147857 A147858 this_sequence A027455 A152729 
               A055268
%Y A005917 Adjacent sequences: A005914 A005915 A005916 this_sequence A005918 A005919 
               A005920
%K A005917 nonn,easy,nice
%O A005917 0,2
%A A005917 N. J. A. Sloane (njas(AT)research.att.com).
%E A005917 More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

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