0,2

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

a(n) = VarScheme(n,2) in the scheme displayed in A128195. - Peter Luschny (peter(AT)luschny.de), 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

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]

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.

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).

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.

Index entries for sequences related to linear recurrences with constant coefficients

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Nexus Number

a(n) = (2n-1)(2n^2 - 2n +1).

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

Sum a(i), {i=0, .., n)=(n+1)^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)=12*(sum(n^2))+(2*n+1) - Xavier Acloque Oct 16 2003

a(n)=(4*binomial(n+1, 2)+1)sqrt(8*binomial(n+1, 2)+1). - Paul Barry (pbarry(AT)wit.ie), Mar 14 2004

Binomial transform of [1, 14, 36, 24, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 20 2007

A005917:=(z+1)*(z**2+10*z+1)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]

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]

(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. A000538, A000583. First differences of A000583.

A row of A047969.

Cf. A128195.

Sequence in context: A096905 A147857 A147858 this_sequence A027455 A152729 A055268

Adjacent sequences: A005914 A005915 A005916 this_sequence A005918 A005919 A005920

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000