%I M4968 #214 Jan 22 2024 06:01:35

%S 1,15,65,175,369,671,1105,1695,2465,3439,4641,6095,7825,9855,12209,

%T 14911,17985,21455,25345,29679,34481,39775,45585,51935,58849,66351,

%U 74465,83215,92625,102719,113521,125055,137345,150415,164289,178991

%N Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.

%C Final digits of a(n), i.e., a(n) mod 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), i.e., a(n) mod 100, are repeated periodically with period of length 50. - _Alexander Adamchuk_, Aug 11 2006

%C a(n) = VarScheme(n,2) in the scheme displayed in A128195. - _Peter Luschny_, Feb 26 2007

%C 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 Janjic_, Nov 18 2007

%C 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. - _David Quentin Dauthier_, Nov 07 2008

%C Two times the area of the triangle with vertices at (0,0), ((n - 1)^2, n^2), and (n^2, (n - 1)^2). - _J. M. Bergot_, Jun 25 2013

%C Bisection of A006003. - _Omar E. Pol_, Sep 01 2018

%C Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals a(n+1). The first five rows of array M are [1, 7, 17, 31, 49, ...]; [3, 5, 15, 29, 47, ...]; [9, 11, 13, 27, 45, ...]; [19, 21, 23, 25, 43, ...]; [33, 35, 37, 39, 41, ...]. - _J. M. Bergot_, Jul 16 2013 [This contribution was moved here from A047926 by _Petros Hadjicostas_, Mar 08 2021.]

%C For n>=2, these are the primitive sides s of squares of type 2 described in A344332. - _Bernard Schott_, Jun 04 2021

%C (a(n) + 1) / 2 = A212133(n) is the number of cells in the n-th rhombic-dodecahedral polycube. - _George Sicherman_, Jan 21 2024

%D J. H. Conway and R. K. Guy, The Book of Numbers, p. 53.

%D E. Deza and M. M. Deza, Figurate Numbers, World Scientific Publishing, 2012, pp. 123-124.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005917/b005917.txt">Table of n, a(n) for n = 1..10000</a>

%H Mario Defranco and Paul E. Gunnells, <a href="https://arxiv.org/abs/2204.11361">Hypergraph matrix models and generating functions</a>, arXiv:2204.11361 [math.CO], 2022.

%H Milan Janjic, <a href="http://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>

%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (9).

%H Andy Nicol, <a href="/A005917/a005917.png">Illustration of Rhombic Dodecahedral Numbers</a>

%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RhombicDodecahedralNumber.html">Rhombic Dodecahedral Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NexusNumber.html">Nexus Number</a>.

%H D. Zeitlin, <a href="http://www.jstor.org/stable/2319798">A family of Galileo sequences</a>, Amer. Math. Monthly 82 (1975), 819-822.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (2*n - 1)*(2*n^2 - 2*n + 1).

%F Sum_{i=1..n} a(i) = n^4 = A000583(n). First differences of A000583.

%F G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^4. - _Simon Plouffe_ in his 1992 dissertation

%F 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_, May 08 2002

%F 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 a(n) = 2*n - 1 + 12*Sum_{i = 1..n} (i - 1)^2. - Xavier Acloque, Oct 16 2003

%F a(n) = (4*binomial(n,2) + 1)*sqrt(8*binomial(n,2) + 1). - _Paul Barry_, Mar 14 2004

%F Binomial transform of [1, 14, 36, 24, 0, 0, 0, ...], if the offset is 0. - _Gary W. Adamson_, Dec 20 2007

%F Sum_{i=1..n-1}(a(i) + a(i+1)) = 8*Sum_{i=1..n}(i^3 + i) = 16*A002817(n-1) for n > 1. - _Bruno Berselli_, Mar 04 2011

%F a(n+1) = a(n) + 2*(6*n^2 + 1) = a(n) + A005914(n). - _Vincenzo Librandi_, Mar 16 2011

%F a(n) = -a(-n+1). a(n) = (1/6)*(A181475(n) - A181475(n-2)). - _Bruno Berselli_, Sep 26 2011

%F a(n) = A045975(2*n-1,n) = A204558(2*n-1)/(2*n - 1). - _Reinhard Zumkeller_, Jan 18 2012

%F a(n+1) = Sum_{k=0..2*n+1} (A176850(n,k) - A176850(n-1,k))*(2*k + 1), n >= 1. - _L. Edson Jeffery_, Nov 02 2012

%F a(n) = A005408(n-1) * A001844(n-1) = (2*(n - 1) + 1) * (2*(n - 1)*n + 1) = A000290(n-1)*12 + 2 + a(n-1). - _Bruce J. Nicholson_, May 17 2017

%F a(n) = A007588(n) + A007588(n-1) = A000292(2n-1) + A000292(2n-2) + A000292(2n-3) = A002817(2n-1) - A002817(2n-2). - _Bruce J. Nicholson_, Oct 22 2017

%F a(n) = A005898(n-1) + 6*A000330(n-1) (cf. Deza, Deza, 2012, p. 123, Section 2.6.2). - _Felix Fröhlich_, Oct 01 2018

%F a(n) = A300758(n-1) + A005408(n-1). - _Bruce J. Nicholson_, Apr 23 2020

%F G.f.: polylog(-4, x)*(1-x)/x. See the _Simon Plouffe_ formula above (with expanded numerator), and the g.f. of the rows of A008292 by _Vladeta Jovovic_, Sep 02 2002. - _Wolfdieter Lang_, May 10 2021

%t Table[n^4-(n-1)^4,{n,40}] (* _Harvey P. Dale_, Apr 01 2011 *)

%t #[[2]]-#[[1]]&/@Partition[Range[0,40]^4,2,1] (* More efficient than the above Mathematica program because it only has to calculate each 4th power once *) (* _Harvey P. Dale_, Feb 07 2015 *)

%t Differences[Range[0,40]^4] (* _Harvey P. Dale_, Aug 11 2023 *)

%o (PARI) a(n)=n^4-(n-1)^4 \\ _Charles R Greathouse IV_, Jul 31 2011

%o (Magma) [n^4 - (n-1)^4: n in [1..50]]; // _Vincenzo Librandi_, Aug 01 2011

%o (Haskell)

%o a005917 n = a005917_list !! (n-1)

%o a005917_list = map sum $ f 1 [1, 3 ..] where

%o f x ws = us : f (x + 2) vs where (us, vs) = splitAt x ws

%o -- _Reinhard Zumkeller_, Nov 13 2014

%o (Python)

%o A005917_list, m = [], [24, -12, 2, 1]

%o for _ in range(10**2):

%o A005917_list.append(m[-1])

%o for i in range(3):

%o m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015

%Y (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, A063493, A063494, A063495, A063496.

%Y Column k=3 of A047969.

%Y Cf. A128195, A176850, A005408, A176271, A212133.

%Y Cf. A001844, A000583, A000290.

%Y Cf. A007588, A000292, A000332, A002817, A342354.

%Y Cf. A031215, A008292.

%Y Cf. A016754, A344330, A344332.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_