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A005917 Rhombic dodecahedral numbers: n^4 - (n-1)^4.
(Formerly M4968)
41
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; text; internal format)
OFFSET

1,2

COMMENTS

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

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 Janjic, 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. - David Quentin Dauthier, Nov 07 2008

a(n) = A045975(2*n-1,n) = A204558(2*n-1)/(2*n-1). - Reinhard Zumkeller, Jan 18 2012

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

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

REFERENCES

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

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Milan Janjic, Two Enumerative Functions

T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (9).

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number, MathWorld.

Eric Weisstein's World of Mathematics, Nexus Number

D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

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

Sum_{i=1..n} a(i) = n^4 = A000583(n). 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, May 08 2002

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_{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

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

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

MAPLE

A005917:=(z+1)*(z**2+10*z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation, for offset 0

MATHEMATICA

Table[n^4-(n-1)^4, {n, 40}]  (* Harvey P. Dale, Apr 01 2011 *)

#[[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 *)

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

(Haskell)

a005917 n = a005917_list !! (n-1)

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

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

-- Reinhard Zumkeller, Nov 13 2014

(Python)

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

for _ in range(10**2):

    A005917_list.append(m[-1])

    for i in range(3):

        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

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, A063493, A063494, A063495, A063496.

A row of A047969.

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

Cf. A001844, A000583, A000290.

Sequence in context: A096905 A147857 A147858 * A218216 A027455 A152729

Adjacent sequences:  A005914 A005915 A005916 * A005918 A005919 A005920

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified August 21 02:25 EDT 2017. Contains 290855 sequences.