OFFSET
0,3
COMMENTS
Recurrence a(n) = a(n-2) + n^3, starting with a(0)=0, a(1)=1. Also, in physics, a(n)/4 is the trace of the spin operator |S_z|^3 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of the absolute values of their 3rd powers is 2*28/8 = a(3)/4. - Stanislav Sykora, Nov 07 2013
Also the number of 3-cycles in the (n+1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
With zero prepended and offset 1, the sequence starts 0,0,1,8,28,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q). Then b(n) is the total volume of the family of cubes with side length |q - p|. - Wesley Ivan Hurt, Apr 14 2018
LINKS
Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).
FORMULA
G.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^5).
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
a(n) = (2*n^4 + 8*n^3 + 8*n^2 - 1 + (-1)^n)/16.
a(n) = Sum_{k=0..floor((n-1)/2)} (n-2*k)^3.
a(n+1) = Sum_{k=0..n} k^3*(1 - (-1)^(n+k-1))/2.
a(n) = ((((x^2 - (x mod 2) - 4)/4)^2 - (((x^2 - (x mod 2) - 4)/4) mod 2))/8) = floor(((floor(x^2/4) - 1)^2)/8) where x = 2*n + 2. Replace x with 2*n - 1 to obtain A050534(n) = 3*A000332(n+1). Note that a(2*n) = A060300(n)/2 and a(2*n + 1) = A002593(n+1). - Raphie Frank, Jan 30 2014
a(n) = floor(1/(exp(2/n^2) - 1)^2)/2. Also a(n) = A007590(n+1)*A074148(n-1)/2. - Richard R. Forberg, Oct 26 2014
Sum_{n>=1} 1/a(n) = -cot(Pi/sqrt(2))*Pi/sqrt(2) - 1/2. - Amiram Eldar, Aug 25 2022
MATHEMATICA
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 1, 8, 28, 72, 153}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
CoefficientList[Series[x (1 + 4 x + x^2)/((1 + x) (1 - x)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2012 *)
Table[((-1)^n + 2 n^2 (n + 2)^2 - 1)/16, {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
PROG
(Magma) [(2*n^4+8*n^3+8*n^2-1)/16+(-1)^n/16: n in [0..50]]; // Vincenzo Librandi, Oct 27 2014
(PARI) my(x='x+O('x^99)); concat(0, Vec(x*(1+4*x+x^2)/((1+x)*(1-x)^5))) \\ Altug Alkan, Apr 16 2018
(Sage) [(2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16 for n in range(30)] # G. C. Greubel, Dec 16 2018
(GAP) List([0..30], n -> (2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16); # G. C. Greubel, Dec 16 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Apr 16 2005
STATUS
approved