A161712 a(n) = (4*n^3 - 6*n^2 + 8*n + 3)/3.

1, 3, 9, 27, 65, 131, 233, 379, 577, 835, 1161, 1563, 2049, 2627, 3305, 4091, 4993, 6019, 7177, 8475, 9921, 11523, 13289, 15227, 17345, 19651, 22153, 24859, 27777, 30915, 34281, 37883, 41729, 45827, 50185, 54811, 59713, 64899, 70377, 76155

Offset: 0

Author

Reinhard Zumkeller, Jun 17 2009

Keywords

• nonn
• easy

Comments

{a(k): 0 <= k < 4} = divisors of 27:

a(n) = A027750(A006218(26) + k + 1), 0 <= k < A000005(27).

a(n), n > 0 is the number of points of the half-integer lattice in R^n that lie in the open unit ball. - Tom Harris, Jun 15 2021

Examples

			Differences of divisors of 27 to compute the coefficients of their interpolating polynomial, see formula:
  1     3     9    27
     2     6    18
        4    12
           8
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Reinhard Zumkeller, Enumerations of Divisors
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Programs

Magma

[(4*n^3 - 6*n^2 + 8*n + 3)/3: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Mathematica

Table[(4n^3-6n^2+8n+3)/3,{n,0,80}] (* Harvey P. Dale, Apr 13 2011 *)

PARI

a(n)=(4*n^3-6*n^2+8*n)/3+1 \\ Charles R Greathouse IV, Jul 16 2011

Formula

a(n) = C(n,0) + 2*C(n,1) + 4*C(n,2) + 8*C(n,3).

G.f.: (x+1)*(1+x*(5*x-2))/(x-1)^4. - Harvey P. Dale, Apr 13 2011

E.g.f.: (1/3)*(4*x^3 + 6*x^2 + 6*x + 3)*exp(x). - G. C. Greubel, Jul 16 2017

a(n) -a(n-1) = A005899(n-1), n>=2. - R. J. Mathar, Aug 03 2025