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%I #37 Sep 08 2022 08:45:45
%S 1,3,9,27,65,131,233,379,577,835,1161,1563,2049,2627,3305,4091,4993,
%T 6019,7177,8475,9921,11523,13289,15227,17345,19651,22153,24859,27777,
%U 30915,34281,37883,41729,45827,50185,54811,59713,64899,70377,76155
%N a(n) = (4*n^3 - 6*n^2 + 8*n + 3)/3.
%C {a(k): 0 <= k < 4} = divisors of 27:
%C a(n) = A027750(A006218(26) + k + 1), 0 <= k < A000005(27).
%C 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
%H Vincenzo Librandi, <a href="/A161712/b161712.txt">Table of n, a(n) for n = 0..10000</a>
%H Reinhard Zumkeller, <a href="/A161700/a161700.txt">Enumerations of Divisors</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = C(n,0) + 2*C(n,1) + 4*C(n,2) + 8*C(n,3).
%F G.f.: ((x+1)(1+x(5x-2)))/(x-1)^4. - _Harvey P. Dale_, Apr 13 2011
%F E.g.f.: (1/3)*(4*x^3 + 6*x^2 + 6*x + 3)*exp(x). - _G. C. Greubel_, Jul 16 2017
%e Differences of divisors of 27 to compute the coefficients of their interpolating polynomial, see formula:
%e 1 3 9 27
%e 2 6 18
%e 4 12
%e 8
%t Table[(4n^3-6n^2+8n+3)/3,{n,0,80}] (* _Harvey P. Dale_, Apr 13 2011 *)
%o (PARI) a(n)=(4*n^3-6*n^2+8*n)/3+1 \\ _Charles R Greathouse IV_, Jul 16 2011
%o (Magma) [(4*n^3 - 6*n^2 + 8*n + 3)/3: n in [0..40]]; // _Vincenzo Librandi_, Jul 17 2011
%K nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Jun 17 2009
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