A173961 Averages of two consecutive even cubes: (n^3 + (n+2)^3)/2.
4, 36, 140, 364, 756, 1364, 2236, 3420, 4964, 6916, 9324, 12236, 15700, 19764, 24476, 29884, 36036, 42980, 50764, 59436, 69044, 79636, 91260, 103964, 117796, 132804, 149036, 166540, 185364, 205556, 227164, 250236, 274820, 300964, 328716, 358124, 389236, 422100
Offset: 1
Examples
(0^3+2^3)/2=4, (2^3+4^3)/2=36, ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
I:=[4, 36, 140, 364]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012
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Mathematica
f[n_]:=(n^3+(n+2)^3)/2;Table[f[n],{n,0,5!,2}] CoefficientList[Series[(4+20*x+20*x^2+4*x^3)/(1-4*x+6*x^2-4*x^3+x^4),{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *) -
PARI
a(n)=4*n*(2*n^2-3*n+3)-4 \\ Charles R Greathouse IV, Jan 02 2012
Formula
G.f.: x*(4+20*x+20*x^2+4*x^3)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 04 2012
a(n) = 8*n^3 - 12*n^2 + 12*n - 4. - Charles R Greathouse IV, Jan 02 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012
a(n) = 4*A005898(n-1).
E.g.f.: 4 + 4*exp(x)*(-1 + 2*x + 3*x^2 + 2*x^3). - Elmo R. Oliveira, Aug 23 2025