A301696 - OEIS

%I #23 Jun 08 2020 06:17:11

%S 1,6,17,33,54,81,113,150,193,241,294,353,417,486,561,641,726,817,913,

%T 1014,1121,1233,1350,1473,1601,1734,1873,2017,2166,2321,2481,2646,

%U 2817,2993,3174,3361,3553,3750,3953,4161,4374,4593,4817,5046,5281,5521,5766

%N Partial sums of A219529.

%H Colin Barker, <a href="/A301696/b301696.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F From _Colin Barker_, Mar 26 2018: (Start)

%F G.f.: (1 + x)^4 / ((1 - x)^3*(1 + x + x^2)).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4. (End)

%F From _G. C. Greubel_, May 27 2020: (Start)

%F a(n) = (ChebyshevU(n, -1/2) - ChebyshevU(n-1, -1/2) + 8*(3*n*(n+1) +1))/9.

%F a(n) = ( A131713(n) + 8*A028896(n) + 8 )/9. (End)

%p A301696:= n-> (8*(3*n*(n+1) +1) + `mod`(n+2, 3) - `mod`(n+1, 3))/9;

%p seq(A301696(n), n=0..60); # _G. C. Greubel_, May 27 2020

%t Table[(Mod[n+2, 3] - Mod[n+1, 3] + 8*(3*n*(n+1) +1))/9, {n,0,60}] (* _G. C. Greubel_, May 27 2020 *)

%o (PARI) Vec((1 + x)^4 / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ _Colin Barker_, Mar 26 2018

%o (Sage) [(24*n*(n+1)+8 + (n+2)%3 - (n+1)%3 )/9 for n in (0..60)] # _G. C. Greubel_, May 27 2020

%Y Cf. A219529.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Mar 25 2018