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Interlacing n^3/2 and n^2(n + 1)/2.
2

%I #19 Mar 15 2024 05:52:18

%S 1,4,18,32,75,108,196,256,405,500,726,864,1183,1372,1800,2048,2601,

%T 2916,3610,4000,4851,5324,6348,6912,8125,8788,10206,10976,12615,13500,

%U 15376,16384,18513,19652,22050,23328,26011,27436,30420,32000,35301,37044

%N Interlacing n^3/2 and n^2(n + 1)/2.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

%F a(n) = n^2 * floor((n + 1)/2).

%F G.f.: x*(1+3*x+11*x^2+5*x^3+4*x^4)/((1-x)^4*(1+x)^3). - _R. J. Mathar_, Sep 09 2008

%F a(n) = a(n-1)+ 3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7), a(1)=1, a(2)=4, a(3)=18, a(4)=32, a(5)=75, a(6)=108, a(7)=196. - _Harvey P. Dale_, Feb 18 2015

%F Sum_{n>=1} 1/a(n) = zeta(3)/4 + Pi^2/4 - 2*log(2). - _Amiram Eldar_, Mar 15 2024

%p A130656:=n->n^2 * floor((n + 1)/2): seq(A130656(n), n=1..100); # _Wesley Ivan Hurt_, Jan 21 2017

%t a[n_Integer] := n^2 * Floor[(n + 1)/2]

%t LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,4,18,32,75,108,196},50] (* _Harvey P. Dale_, Feb 18 2015 *)

%Y Cf. A093005 (quadratic equivalent), A065423 (linear equivalent).

%K easy,nonn

%O 1,2

%A _Olivier GĂ©rard_, Jun 21 2007