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A236773 a(n) = n + floor( n^2/2 + n^3/3 ). 2

%I

%S 0,1,6,16,33,59,96,145,210,292,393,515,660,829,1026,1252,1509,1799,

%T 2124,2485,2886,3328,3813,4343,4920,5545,6222,6952,7737,8579,9480,

%U 10441,11466,12556,13713,14939,16236,17605,19050,20572,22173,23855,25620,27469

%N a(n) = n + floor( n^2/2 + n^3/3 ).

%C This sequence follows A074148 and A042965, A236771.

%C The prime terms are 59, 829, 14939, 35759, 93719, 132409, 155219, 290399, 414179, 487463, ... .

%C If a(k) is prime then k == 1, 5, 7 or 11 (mod 12).

%C Third differences: 1, 2, 2, 2, 1, 4 repeated (unsigned terms of A181982).

%C Fourth differences: 1, 0, 0, -1, 3, -3 repeated (see A131193).

%H Bruno Berselli, <a href="/A236773/b236773.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,1,-3,3,-1).

%F G.f.: x*(1+3*x+x^2+2*x^3+2*x^4+2*x^5+x^7) / ((1+x)*(1-x+x^2)*(1+x+x^2)*(1-x)^4).

%F a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-6) -3*a(n-7) +3*a(n-8) -a(n-9).

%F Also, for h>=0:

%F a(6h) = 6*h*( 12*h^2 + 3*h + 1 ),

%F a(6h+1) = 72*h^3 + 54*h^2 + 18*h + 1,

%F a(6h+2) = 6*( 4*h + 1 )*( 3*h^2 + 3*h + 1 ),

%F a(6h+3) = 2*( 36*h^3 + 63*h^2 + 39*h + 8 ),

%F a(6h+4) = 3*( 24*h^3 + 54*h^2 + 42*h + 11 ),

%F a(6h+5) = 72*h^3 + 198*h^2 + 186*h + 59.

%p seq(n+floor(n^2/2+n^3/3),n=0..43); # _Paolo P. Lava_, Aug 24 2018

%t Table[n + Floor[n^2/2 + n^3/3], {n, 0, 50}]

%t CoefficientList[Series[x (1 + 3 x + x^2 + 2 x^3 + 2 x^4 + 2 x^5 + x^7)/((1 + x) (1 - x + x^2) (1 + x + x^2) (1 - x)^4), {x, 0, 50}], x] (* _Vincenzo Librandi_, Feb 08 2014 *)

%o (MAGMA) [n+Floor(n^2/2+n^3/3): n in [0..50]];

%o (MAGMA) I:=[0,1,6,16,33,59,96,145,210]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-6)-3*Self(n-7)+3*Self(n-8)-Self(n-9): n in [1..50]]; // _Vincenzo Librandi_, Feb 08 2014

%o (PARI) vector(60, n, n--; n+floor(n^2/2 +n^3/3)) \\ _G. C. Greubel_, Aug 12 2018

%Y Cf. A074148: n+floor(n^2/2).

%Y Cf. A042965: n+floor(1/2+n/3); A236771: n+floor(n/2+n^2/3).

%Y Cf. A236772: floor(sum(i=1..n, n^i/i)).

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Feb 07 2014

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Last modified January 26 20:11 EST 2020. Contains 331288 sequences. (Running on oeis4.)