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A281333 a(n) = 1 + floor(n/2) + floor(n^2/3).

%I A281333 #36 Sep 29 2024 12:11:56
%S A281333 1,1,3,5,8,11,16,20,26,32,39,46,55,63,73,83,94,105,118,130,144,158,
%T A281333 173,188,205,221,239,257,276,295,316,336,358,380,403,426,451,475,501,
%U A281333 527,554,581,610,638,668,698,729,760,793,825,859,893,928,963,1000,1036,1074,1112,1151,1190
%N A281333 a(n) = 1 + floor(n/2) + floor(n^2/3).
%H A281333 Bruno Berselli, <a href="/A281333/b281333.txt">Table of n, a(n) for n = 0..1000</a>
%H A281333 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A281333 G.f.: (1 + x^2 + x^3 + x^4)/((1 + x)*(1 + x + x^2)*(1 - x)^3).
%F A281333 a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
%F A281333 a(n) = 1 + floor(n/2 + n^2/3).
%F A281333 a(n) = (12*n^2 + 18*n + 4*(-1)^(2*n/3) + 4*(-1)^(-2*n/3) + 9*(-1)^n + 19)/36.
%F A281333 a(n) - n = a(-n).
%F A281333 a(6*k+r) = 12*k^2 + (4*r+3)*k + a(r), where 0 <= r <= 5. Particular cases:
%F A281333 a(6*k) = A244805(k+1), a(6*k+1) = A033577(k).
%F A281333 a(n+2) -  a(n) =   A004773(n+2).
%F A281333 a(n+3) -  a(n) =   A014601(n+2).
%F A281333 a(n+4) -  a(n) =   A047480(n+3).
%F A281333 a(n) - a(-n+3) = 2*A001651(n-1).
%F A281333 a(n) + a(-n+3) = 2*A097922(n-1).
%F A281333 a(n) = 1 + A004526(n) + A000212(n) = A008619(n) + A000212(n). - _Omar E. Pol_, Dec 23 2020
%p A281333 A281333:=n->1 + floor(n/2) + floor(n^2/3): seq(A281333(n), n=0..100); # _Wesley Ivan Hurt_, Feb 09 2017
%t A281333 Table[1 + Floor[n/2] + Floor[n^2/3], {n, 0, 60}]
%t A281333 LinearRecurrence[{1,1,0,-1,-1,1},{1,1,3,5,8,11},80] (* _Harvey P. Dale_, Sep 29 2024 *)
%o A281333 (PARI) vector(60, n, n--; 1+floor(n/2)+floor(n^2/3))
%o A281333 (Python) [1+int(n/2)+int(n**2/3) for n in range(60)]
%o A281333 (Sage) [1+floor(n/2)+floor(n^2/3) for n in range(60)]
%o A281333 (Maxima) makelist(1+floor(n/2)+floor(n^2/3), n, 0, 60);
%o A281333 (Magma) [1 + n div 2 + n^2 div 3: n in [0..60]];
%Y A281333 Subsequences: A033577, A244805 (numbers of the form 1 + k/2 + k^2/3), A212978 (second bisection).
%Y A281333 Cf. A236771: n + floor(n/2) + floor(n^2/3).
%Y A281333 Cf. A008619: 1 + floor(n/2); A087483: 1 + floor(n^2/3).
%Y A281333 Cf. A000212, A004526.
%K A281333 nonn,easy
%O A281333 0,3
%A A281333 _Bruno Berselli_, Jan 20 2017