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

1, 1, 3, 5, 8, 11, 16, 20, 26, 32, 39, 46, 55, 63, 73, 83, 94, 105, 118, 130, 144, 158, 173, 188, 205, 221, 239, 257, 276, 295, 316, 336, 358, 380, 403, 426, 451, 475, 501, 527, 554, 581, 610, 638, 668, 698, 729, 760, 793, 825, 859, 893, 928, 963, 1000, 1036, 1074, 1112, 1151, 1190

Offset: 0

Author

Bruno Berselli, Jan 20 2017

Keywords

• nonn
• easy

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

Subsequences: A033577, A244805 (numbers of the form 1 + k/2 + k^2/3), A212978 (second bisection).

Cf. A236771: n + floor(n/2) + floor(n^2/3).

Cf. A008619: 1 + floor(n/2); A087483: 1 + floor(n^2/3).

Cf. A000212, A004526.

Programs

Magma

[1 + n div 2 + n^2 div 3: n in [0..60]];

Maple

A281333:=n->1 + floor(n/2) + floor(n^2/3): seq(A281333(n), n=0..100); # Wesley Ivan Hurt, Feb 09 2017

Mathematica

Table[1 + Floor[n/2] + Floor[n^2/3], {n, 0, 60}]
LinearRecurrence[{1,1,0,-1,-1,1},{1,1,3,5,8,11},80] (* Harvey P. Dale, Sep 29 2024 *)

Maxima

makelist(1+floor(n/2)+floor(n^2/3), n, 0, 60);

PARI

vector(60, n, n--; 1+floor(n/2)+floor(n^2/3))

Python

[1+int(n/2)+int(n**2/3) for n in range(60)]

Sage

[1+floor(n/2)+floor(n^2/3) for n in range(60)]

Formula

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

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).

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

a(n) = (12*n^2 + 18*n + 4*(-1)^(2*n/3) + 4*(-1)^(-2*n/3) + 9*(-1)^n + 19)/36.

a(n) - n = a(-n).

a(6*k+r) = 12*k^2 + (4*r+3)*k + a(r), where 0 <= r <= 5. Particular cases:

a(6*k) = A244805(k+1), a(6*k+1) = A033577(k).

a(n+2) - a(n) = A004773(n+2).

a(n+3) - a(n) = A014601(n+2).

a(n+4) - a(n) = A047480(n+3).

a(n) - a(-n+3) = 2*A001651(n-1).

a(n) + a(-n+3) = 2*A097922(n-1).

a(n) = 1 + A004526(n) + A000212(n) = A008619(n) + A000212(n). - Omar E. Pol, Dec 23 2020