A236771 -id:A236771 - cp's OEIS frontend

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

Bruno Berselli, Jan 20 2017

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).

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.

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

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

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 *)

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

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

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

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

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

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

Original entry on oeis.org

0, 1, 6, 16, 33, 59, 96, 145, 210, 292, 393, 515, 660, 829, 1026, 1252, 1509, 1799, 2124, 2485, 2886, 3328, 3813, 4343, 4920, 5545, 6222, 6952, 7737, 8579, 9480, 10441, 11466, 12556, 13713, 14939, 16236, 17605, 19050, 20572, 22173, 23855, 25620, 27469

Offset: 0

Bruno Berselli, Feb 07 2014

This sequence follows A074148 and A042965, A236771.
The prime terms are 59, 829, 14939, 35759, 93719, 132409, 155219, 290399, 414179, 487463, ... .
If a(k) is prime then k == 1, 5, 7 or 11 (mod 12).
Third differences: 1, 2, 2, 2, 1, 4 repeated (unsigned terms of A181982).
Fourth differences: 1, 0, 0, -1, 3, -3 repeated (see A131193).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,1,-3,3,-1).

Cf. A074148: n+floor(n^2/2).
Cf. A042965: n+floor(1/2+n/3); A236771: n+floor(n/2+n^2/3).
Cf. A236772: floor(sum(i=1..n, n^i/i)).

Magma
```
[n+Floor(n^2/2+n^3/3): n in [0..50]];
```

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

seq(n+floor(n^2/2+n^3/3),n=0..43); # Paolo P. Lava, Aug 24 2018

Table[n + Floor[n^2/2 + n^3/3], {n, 0, 50}]
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 *)

vector(60, n, n--; n+floor(n^2/2 +n^3/3)) \\ G. C. Greubel, Aug 12 2018

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).
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).
Also, for h>=0:
a(6h)   = 6*h*( 12*h^2 + 3*h + 1 ),
a(6h+1) = 72*h^3 + 54*h^2 + 18*h + 1,
a(6h+2) = 6*( 4*h + 1 )*( 3*h^2 + 3*h + 1 ),
a(6h+3) = 2*( 36*h^3 + 63*h^2 + 39*h + 8 ),
a(6h+4) = 3*( 24*h^3 + 54*h^2 + 42*h + 11 ),
a(6h+5) = 72*h^3 + 198*h^2 + 186*h + 59.

cp's OEIS Frontend

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

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

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Magma

Magma

Maple

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