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

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

Author

Bruno Berselli, Feb 07 2014

Keywords

• nonn
• easy

Comments

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).

Links

• 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).

Crossrefs

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)).

Programs

Magma

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

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

Maple

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

Mathematica

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 *)

PARI

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

Formula

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.