A131476 -id:A131476 - cp's OEIS frontend

A287335 Nonnegative numbers k such that 3*k + 2 is a cube.

2, 41, 170, 443, 914, 1637, 2666, 4055, 5858, 8129, 10922, 14291, 18290, 22973, 28394, 34607, 41666, 49625, 58538, 68459, 79442, 91541, 104810, 119303, 135074, 152177, 170666, 190595, 212018, 234989, 259562, 285791, 313730, 343433, 374954, 408347, 443666, 480965

Offset: 1

Bruno Berselli, May 23 2017

Corresponding cubes are listed in A016791.

Primes in the sequence: 2, 41, 443, 1637, 22973, 34607, 91541, 234989, ...

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A047292.
Cf. A016791, A131476, A212069.
Cf. A244728: nonnegative k such that 3*k is a cube.
Cf. A121628: nonnegative k such that 3*k + 1 is a cube.

Magma
```
[9*n^3-9*n^2+3*n-1: n in [1..40]];
```

Table[9 n^3 - 9 n^2 + 3 n - 1, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{2,41,170,443},40] (* Harvey P. Dale, Aug 28 2021 *)

Maxima
```
makelist(9*n^3-9*n^2+3*n-1, n, 1, 40);
```

[9*n**3-9*n**2+3*n-1 for n in range(1,40)]

Sage
```
[9*n^3-9*n^2+3*n-1 for n in (1..40)]
```

O.g.f.: x*(2 + 33*x + 18*x^2 + x^3)/(1 - x)^4.
E.g.f.: 1 - (1 - 3*x - 18*x^2 - 9*x^3)*exp(x).
a(n) = 9*n^3 - 9*n^2 + 3*n - 1.
a(n) = A131476(3*n-1) = A212069(3*n-1).

A173707 Partial sums of floor(n^3/3).

Original entry on oeis.org

0, 0, 2, 11, 32, 73, 145, 259, 429, 672, 1005, 1448, 2024, 2756, 3670, 4795, 6160, 7797, 9741, 12027, 14693, 17780, 21329, 25384, 29992, 35200, 41058, 47619, 54936, 63065, 72065, 81995, 92917, 104896, 117997, 132288, 147840, 164724, 183014, 202787, 224120, 247093, 271789, 298291, 326685, 357060, 389505, 424112, 460976, 500192, 541858

Offset: 0

Mircea Merca, Nov 25 2010

Partial sums of A131476.

			a(4) = floor(1/3) + floor(8/3) + floor(27/3) + floor(64/3) = 32.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Cf. A131476.

[Floor((n^4+2*n^3+n^2-4*n)/12): n in [0..60]]; // Vincenzo Librandi, May 08 2011

A061347 := proc(n) op(1+(n mod 3),[-2,1,1]) ; end proc:
A173707 := proc(n) n^4/12+n^3/6+n^2/12-n/3-1/9 ; %+A061347(n+1)/9 ; end proc:
# program replaced by a structured version by R. J. Mathar, Nov 26 2010

Table[Sum[Floor[k^3/3],{k,0,n}],{n,0,60}] (* G. C. Greubel, Nov 23 2016 *)
Accumulate[Table[Floor[n^3/3],{n,0,60}]] (* or *) LinearRecurrence[{4,-6,5,-5,6,-4,1},{0,0,2,11,32,73,145},60] (* Harvey P. Dale, May 29 2018 *)

a(n)=(n^4+2*n^3+n^2-4*n)\12 \\ Charles R Greathouse IV, May 08 2011

def A173707(n): return n*(n*(n*(n + 2) + 1) - 4)//12 # Chai Wah Wu, Feb 02 2023

[floor(n*(n^3 +2*n^2 +n -4)/12) for n in (0..60)] # G. C. Greubel, Jul 02 2019

a(n) = Sum_{k=0..n} floor(k^3/3).
a(n) = round((n^4 + 2*n^3 + n^2 - 4*n)/12).
a(n) = round((n^4 + 2*n^3 + n^2 - 4*n - 2)/12).
a(n) = floor((n^4 + 2*n^3 + n^2 - 4*n)/12).
a(n) = ceiling((n+1)*(n^3 + n^2 - 4)/12).
a(n) = a(n-3) + n^3 - 3*n^2 + 5*n - 4, n > 2.
From R. J. Mathar, Nov 26 2010: (Start)
G.f.: x^2*(2 + 3*x + x^3) / ( (1+x+x^2)*(1-x)^5 ).
a(n) =  n^4/12 + n^3/6 + n^2/12 - n/3 - 1/9 + A061347(n+1)/9. (End)

cp's OEIS Frontend

A287335 Nonnegative numbers k such that 3*k + 2 is a cube.

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