A016791 -id:A016791 - 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).

A370753 Antidiagonal products of A319840.

Original entry on oeis.org

1, 1, 4, 36, 576, 12800, 360000, 12192768, 481890304, 21743271936, 1101996057600, 61952000000000, 3824628881965056, 257164113195565056, 18704075505689706496, 1462975070062038220800, 122444006400000000000000, 10918111308394619734065152, 1033255398127440061257744384

Offset: 0

Stefano Spezia, Jun 22 2024

nonn

a(n) has trailing zeros iff n is congruent to 0 or 1 mod 5. Cf. A008851.
a(n) is a square iff n = 1 or congruent to {1, 3, 4} mod 5. Cf. A047206.
It appears that: (Start)
a(n) is a cube iff n = 0, 1, or is of the form (3*m - 4)^3 with m > 1 (A016791);
the only fourth powers in the sequence are 1 and a(9) = 21743271936 = 384^4;
the only fifth powers in the sequence are 1 and a(32) = 227200942336^5;
a(n) is a sixth power iff n = 0, 1, or is of the form (6*m - 10)^3 with m > 1;
the only seventh powers in the sequence are 1 and a(128) = 77458109039896212820250015287665035595218944^7. (End)

Cf. A000079, A000169, A000290, A008851, A016791, A047206, A319840.

a[0]=a[1]=1; a[n_]:=n^2*2^(n-2)*(n-1)^(n-2); Array[a,19,0]

a(0) = a(1) = 1, and a(n) = n^2*2^(n-2)*(n - 1)^(n-2) for n > 1.

A016800 a(n) = (3*n + 2)^12.

Original entry on oeis.org

4096, 244140625, 68719476736, 3138428376721, 56693912375296, 582622237229761, 4096000000000000, 21914624432020321, 95428956661682176, 353814783205469041, 1152921504606846976, 3379220508056640625, 9065737908494995456, 22563490300366186081, 52654090776777588736

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795, A016796, A016797, A016798, A016799.

Subsequence of A008456.

Table[(3n+2)^12,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016789(n)^12 = A016790(n)^6 = A016791(n)^4 = A016792(n)^3 = A016794(n)62.
Sum_{n>=0} 1/a(n) = PolyGamma(11, 2/3)/21213424108800. (End)

A017619 a(n) = (12*n + 8)^3.

Original entry on oeis.org

512, 8000, 32768, 85184, 175616, 314432, 512000, 778688, 1124864, 1560896, 2097152, 2744000, 3511808, 4410944, 5451776, 6644672, 8000000, 9528128, 11239424, 13144256, 15252992, 17576000, 20123648

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[(12*n+8)^3 : n in [0..30]]; // Vincenzo Librandi, Sep 29 2011

(12*Range[0,30]+8)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{512,8000,32768,85184},30] (* Harvey P. Dale, Dec 14 2015 *)

From R. J. Mathar, Jun 24 2009: (Start)
G.f.: 64*(8 + 93*x + 60*x^2 + x^3)/(-1+x)^4.
a(n) = 64*A016791(n). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=512, a(1)=8000, a(2)=32768, a(3)=85184. - Harvey P. Dale, Dec 14 2015

A370000 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+2)^3.

Original entry on oeis.org

1, 1, 8, 4, 3, 8, 7, 7, 8, 4, 2, 5, 0, 5, 7, 5, 2, 9, 6, 2, 5, 6, 1, 6, 8, 6, 1, 9, 4, 3, 0, 2, 5, 4, 3, 8, 7, 3, 2, 8, 8, 7, 9, 8, 2, 9, 7, 8, 3, 5, 6, 6, 8, 2, 4, 8, 0, 7, 9, 6, 8, 5, 0, 3, 4, 4, 7, 5, 5, 7, 1, 7, 5, 8, 4, 1, 0, 2, 8, 2, 0, 4, 2, 9, 0, 0, 4, 5, 9, 6, 2, 1, 1, 9

Offset: 0

R. J. Mathar, Feb 07 2024

			1/8 - 1/125 + 1/512 - 1/1331 + ... = 0.118438778425057529625616861943025438732887982...

R. W. Gosper, Acceleration of Series, Artificial Intelligence (1974) # 304.

Cf. A226735, A016791.

5*Pi^3/2/3^(9/2)-13*Zeta(3)/36 ; evalf(%) ;

RealDigits[5*Pi^3/(2*3^(9/2)) - 13*Zeta[3]/36, 10, 120][[1]] (* Amiram Eldar, Feb 09 2024 *)

sumalt(k=0, (-1)^k/(3*k+2)^3) \\ Michel Marcus, Feb 07 2024

Equals Sum_{n>=0} (-1)^n/A016791(n).

Equals A226735 - 13*zeta(3)/18 = 5*Pi^3/(2*3^(9/2)) - 13*zeta(3)/36.

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

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A370753 Antidiagonal products of A319840.

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A016800 a(n) = (3*n + 2)^12.

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A017619 a(n) = (12*n + 8)^3.

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A370000 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+2)^3.

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