A117642 -id:A117642 - cp's OEIS frontend

A033431 a(n) = 2*n^3.

0, 2, 16, 54, 128, 250, 432, 686, 1024, 1458, 2000, 2662, 3456, 4394, 5488, 6750, 8192, 9826, 11664, 13718, 16000, 18522, 21296, 24334, 27648, 31250, 35152, 39366, 43904, 48778, 54000, 59582, 65536, 71874, 78608, 85750, 93312, 101306, 109744, 118638, 128000, 137842

Offset: 0

Jeff Burch

Also the largest determinant of a 3 X 3 matrix with entries from {0..n}. - Jud McCranie, Aug 12 2001
4*a(n) is a perfect cube.
The positive terms comprise the principal diagonal of the convolution array A213821. - Clark Kimberling, Jul 04 2012
Volume of a pyramid (square base) with side n and height 6*n. - Wesley Ivan Hurt, Aug 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Cardano Formula and Some Figurate Numbers, Politecnico di Torino (Italy, 2021).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A002378, A033581, A117642, A213821.

[2*n^3: n in [0..30]]; // Vincenzo Librandi, Jun 26 2011

seq(2*n^3, n=0..39); # Nathaniel Johnston, Jun 26 2011

2 Range[0, 50]^3 (* Wesley Ivan Hurt, Aug 25 2014 *)

a(n)=2*n^3 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 2*x*(1 + 4*x + x^2) / (1 - x)^4. - R. J. Mathar, Feb 04 2011
a(n) = 2*A000578(n). - Omar E. Pol, May 14 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Aug 25 2014
a(n) = A002378(n)^2 - A002378(n^2). - Bruno Berselli, Oct 20 2016
E.g.f.: 2*x*(1 + 3*x + x^2)*exp(x). - G. C. Greubel, Jul 15 2017
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = zeta(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/8. (End)

A244725 a(n) = 5*n^3.

Original entry on oeis.org

0, 5, 40, 135, 320, 625, 1080, 1715, 2560, 3645, 5000, 6655, 8640, 10985, 13720, 16875, 20480, 24565, 29160, 34295, 40000, 46305, 53240, 60835, 69120, 78125, 87880, 98415, 109760, 121945, 135000, 148955, 163840, 179685, 196520, 214375, 233280, 253265

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences of the type k*n^3: A000578 (k=1), A033431 (k=2), A117642 (k=3), A033430 (k=4), this sequence (k=5), A244726 (k=6), A244727 (k=7), A016743 (k=8), A244728 (k=9), A244729 (k=10), A016767 (k=27), A016803 (k=64), A016851 (k=125), A016911 (k=216), A016983 (k=343), A017067 (k=512), A017163 (k=729), A017271 (k=1000), A017391 (k=1331), A017523 (k=1728).

Magma
```
[5*n^3: n in [0..40]];
```

I:=[0,5,40,135]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

Table[5 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[5 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]

a(n)=5*n^3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 5*x*(1 + 4*x + x^2)/(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

Original entry on oeis.org

3, 4, 7, 10, 13, 18, 19, 22, 24, 25, 29, 32, 34, 37, 42, 43, 45, 46, 53, 55, 56, 60, 61, 62, 71, 78, 79, 80, 81, 82, 85, 89, 94, 98, 99, 101, 104, 105, 107, 108, 113, 114, 115, 118, 121, 131, 132, 134, 139, 140, 144, 146, 150, 151, 152, 153, 155, 163, 166, 173, 174, 176, 181, 182, 187, 189, 192, 193, 194, 195, 199, 200, 204

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, A332821 and this sequence.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332821. This sequence has the primes with even indexes, those in A031215.
The terms are the even numbers in A332820 halved. The terms are also the numbers m such that 5m is in A332820, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332821, and so on for alternate primes: 7, 13, 19, 29 etc.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, we get the same set of numbers as we get from halving the even terms of this sequence, and A332821 consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332820, which consists exactly of those numbers. The numbers that are one fifth of the terms that are multiples of 5 constitute A332821, and for larger primes, an alternating pattern applies as described in the previous paragraph.
The product of any 2 terms of this sequence is in A332821, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332821, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of terms valued -1 in A332823; equivalently, numbers in row 3k-1 of A277905 for some k >= 1.
Cf. A048675, A332820, A332821.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066207, A031215 (prime terms), A033430\{0}, A117642\{0}, A169604, A244727\{0}, A244729\{0}, A338910 (semiprime terms).

Select[Range@ 204, Mod[Total@ #, 3] == 2 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332822(n) =  { my(f = factor(n)); (2==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332821(k) : k >= 1} U {A003961(A332821(k)) : k >= 1}.

{a(n) : n >= 1} = {A332821(k)^2 : k >= 1} U {A331590(2, A332821(k)) : k >= 1}.

A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 18, 24, 5, 5, 32, 81, 80, 8, 6, 50, 192, 405, 256, 13, 7, 72, 375, 1280, 1944, 832, 21, 8, 98, 648, 3125, 8192, 9477, 2688, 34, 9, 128, 1029, 6480, 25000, 53248, 45927, 8704, 55, 10, 162, 1536, 12005, 62208, 203125, 344064, 223074, 28160, 89, 11, 200, 2187

Offset: 0

Ralf Stephan, Dec 24 2013

			Array starts:
1,  2,   3,    5,     8,     13,    21,   34, 55, 89,...    (A000045)
2,  8,  24,   80,   256,    832,  2688, 8704,...   (A063727, A085449)
3, 18,  81,  405,  1944,   9477, 45927,...         (A122069, A099012)
4, 32, 192, 1280,  8192,  53248,...                         (A099133)
5, 50, 375, 3125, 25000, 203125,...
6, 72, 648, 6480, 62208, 606528,...
...
Columns: A000027, A001105, A117642.

PARI
```
T(n,k)=n^k*fibonacci(k)
```

T(n,k)=polcoeff(Ser(1/(1-n*x-n^2*x^2)),k)

G.f. of n-th row: 1/(1 - n*x - n^2*x^2).

Recurrence: T(n,k) = n*T(n,k-1) + n^2*T(n,k-2), starting n, 2*n^2.

A171607 Expressible as A*B^A in a nontrivial way.

Original entry on oeis.org

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592

Offset: 1

Robert Munafo, Dec 12 2009

nonn

			8=2*2^2. 24=3*2^3. 375=3*5^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. Munafo, Generalized Cullen and Woodall Numbers

Cf. A171606. Union of the "KN^K" sequences A001105, A117642, A141046, ... or of the "NK^N" sequences A036289, A036290, A018215, A036291, ... but omitting the trivial initial terms.

is(n)=if(n<8, return(0)); for(a=2,logint(n\2,2), if(n%a==0 && ispower(n/a,a), return(1))); 0 \\ Charles R Greathouse IV, Feb 19 2017

list(lim)=my(v=List()); if(lim<8,return([])); for(a=2,logint(lim\2,2), for(b=2,sqrtnint(lim\a,a), listput(v,a*b^a))); Set(v) \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 2n^2 - O(n^(5/3)). - Charles R Greathouse IV, Feb 19 2017

A269792 a(n) = 5*n^4.

Original entry on oeis.org

0, 5, 80, 405, 1280, 3125, 6480, 12005, 20480, 32805, 50000, 73205, 103680, 142805, 192080, 253125, 327680, 417605, 524880, 651605, 800000, 972405, 1171280, 1399205, 1658880, 1953125, 2284880, 2657205, 3073280, 3536405, 4050000, 4617605, 5242880, 5929605

Offset: 0

Ilya Gutkovskiy, Mar 31 2016

More generally, the ordinary generating function for the sequences of the form k*n^m, is k*Sum_{j>=1}x^j*j^m (when abs(x)<1).

More generally, the ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r, is (r + (p + q + k + m - 4*r)*x + (11*p + 3*q - k - 3*m + 6*r)*x^2 + (11*p - 3*q - k + 3*m - 4*r)*x^3 + (p - q + k - m + r)*x^4)/(1 - x)^5.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quartic polynomial
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. similar sequences of the form k*n^m, for k = 1...5, m = 1...10: A001477(k = 1, m = 1), A005843 (k = 2, m = 1), A008585 (k = 3, m = 1), A008586 (k = 4, m = 1), A008587 (k = 5, m = 1), A000290 (k = 1, m = 2), A001105 (k = 2, m = 2), A033428 (k = 3, m = 2), A016742 (k = 4, m = 2), A033429 (k = 5, m = 2), A000578 (k = 1, m = 3), A033431 (k = 2, m = 3), A117642 (k = 3, m = 3), A033430 (k = 4, m = 3), A244725 (k = 5, m = 3), A000583 (k = 1, m = 4), A244730 (k = 2, m = 4), A219056 (k = 3, m = 4), A141046 (k = 4, m = 4), this sequence(k = 5, m = 4), A000584 (k = 1, m = 5), A001014 (k = 1, m = 6), A106318 (k = 2, m = 6), A001015 (k = 1, m = 7), A001016 (k = 1, m = 8), A001017 (k = 1, m = 9), A008454 (k = 1, m = 10).

A269792:=n->5*n^4: seq(A269792(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[5 n^4, {n, 0, 33}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 80, 405, 1280}, 34]

x='x+O('x^99); concat(0, Vec(5*x*(1+11*x+11*x^2+x^3)/(1-x)^5)) \\ Altug Alkan, Mar 31 2016

G.f.: 5*x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5.
E.g.f.: 5*exp(x)^x*x*(1 + 7*x + 6*x^2 + x^3).
a(n) = 5*a(n-1) - 10*(9n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 5*A000583(n) = A008587(n)*A000578(n).
Sum_{n>=1} 1/a(n) = Pi^4/450 = (1/450)*A092425 = 0.216464646742...

A027903 a(n) = n(n + 1)(3*n + 1).

Original entry on oeis.org

0, 8, 42, 120, 260, 480, 798, 1232, 1800, 2520, 3410, 4488, 5772, 7280, 9030, 11040, 13328, 15912, 18810, 22040, 25620, 29568, 33902, 38640, 43800, 49400, 55458, 61992, 69020, 76560, 84630, 93248, 102432, 112200, 122570, 133560, 145188, 157472, 170430

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000567, A001477, A002378, A016742, A016777, A049451, A117642.

A027903:=n->n*(n + 1)*(3*n + 1); seq(A027903(n), n=0..100); # Wesley Ivan Hurt, Dec 02 2013

Table[n (n + 1) (3*n + 1), {n, 0, 100}] (* Wesley Ivan Hurt, Dec 02 2013 *)

From Wesley Ivan Hurt, Dec 02 2013: (Start)
a(n) = n*(n + 1)*(3*n + 1).
a(n) = 3*n^3 + 4*n^2 + n.
a(n) = A002378(n) * A016777(n).
a(n) = A049451(n) * A001477(n+1).
a(n) = A001477(n) * A000567(n-1).
a(n) = A001477(n) * A001477(n+1) * A016777(n).
a(n) = A117642(n) + A016742(n) + A001477(n). (End)
From Amiram Eldar, Aug 15 2025: (Start)
Sum_{n>=1} 1/a(n) = 4 - sqrt(3)*Pi/4 - 9*log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/2 + 2*log(2) - 4. (End)
From Elmo R. Oliveira, Aug 29 2025: (Start)
G.f.: 2*x*(4 + 5*x)/(1 - x)^4.
E.g.f.: x*(8 + 13*x + 3*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A251781 Numbers whose square is the sum of two distinct positive cubes.

Original entry on oeis.org

3, 24, 81, 98, 168, 192, 228, 312, 375, 525, 588, 648, 671, 784, 847, 1014, 1029, 1183, 1225, 1261, 1323, 1344, 1536, 1824, 2187, 2496, 2646, 2888, 3000, 3993, 4200, 4225, 4536, 4563, 4644, 4704, 5184, 5368, 6156, 6272, 6292, 6371, 6591, 6696, 6776, 6877, 8112

Offset: 1

Daniel Arribas, Dec 08 2014

nonn

This list contains A117642 (if n=3*k^3, then n^2 = 9*k^6 = 8*k^6 + k^6 = (2*k^2)^3 + (k^2)^3). (Old comment rewritten as suggested by Michel Marcus, Dec 10 2014.)

Subsequence of A050801 and A217248. - Wolfdieter Lang, Jan 04 2015

			3^2 = 1^3 + 2^3; 24^2 = 4^3 + 8^3.

Daniel Arribas, Table of n, a(n) for n = 1..575

Cf. A024670, A117642, A050801, A217248, A099426 (coprime positive cubes).

def aupto(limit):
  c = [i**3 for i in range(1, int(limit**(2/3))+2) if i**3 <= limit**2]
  cc = [c1 + c2 for i, c1 in enumerate(c) for c2 in c[i+1:]]
  return sorted([i for i in range(1, limit+1) if i*i in cc])
print(aupto(8122)) # Michael S. Branicky, Mar 24 2021

L = []
for k in range(1,10^3):
    for l in range(k + 1,10^3):
        if is_square(k**3+l**3):
            L.append(sqrt(k**3+l**3))

A267983 Integers n such that n^3 = (x^2 + y^2 + z^2) / 3 where x > y > z > 0, is soluble.

Original entry on oeis.org

3, 6, 7, 9, 10, 11, 12, 14, 15, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 86, 87, 88, 89, 90, 91, 92, 94

Offset: 1

Altug Alkan, Jan 23 2016

nonn

Motivation was this simple question: What are the cubes that are the averages of 3 nonzero distinct squares?
Corresponding cubes are 27, 216, 343, 729, 1000, 1331, 1728, 2744, 3375, 4913, 5832, 6859, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 27000, ...
Complement of this sequence for positive integers is 1, 2, 4, 5, 8, 13, 16, 20, 21, 29, 32, 37, 45, 52, 53, 61, 64, 69, 77, ...
The positive cubes that are not the averages of 3 nonzero distinct squares are 1, 8, 64, 125, 512, 2197, 4096, 8000, 9261, 24389, 32768, 50653, 91125, ...

			3 is a term since 3^3 is the average of 1^2, 4^2, 8^2. 3^3 = (1^2 + 4^2 + 8^2) / 3.

Cf. A004432, A117642.

Select[Range@ 94, Resolve[Exists[{x, y, z}, Reduce[#^3 == (x^2 + y^2 + z^2)/3, {x, y, z}, Integers], x > y > z > 0]] &] (* Michael De Vlieger, Jan 24 2016 *)

isA004432(n) = for(x=1, sqrtint(n\3), for(y=x+1, sqrtint((n-1-x^2)\2), issquare(n-x^2-y^2) && return(1)));
for(n=1, 1e2, if(isA004432(3*n^3), print1(n, ", ")));

A294315 a(n) = 3*n^3 + n^2.

Original entry on oeis.org

0, 4, 28, 90, 208, 400, 684, 1078, 1600, 2268, 3100, 4114, 5328, 6760, 8428, 10350, 12544, 15028, 17820, 20938, 24400, 28224, 32428, 37030, 42048, 47500, 53404, 59778, 66640, 74008, 81900, 90334, 99328, 108900, 119068, 129850, 141264, 153328, 166060, 179478

Offset: 0

Jason Morgan, Oct 28 2017

All terms are even.

			a(3)=90 because 3*3^3 + 3^2 = 3*27 + 9 = 90.

Muniru A Asiru, Table of n, a(n) for n = 0..10000 (first 1000 terms from Colin Barker)
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A117642, A000290, A036659.

A294315:=List([0..10^4],n -> 3 *n^3 + n^2 ); # Muniru A Asiru, Dec 11 2017

Array[3 #^3 + #^2 &, 40, 0] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 28, 90}, 40] (* or *)
CoefficientList[Series[2 x (2 + 6 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Dec 12 2017 *)

PARI
```
a(n) = 3*n^3 + n^2;
```

concat(0, Vec(2*x*(2 + 6*x + x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 11 2017

a(n) = 3*n^3 + n^2.
a(n) = A117642(n) + A000290(n).
a(n) = 2*A036659(n).
From Colin Barker, Dec 11 2017: (Start)
G.f.: 2*x*(2 + 6*x + x^2) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + sqrt(3)*Pi/2 + 9*log(3)/2 - 9.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - sqrt(3)*Pi - 6*log(2) + 9. (End)

cp's OEIS Frontend

A033431 a(n) = 2*n^3.

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A244725 a(n) = 5*n^3.

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A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

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Mathematica

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A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

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PARI

PARI

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A171607 Expressible as A*B^A in a nontrivial way.

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PARI

PARI

Formula

A269792 a(n) = 5*n^4.

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Maple

Mathematica

PARI

Formula

A027903 a(n) = n*(n + 1)*(3*n + 1).

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Maple

Mathematica

Formula

A251781 Numbers whose square is the sum of two distinct positive cubes.

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A027903 a(n) = n(n + 1)(3*n + 1).