A011379 -id:A011379 - cp's OEIS frontend

A133070 a(n) = n^5 - n^3 - n^2.

0, -1, 20, 207, 944, 2975, 7524, 16415, 32192, 58239, 98900, 159599, 246960, 368927, 534884, 755775, 1044224, 1414655, 1883412, 2468879, 3191600, 4074399, 5142500, 6423647, 7948224, 9749375, 11863124, 14328495, 17187632, 20485919, 24272100, 28598399, 33520640, 39098367, 45394964

Offset: 0

Omar E. Pol, Nov 01 2007

Exponents are prime numbers in decreasing order.

			a(7)=16415 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807-343-49=16415.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133071, A133072, A133073.

[n^5-n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Table[n^5-n^3-n^2,{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,-1,20,207,944,2975},41] (* Harvey P. Dale, Jul 23 2011 *)

for(n=0,50, print1(n^5 - n^3 - n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

a(n) = n^5 - n^3 - n^2.
G.f.: x*(-1 +26*x + 72*x^2 + 22*x^3 + x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007
a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6), with a(0)=0, a(1)=-1, a(2)=20, a(3)=207, a(4)=944, a(5)=2975. - Harvey P. Dale, Jul 23 2011

More terms from Vincenzo Librandi, Dec 15 2010

A133071 a(n) = n^5 - n^3 + n^2.

Original entry on oeis.org

0, 1, 28, 225, 976, 3025, 7596, 16513, 32320, 58401, 99100, 159841, 247248, 369265, 535276, 756225, 1044736, 1415233, 1884060, 2469601, 3192400, 4075281, 5143468, 6424705, 7949376, 9750625, 11864476, 14329953, 17189200, 20487601, 24273900, 28600321, 33522688, 39100545, 45397276

Offset: 0

Omar E. Pol, Nov 01 2007

nonn

Exponents are prime numbers in decreasing order.

			a(7)=16513 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807-343+49=16513.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133070, A133072, A133073.

[n^5-n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Table[n^5 - n^3 + n^2, {n,0,50}] (* G. C. Greubel, Oct 20 2017 *)

for(n=0,50, print1(n^5 - n^3 + n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

a(n) = n^5 - n^3 + n^2.

G.f.: x*(1 + 22*x + 72*x^2 + 26*x^3 - x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

More terms from Vincenzo Librandi, Dec 15 2010

A133072 a(n) = n^5 + n^3 - n^2.

Original entry on oeis.org

0, 1, 36, 261, 1072, 3225, 7956, 17101, 33216, 59697, 100900, 162261, 250416, 373321, 540372, 762525, 1052416, 1424481, 1895076, 2482597, 3207600, 4092921, 5163796, 6447981, 7975872, 9780625, 11898276, 14367861, 17231536, 20534697, 24326100, 28657981, 33586176, 39170241, 45473572

Offset: 0

Omar E. Pol, Nov 01 2007

nonn

Exponents are the prime numbers in decreasing order.

			a(7)=17101 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807+343-49=17101.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133070, A133071, A133073.

[n^5+n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Table[n^5 + n^3 - n^2, {n, 0, 50}] (* G. C. Greubel, Oct 20 2017 *)

for(n=0,50, print1(n^5 + n^3 - n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

a(n) = n^5 + n^3 - n^2.

G.f.: x*(1 + 30*x + 60*x^2 + 26*x^3 + 3*x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

More terms from Vincenzo Librandi, Dec 15 2010

A133073 a(n) = n^5 + n^3 + n^2.

Original entry on oeis.org

0, 3, 44, 279, 1104, 3275, 8028, 17199, 33344, 59859, 101100, 162503, 250704, 373659, 540764, 762975, 1052928, 1425059, 1895724, 2483319, 3208400, 4093803, 5164764, 6449039, 7977024, 9781875, 11899628, 14369319, 17233104, 20536379, 24327900, 28659903, 33588224, 39172419, 45475884

Offset: 0

Omar E. Pol, Nov 01 2007

nonn

Exponents are prime numbers in decreasing order.

			a(7) = 17199 because 7^5 = 16807, 7^3 = 343, 7^2 = 49 and we can write 16807 + 343 + 49 = 17199.

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A000584, A011379, A045991, A100019.

Cf. A133070, A133071, A133072.

[n^5+n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

Total[#^{5,3,2}]&/@Range[0,40]  (* Harvey P. Dale, Jan 18 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,3,44,279,1104,3275},35] (* James C. McMahon, Mar 10 2025 *)

for(n=0,50, print1(n^5 + n^3 + n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

G.f.: x*(3 + 26*x + 60*x^2 + 30*x^3 + x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

a(n) = n^2*(n^3 + n + 1). - Wesley Ivan Hurt, Mar 02 2023

More terms from Vincenzo Librandi, Dec 15 2010

A270205 Number of 2 X 2 planar subsets in an n X n X n cube.

Original entry on oeis.org

0, 0, 6, 36, 108, 240, 450, 756, 1176, 1728, 2430, 3300, 4356, 5616, 7098, 8820, 10800, 13056, 15606, 18468, 21660, 25200, 29106, 33396, 38088, 43200, 48750, 54756, 61236, 68208, 75690, 83700, 92256

Offset: 0

Craig Knecht, Mar 13 2016

William H. Press looked at the hybrid structure of a most-perfect magic square and the Hilbert space filling curve and thought it might be the "most uniform" way of putting the consecutive integers in a 2-d square. He thought a definition of "most uniform" would be useful.
Al Zimmermann suggested this: Start by defining the "non-uniformity of a distribution of integers among the cells of a square [or cube or hypercube]" to be the standard deviation of the sums of the 2 X 2 planar subsets. Then define a "most uniform distribution of integers" to be a distribution with the smallest non-uniformity. For both the most-perfect square and most-perfect cube the non-uniformity is 0 and so each is a most uniform distribution. (Of course, you'd want a better word for "non-uniformity". Skewness?) Perhaps use "2 X 2 planar subset" instead of "2 X 2 partition"?
Comment from Dwane Campbell: For cubes, the definition of compact is that all 2 X 2 X 2 subcubes add to the same sum. That definition also includes wrap around. Your most perfect space cube is compact. It has the additional constraint that each orthogonal plane is also compact. There are 64 2 X 2 X 2 subcubes that add to 260 and 192 2 X 2 subsquares that add to 130 in your cube. I did not think either result was possible. Congratulations!
The most-perfect order 4 cube and the reversible order 4 cube are the new findings to look at in the link section.
Most-perfect magic squares require every 2 X 2 cell block to have the same sum. This sequence looks at that same subset in the cube.
Most-perfect space is defined as a structure where all these 2 X 2 subsets have the same sum.
What structure provides the most uniform distribution of integers in a cube?
a(n+1) is the number of unit faces required to make an n X n X n cubic lattice. Number of unit edges required for the same is A059986(n). - Mohammed Yaseen, Aug 22 2021
a(n-3) is the maximum sigma irregularity over all maximal 3-degenerate graphs with n vertices.  The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices).  (The sigma irregularity of a graph is the sum of the squares of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 14 2023

			The 2 X 2 X 2 cube labeled with the integers 1 to 8 has the following six 2 X 2 planar subsets each containing 4 cells: 1,2,3,4; 5,6,7,8; 1,2,5,6; 3,4,7,8; 1,4,5,8; 2,3,6,7.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Craig Knecht, Cube assembly from different 2x2 planar criteria.
Craig Knecht, F1 code most-perfect magic cube 960 examples.
Craig Knecht, F1 code reversible cube 960 examples.
Craig Knecht, magic space.
Craig Knecht, Most-perfect space.
Walter Trump, Most-Perfect magic cube.
Walter Trump, 6 unique neighbors for the most-perfect magic cube.
Wikipedia, Most-perfect magic square translated to a cube via the Hilbert space filling curve.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002411, A059986, A118659.

Cf. A011379, A181617, A270205 (sigma irregularities of maximal k-degenerate graphs).

[3*n^3 - 6*n^2 + 3*n: n in [0..50]]; // Wesley Ivan Hurt, Mar 13 2016

A270205:=n->3*n^3-6*n^2+3*n: seq(A270205(n), n=0..50); # Wesley Ivan Hurt, Mar 13 2016

Table[3*n^3 - 6*n^2 + 3*n, {n, 0, 50}] (* Wesley Ivan Hurt, Mar 13 2016 *)
CoefficientList[Series[(6 (x^2 + 2 x^3))/(-1 + x)^4, {x, 0, 32}], x] (* Michael De Vlieger, Mar 15 2016 *)

concat([0, 0], Vec(6*x^2*(1+2*x)/(x-1)^4 + O(x^100))) \\ Altug Alkan, Mar 14 2016

a(n) = 3*n^3 - 6*n^2 + 3*n \\ Charles R Greathouse IV, Mar 15 2016

a(n) = 3*n^3 - 6*n^2 + 3*n.
From Wesley Ivan Hurt, Mar 13 2016: (Start)
G.f.: 6*x^2*(1+2*x)/(x-1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
E.g.f.: 3*x^2*(1+x)*exp(x). - G. C. Greubel, May 10 2016
a(n) = 6 * A002411(n-1) for n>=1. - Joerg Arndt, May 11 2016
a(n) = A118659((n-1)^3), n>1. - Mohammed Yaseen, Aug 22 2021
From Amiram Eldar, Jul 02 2023: (Start)
Sum_{n>=2} 1/a(n) = Pi^2/18 - 1/3.
Sum_{n>=2} (-1)^n/a(n) = Pi^2/36 - 2*log(2)/3 + 1/3. (End)

A153978 a(n) = n(n-1)(n+1)(3n-2)/12.

Original entry on oeis.org

0, 2, 14, 50, 130, 280, 532, 924, 1500, 2310, 3410, 4862, 6734, 9100, 12040, 15640, 19992, 25194, 31350, 38570, 46970, 56672, 67804, 80500, 94900, 111150, 129402, 149814, 172550, 197780, 225680, 256432, 290224, 327250, 367710, 411810, 459762

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Partial sums of A011379.

Antidiagonal sums of the convolution array A213819. - Clark Kimberling, Jul 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819.

With[{r=Range[0,50]},Accumulate[r^2+r^3]] (* Harvey P. Dale, Jan 16 2011 *)
Rest[CoefficientList[Series[-2 x^2 * (2 x + 1)/(x - 1)^5, {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 30 2014 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,2,14,50,130}, 25] (* G. C. Greubel, Sep 01 2016 *)

concat(0, Vec(-2*x^2*(2*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Jun 28 2014

a(n) = n*(n-1)*(n+1)*(3*n-2)/12 \\ Charles R Greathouse IV, Sep 01 2016

a(n) = 2 * A001296(n-1) = (n-1)*n*(n+1)*(3*n-2)/12 (n>0). - Bruno Berselli, Apr 21 2010
a(n) = Sum_{i=1..n-1} binomial(i+1,i)*i^2. - Enrique Pérez Herrero, Jun 28 2014
G.f.: 2*x^2*(2*x+1) / (1 - x)^5. - Colin Barker, Jun 28 2014
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - Vincenzo Librandi, Jun 30 2014
a(n) = Sum_{k=1..n-1}k*((n-1)*n/2 + k) for n > 1. - J. M. Bergot, Feb 16 2018
From Amiram Eldar, Aug 23 2022: (Start)
Sum_{n>=2} 1/a(n) = 141/5 - 9*sqrt(3)*Pi/5 - 81*log(3)/5.
Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 + 48*log(2)/5 - 129/5. (End)

Edited by Bruno Berselli, Jun 15 2010

Simpler definition as suggested by Wesley Ivan Hurt, Jun 29 2014

A215190 T(n,k)=Number of arrays of n 0..k integers with no sum of consecutive elements equal to a disjoint adjacent sum of an equal number of elements.

Original entry on oeis.org

2, 3, 2, 4, 6, 2, 5, 12, 12, 0, 6, 20, 36, 18, 0, 7, 30, 80, 88, 30, 0, 8, 42, 150, 276, 216, 30, 0, 9, 56, 252, 664, 954, 440, 18, 0, 10, 72, 392, 1366, 2940, 2898, 896, 0, 0, 11, 90, 576, 2512, 7404, 11756, 8808, 1626, 0, 0, 12, 110, 810, 4264, 16092, 36864, 46972, 24014

Offset: 1

R. H. Hardin Aug 05 2012

Table starts
.2..3....4......5.......6........7.........8..........9.........10........11
.2..6...12.....20......30.......42........56.........72.........90.......110
.2.12...36.....80.....150......252.......392........576........810......1100
.0.18...88....276.....664.....1366......2512.......4264.......6800.....10330
.0.30..216....954....2940.....7404.....16092......31560......57072.....96990
.0.30..440...2898...11756....36864.....95832.....219092.....452368....864810
.0.18..896...8808...46972...183438....570460....1520506....3584736...7709744
.0..0.1626..24014..172046...848802...3191034....9990182...27052236..65759590
.0..0.2980..65462..630456..3931086..17862744...65678336..204247760.561117076
.0..0.4692.160670.2139436.17086156..94691966..411561564.1477403080
.0..0.7214.394750.7274062.74389138.502572562.2581475090
Column 1 is zero for n>=4
Column 2 is zero for n>=8
Column 3 is zero for n>=51

			Some solutions for n=6 k=4
..1....4....0....0....0....0....4....3....0....3....1....4....3....3....1....2
..3....3....3....3....4....4....2....4....3....0....3....2....0....4....0....3
..2....2....4....0....0....3....0....0....1....4....1....4....2....2....3....0
..0....1....3....4....2....2....4....2....4....3....2....3....4....0....4....4
..4....2....0....2....0....4....3....3....3....0....1....4....0....3....1....0
..3....0....1....3....4....2....4....4....1....2....4....1....3....2....4....3

R. H. Hardin, Table of n, a(n) for n = 1..164

Row 2 is A002378

Row 3 is A011379

A096038 Triangle T(n,m) = (3n^2-3m^2+5*m-4+n)/2 read by rows.

Original entry on oeis.org

1, 6, 4, 14, 12, 7, 25, 23, 18, 10, 39, 37, 32, 24, 13, 56, 54, 49, 41, 30, 16, 76, 74, 69, 61, 50, 36, 19, 99, 97, 92, 84, 73, 59, 42, 22, 125, 123, 118, 110, 99, 85, 68, 48, 25, 154, 152, 147, 139, 128, 114, 97, 77, 54, 28, 186, 184, 179, 171, 160, 146, 129, 109, 86, 60, 31

Offset: 1

Gary W. Adamson, Jun 17 2004

The triangle is obtained by subtracting the triangle A094930 from
its square root (also described in A094930) and then dividing each element of column m through 3*m-1.
For the first three rows n=1 to 3 this yields for example:
4;.................2;............2......................1;
14,25;......minus..2,5;.......=..12,20;......->.divide..6,4;
30,65,64;..........2,5,8;........28,60,56;..............14;12,7;

Cf. A095794, A011379, A002411, A096037, A096036, A024212.

def A096038(n,m):
    return (3*n**2-3*m**2+5*m-4+n)//2
print( [A096038(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

T(n,1) = A095794(n).
T(n,n) = 3*n-2.
T(n,m) = A094930(n,m)/(3*m-1)-1.

Edited, T(3,2) corrected, and extended by R. J. Mathar, Oct 11 2009

A212972 Number of triples (w,x,y) with all terms in {0,...,n} and w >= floor((x+y)/3).

Original entry on oeis.org

1, 8, 24, 53, 100, 168, 261, 384, 540, 733, 968, 1248, 1577, 1960, 2400, 2901, 3468, 4104, 4813, 5600, 6468, 7421, 8464, 9600, 10833, 12168, 13608, 15157, 16820, 18600, 20501, 22528, 24684, 26973, 29400, 31968, 34681, 37544, 40560, 43733

Offset: 0

Clark Kimberling, Jun 03 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A212959, A212971.

Cf. A011379, A212973.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w >= Floor[(x + y)/3], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212972 *)

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
G.f.: (1 + 5x + 3*x^2 + 3*x^3)/((1 + x + x^2)*(1-x)^4).
a(n) = (n+1)^3 - A212971(n).
From Ayoub Saber Rguez, Dec 11 2023: (Start)
a(n) = A011379(n+1) - A212973(n).
a(n) = (2*n^3 + 8*n^2 + 10*n + 4 - (((n+1) mod 3) mod 2))/3. (End)

Name corrected by Ayoub Saber Rguez, Jan 09 2024

A212973 Number of triples (w,x,y) with all terms in {0,...,n} and w <= floor((x+y)/3).

Original entry on oeis.org

1, 4, 12, 27, 50, 84, 131, 192, 270, 367, 484, 624, 789, 980, 1200, 1451, 1734, 2052, 2407, 2800, 3234, 3711, 4232, 4800, 5417, 6084, 6804, 7579, 8410, 9300, 10251, 11264, 12342, 13487, 14700, 15984, 17341, 18772, 20280, 21867, 23534

Offset: 0

Clark Kimberling, Jun 03 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A011379, A212959, A212972, A212974.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w <= Floor[(x + y)/3], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212973 *)
LinearRecurrence[{3,-3,2,-3,3,-1},{1,4,12,27,50,84},50] (* Harvey P. Dale, Jan 24 2015 *)

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
G.f.: (1 + x + 3*x^2 + x^3)/((1+x+x^2)*(1-x)^4).
a(n) = (n+1)^3 - A212974(n).
From Ayoub Saber Rguez, Dec 11 2023: (Start)
a(n) = A011379(n+1) - A212972(n).
a(n) = (n^3 + 4*n^2 + 5*n + 2 + (((n+1) mod 3) mod 2))/3. (End)

cp's OEIS Frontend

A133070 a(n) = n^5 - n^3 - n^2.

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A133071 a(n) = n^5 - n^3 + n^2.

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A133072 a(n) = n^5 + n^3 - n^2.

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A133073 a(n) = n^5 + n^3 + n^2.

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A270205 Number of 2 X 2 planar subsets in an n X n X n cube.

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A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12.

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A153978 a(n) = n(n-1)(n+1)(3n-2)/12.

A096038 Triangle T(n,m) = (3n^2-3m^2+5*m-4+n)/2 read by rows.