A058895 -id:A058895 - cp's OEIS frontend

A068601 a(n) = n^3 - 1.

0, 7, 26, 63, 124, 215, 342, 511, 728, 999, 1330, 1727, 2196, 2743, 3374, 4095, 4912, 5831, 6858, 7999, 9260, 10647, 12166, 13823, 15624, 17575, 19682, 21951, 24388, 26999, 29790, 32767, 35936, 39303, 42874, 46655, 50652, 54871, 59318, 63999, 68920

Offset: 1

Naohiro Nomoto, Mar 28 2002

a(n) is the least positive integer k such that k can only contain 'n-1' in exactly 2 different bases B, where 1 < B <= k.
Apart from the first term, the same as A135300. - R. J. Mathar, Apr 29 2008
A058895(n)^3 + a(n)^3 + A033562(n)^3 = A185065(n)^3. - Vincenzo Librandi, Mar 13 2012
Numbers k such that for every nonnegative integer m, k^(3*m+1) + k^(3*m) is a cube. - Arkadiusz Wesolowski, Aug 10 2013

			For n=6; 215 written in bases 6 and 42 is 555, 55 and (555, 55) are exactly 2 different bases.

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

Cf. A000217, A003215, A005448, A016921, A033562, A058895, A129294, A135300, A185065, A339604.

List([1..45],n->n^3-1); # Muniru A Asiru, Oct 23 2018

[n^3-1: n in [1..40]]; // Vincenzo Librandi, Mar 11 2012

A068601:=n->n^3-1: seq(A068601(n), n=1..50); # Wesley Ivan Hurt, Jul 23 2014

f[n_]:=n^3-1;f[Range[60]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
LinearRecurrence[{4,-6,4,-1},{0,7,26,63},50] (* Vincenzo Librandi, Mar 11 2012 *)
Range[50]^3 - 1 (* Wesley Ivan Hurt, Jul 23 2014 *)

PARI
```
a(n)=n^3-1
```

for n in range(1,50): print(n**3-1, end=', ') # Stefano Spezia, Nov 21 2018

Partial sums of A003215, hex (or centered hexagonal) numbers: 3*n(n+1)+1. - Jonathan Vos Post, Mar 16 2006
G.f.: x^2*(7-2*x+x^2)/(1-x)^4. - Colin Barker, Feb 12 2012
4*a(m^2-2*m+2) = (m^2-m+1)^3 + (m^2-m-1)^3 + (m^2-3*m+3)^3 + (m^2-3*m+1)^3. - Bruno Berselli, Jun 23 2014
a(n) = Sum_{i=1..n-1} (i+1)^3 - i^3. - Wesley Ivan Hurt, Jul 23 2014
Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(3*n) - 1) = A339604. - Amiram Eldar, Nov 06 2020
Product_{n>=2} (1 + 1/a(n)) = 3*Pi*sech(sqrt(3)*Pi/2). - Amiram Eldar, Jan 20 2021
E.g.f.: 1 + exp(x)*(x^3 + 3*x^2 + x - 1). - Stefano Spezia, Jul 06 2021

A269776 T(n,k)=Number of length-n 0..k arrays with every repeated value unequal to the previous repeated value plus one mod k+1.

Original entry on oeis.org

2, 3, 4, 4, 9, 8, 5, 16, 27, 14, 6, 25, 64, 78, 24, 7, 36, 125, 252, 222, 40, 8, 49, 216, 620, 984, 624, 66, 9, 64, 343, 1290, 3060, 3816, 1740, 108, 10, 81, 512, 2394, 7680, 15040, 14724, 4824, 176, 11, 100, 729, 4088, 16674, 45600, 73680, 56592, 13320, 286, 12, 121

Offset: 1

R. H. Hardin, Mar 04 2016

Table starts
...2.....3......4.......5........6.........7..........8..........9.........10
...4.....9.....16......25.......36........49.........64.........81........100
...8....27.....64.....125......216.......343........512........729.......1000
..14....78....252.....620.....1290......2394.......4088.......6552.......9990
..24...222....984....3060.....7680.....16674......32592......58824......99720
..40...624...3816...15040....45600....115920.....259504.....527616.....994680
..66..1740..14724...73680...270150....804636....2063880....4728384....9915210
.108..4824..56592..360000..1597500...5577768...16398144...42342912...98779500
.176.13320.216864.1755200..9432000..38621016..130175360..378929664..983566800
.286.36672.829116.8542720.55616250.267152256.1032602872.3389054976.9788946390

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

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A019274(n+2).
Column 2 is A269613.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A000578(n+1).
Row 4 is A058895(n+1).

Empirical for column k (apparently a(n) = 2*k*a(n-1) -k*(k-1)*a(n-2) -k^2*a(n-3)):
k=1: a(n) = 2*a(n-1) -a(n-3)
k=2: a(n) = 4*a(n-1) -2*a(n-2) -4*a(n-3)
k=3: a(n) = 6*a(n-1) -6*a(n-2) -9*a(n-3)
k=4: a(n) = 8*a(n-1) -12*a(n-2) -16*a(n-3)
k=5: a(n) = 10*a(n-1) -20*a(n-2) -25*a(n-3)
k=6: a(n) = 12*a(n-1) -30*a(n-2) -36*a(n-3)
k=7: a(n) = 14*a(n-1) -42*a(n-2) -49*a(n-3)
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 3*n + 1
n=4: a(n) = n^4 + 4*n^3 + 6*n^2 + 3*n
n=5: a(n) = n^5 + 5*n^4 + 10*n^3 + 7*n^2 + n
n=6: a(n) = n^6 + 6*n^5 + 15*n^4 + 14*n^3 + 4*n^2
n=7: a(n) = n^7 + 7*n^6 + 21*n^5 + 25*n^4 + 11*n^3 + n^2

A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 4, 0, 2, 0, 5, 0, 6, 2, 0, 6, 0, 12, 6, 6, 0, 7, 0, 20, 12, 24, 4, 0, 8, 0, 30, 20, 60, 18, 14, 0, 9, 0, 42, 30, 120, 48, 78, 12, 0, 10, 0, 56, 42, 210, 100, 252, 72, 28, 0, 11, 0, 72, 56, 336, 180, 620, 240, 234, 24, 0, 12, 0, 90, 72, 504, 294, 1290, 600, 1008, 216, 62

Offset: 1

Andrew Howroyd, Apr 03 2017

			Table starts:
1  2   3    4    5    6     7     8     9    10 ...
0  0   0    0    0    0     0     0     0     0 ...
0  2   6   12   20   30    42    56    72    90 ...
0  2   6   12   20   30    42    56    72    90 ...
0  6  24   60  120  210   336   504   720   990 ...
0  4  18   48  100  180   294   448   648   900 ...
0 14  78  252  620 1290  2394  4088  6552  9990 ...
0 12  72  240  600 1260  2352  4032  6480  9900 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 24 216  960 3000 7560 16464 32256 58320 99000 ...
...
Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2-6 are A056458, A056459, A056460, A056461, A056462.
Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.
Cf. A284826, A284841.

T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017 *)

a(n,k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print();)

T(n,k) = Sum_{d | n} mu(n/d) * k^(ceiling(d/2)).

A033562 a(n) = 2*n^3 + 1.

Original entry on oeis.org

1, 3, 17, 55, 129, 251, 433, 687, 1025, 1459, 2001, 2663, 3457, 4395, 5489, 6751, 8193, 9827, 11665, 13719, 16001, 18523, 21297, 24335, 27649, 31251, 35153, 39367, 43905, 48779, 54001, 59583, 65537, 71875, 78609, 85751, 93313, 101307, 109745, 118639, 128001

Offset: 0

N. J. A. Sloane

A058895(n)^3 + A068601(n)^3 + a(n)^3 = A185065(n)^3, for n>0. - Vincenzo Librandi, Mar 13 2012

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. A058895, A068601, A185065.

List([0..50], n-> 2*n^3+1); # G. C. Greubel, Oct 12 2019

[2*n^3+1: n in [0..50]]; // G. C. Greubel, Oct 12 2019

seq(2*n^3+1, n=0..50); # G. C. Greubel, Oct 12 2019

2*Range[0,50]^3+1 (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
CoefficientList[Series[1+x*(3+5x+5x^2-x^3)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Mar 13 2012 *)
LinearRecurrence[{4,-6,4,-1},{1,3,17,55},50] (* Harvey P. Dale, Aug 14 2023 *)

a(n)=2*n^3+1 \\ Charles R Greathouse IV, Mar 11 2012

[2*n^3+1 for n in range(50)] # G. C. Greubel, Oct 12 2019

G.f.: 1 + x*(3 + 5*x + 5*x^2 - x^3)/(1-x)^4. - Vincenzo Librandi, Mar 13 2012

E.g.f.: (1 + 2*x + 6*x^2 + 2*x^3)*exp(x). - G. C. Greubel, Oct 12 2019

Terms a(34) onward added by G. C. Greubel, Oct 12 2019

A131471 a(n) = n^5+n.

Original entry on oeis.org

0, 2, 34, 246, 1028, 3130, 7782, 16814, 32776, 59058, 100010, 161062, 248844, 371306, 537838, 759390, 1048592, 1419874, 1889586, 2476118, 3200020, 4084122, 5153654, 6436366, 7962648, 9765650, 11881402, 14348934, 17210396, 20511178, 24300030, 28629182, 33554464

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

Cf. A271208, A271209.

[n^5+n: n in [0..30]]; // Vincenzo Librandi, Oct 01 2011

Table[n^5 + n, {n, 0, 50}] (* Paolo Xausa, Nov 03 2024 *)

a(n) = n^5+n; \\ Michel Marcus, Apr 02 2016

G.f.: 2*x*(1+11*x+36*x^2+11*x^3+x^4)/(-1+x)^6. - R. J. Mathar, Nov 14 2007

a(n) = A271208(n) + 1 = A271209(n) - 1. - Paolo Xausa, Nov 03 2024

A185065 a(n) = n*(n^3 + 2).

Original entry on oeis.org

0, 3, 20, 87, 264, 635, 1308, 2415, 4112, 6579, 10020, 14663, 20760, 28587, 38444, 50655, 65568, 83555, 105012, 130359, 160040, 194523, 234300, 279887, 331824, 390675, 457028, 531495, 614712, 707339, 810060, 923583, 1048640

Offset: 0

Vincenzo Librandi, Mar 31 2011

Numbers a(n) such that a(n)^3 = x^3*(x-2). The values of x are in A084380.

A058895(n)^3 + A068601(n)^3 + A033562(n)^3 = a(n)^3, for n > 0. - Vincenzo Librandi, Mar 13 2012

			20^3 = 10^3*(10-2); 87^3 = 29^3*(29-2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A084380, A058895, A068601, A033562.

Table[n(n^3+2),{n,0,40}]  (* Harvey P. Dale, Apr 11 2011 *)

a(n)=n*(n^3+2) \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(3 + 5*x + 17*x^2 - x^3)/(1-x)^5. - Bruno Berselli, Mar 31 2011

A131472 a(n) = n^6 + n.

Original entry on oeis.org

0, 2, 66, 732, 4100, 15630, 46662, 117656, 262152, 531450, 1000010, 1771572, 2985996, 4826822, 7529550, 11390640, 16777232, 24137586, 34012242, 47045900, 64000020, 85766142, 113379926, 148035912, 191103000, 244140650, 308915802

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6+n: n in [0..30]]; // _Vincenzo Librandi+, Oct 01 2011

Table[n^6+n,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, May 12 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,2,66,732,4100,15630,46662},60] (* Harvey P. Dale, May 03 2012 *)

G.f.: 2*x*(1 + 26*x + 156*x^2 + 146*x^3 + 31*x^4)/(1 - x)^7. - R. J. Mathar, Nov 14 2007
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=0, a(1)=2, a(2)=66, a(3)=732, a(4)=4100, a(5)=15630, a(6)=46662. - Harvey P. Dale, May 03 2012
E.g.f.: exp(x)*x*(2 + 31*x + 90*x^2 + 65*x^3 + 15*x^4 + x^5). - Stefano Spezia, Oct 08 2022

A027482 a(n) = n*(n^3 - 1)/2.

Original entry on oeis.org

7, 39, 126, 310, 645, 1197, 2044, 3276, 4995, 7315, 10362, 14274, 19201, 25305, 32760, 41752, 52479, 65151, 79990, 97230, 117117, 139909, 165876, 195300, 228475, 265707, 307314, 353626, 404985, 461745, 524272, 592944, 668151

Offset: 2

Olivier Gérard

Row sums in an n X n X n pandiagonal magic cube with entries (0..n^3-1).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
S. Gartenhaus, Odd order pandiagonal latin and magic cubes in three and four dimensions, arXiv:math/0210275 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

First subdiagonal of A027478 (Cube of a triangular matrix constructed from the Stirling numbers of the first kind).

[n * (n^3 - 1)/2: n in [2..50]]; // Vincenzo Librandi, Dec 29 2012

Table[(m^4 - m)/2, {m, 44}] (* Zerinvary Lajos, Mar 21 2007 *)
CoefficientList[Series[(7 + 4*x + x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 29 2012 *)

t(n)=n*(n+1)/2;
for(n=0,50,print1(t(n^2)-t(n)","))

a(n) = A027478(n,n-1)
a(n) = A000217(n^2) - A000217(n). - Jon Perry, Jul 21 2003
a(n) = A058895(n)/2. - Zerinvary Lajos, Jan 28 2008
G.f.: x^2*(7 + 4*x + x^2)/(1 - x)^5. - Vincenzo Librandi, Dec 29 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 6. - Chai Wah Wu, Apr 08 2021

A262705 Triangle: Newton expansion of C(n,m)^4, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 14, 1, 0, 36, 78, 1, 0, 24, 978, 252, 1, 0, 0, 4320, 8730, 620, 1, 0, 0, 8460, 103820, 46890, 1290, 1, 0, 0, 7560, 581700, 1159340, 185430, 2394, 1, 0, 0, 2520, 1767360, 13387570, 8314880, 595476, 4088, 1, 0, 0, 0, 3087000, 85806000, 170429490, 44341584, 1642788, 6552, 1

Offset: 0

Giuliano Cabrele, Sep 30 2015

Triangle here T_4(n,m) is such that C(n,m)^4 = Sum_{j=0..n} C(n,j)*T_4(j,m).
Equivalently, lower triangular matrix T_4 such that
|| C(n,m)^4 || = A202750 =  P * T_4 = A007318 * T_4.
T_4(n,m) = 0 for n < m and for 4*m < n.
Refer to comment to A262704.
Example:
C(x,2)^4 = x^4*(x-1)^4 /16 = 1*C(x,2) + 78*C(x,3) + 978*C(x,4) + 4320*C(x,5) + 8460*C(x,6) + 7560*C(x,7) + 2520*C(x,8);
C(5,2)^4 = C(5,3)^4 = 10000 = 1*C(5,2) + 78*C(5,3) + 978*C(5,4) + 4320*C(5,5) = 1*C(5,3) + 252*C(5,4) + 8730*C(5,5).

			Triangle starts:
[1];
[0,  1];
[0, 14,    1];
[0, 36,   78,      1];
[0, 24,  978,    252,     1];
[0,  0, 4320,   8730,   620,    1];
[0,  0, 8460, 103820, 46890, 1290, 1];

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

Row sums are, by definition, the inverse binomial transform of A005260.
Second diagonal (T_4(n+1,n)) is A058895(n+1).
Column T_4(n,2) is A122193(4,n).
Cf. A109983 (transpose of), A262704, A262706.
Cf. A078739, A078741.

[&+[(-1)^(n-j)*Binomial(n,j)*Binomial(j,m)^4: j in [0..n]]: m in [0..n], n in [0..10]]; // Bruno Berselli, Oct 01 2015

T4[n_, m_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[j, m]^4, {j, 0, n}]; Table[T4[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 01 2015 *)

// as a function
T_4:=(n,m)->_plus((-1)^(n-j)*binomial(n,j)*binomial(j,m)^4 $ j=0..n):
// as a matrix h x h
_P:=h->matrix([[binomial(n,m) $m=0..h]$n=0..h]):
_P_4:=h->matrix([[binomial(n,m)^4 $m=0..h]$n=0..h]):
_T_4:=h->_P(h)^-1*_P_4(h):

T_4(nmax) = {for(n=0, nmax, for(m=0, n, print1(sum(j=0, n, (-1)^(n-j)*binomial(n,j)*binomial(j,m)^4), ", ")); print())} \\ Colin Barker, Oct 01 2015

T_4(n,m) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,m)^4.

Also, let S(r,s)(n,m) denote the Generalized Stirling2 numbers as defined in the link above, then T_4(n,m) = n! / (m!)^4 * S(m,m)(4,n).

A131473 a(n) = n^6 - n.

Original entry on oeis.org

0, 0, 62, 726, 4092, 15620, 46650, 117642, 262136, 531432, 999990, 1771550, 2985972, 4826796, 7529522, 11390610, 16777200, 24137552, 34012206, 47045862, 63999980, 85766100, 113379882, 148035866, 191102952, 244140600, 308915750

Offset: 0

Mohammad K. Azarian, Jul 27 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6-n: n in [0..30]]; // Vincenzo Librandi, Aug 11 2011

A131473:=n->n^6-n; seq(A131473(n), n=0..30); # Wesley Ivan Hurt, Feb 25 2014

f[n_]:=n^6-n;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
Array[#^6-#&,60,0] (* Harvey P. Dale, Aug 10 2011 *)

[lucas_number1(3,n^3,n) for n in range(0, 27)] # Zerinvary Lajos, May 16 2009

cp's OEIS Frontend

A068601 a(n) = n^3 - 1.

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A269776 T(n,k)=Number of length-n 0..k arrays with every repeated value unequal to the previous repeated value plus one mod k+1.

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A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

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A033562 a(n) = 2*n^3 + 1.

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A131471 a(n) = n^5+n.

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A185065 a(n) = n*(n^3 + 2).

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A131472 a(n) = n^6 + n.

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