A123865 -id:A123865 - cp's OEIS frontend

A123868 a(n) = n^12 - 1.

0, 4095, 531440, 16777215, 244140624, 2176782335, 13841287200, 68719476735, 282429536480, 999999999999, 3138428376720, 8916100448255, 23298085122480, 56693912375295, 129746337890624, 281474976710655, 582622237229760, 1156831381426175, 2213314919066160

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 13 = 0 iff n mod 13 > 0; a(A008595(n)) = 12; a(A113763(n)) = 0.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A024010, A008456, A005563, A123865, A123866, A123867.

List([1..20], n-> n^12 -1); # G. C. Greubel, Aug 08 2019

[n^12 -1:n in [1..20]]; // Vincenzo Librandi, Dec 27 2010

seq(n^(12) -1, n=1..20); # G. C. Greubel, Aug 08 2019

Range[20]^12 -1 (* G. C. Greubel, Aug 08 2019 *)

vector(20, n, n^12 -1) \\ G. C. Greubel, Aug 08 2019

[n^12 -1 for n in (1..20)] # G. C. Greubel, Aug 08 2019

From Chai Wah Wu, Jun 18 2016: (Start)
a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13) for n > 12.
G.f.: x*(4095 + 478205*x + 10187905*x^2 + 66317979*x^3 + 162513078*x^4 + 162511362*x^5 + 66319266*x^6 + 10187190*x^7 + 478491*x^8 + 4017*x^9 + 13*x^10 - x^11)/(1 - x)^13. (End)

A123867 a(n) = n^10 - 1.

Original entry on oeis.org

0, 1023, 59048, 1048575, 9765624, 60466175, 282475248, 1073741823, 3486784400, 9999999999, 25937424600, 61917364223, 137858491848, 289254654975, 576650390624, 1099511627775, 2015993900448, 3570467226623, 6131066257800, 10239999999999, 16679880978200

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 11 = 0 iff n mod 11 > 0; a(A008593(n)) = 10.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A024008, A008454, A005563, A123865, A123866, A123868.

List([1..25], n-> n^10 -1); # G. C. Greubel, Aug 08 2019

[n^10 - 1:n in [1..40]]; // Vincenzo Librandi, Dec 27 2010

A123867:=n->n^10-1; seq(A123867(n), n=1..40); # Wesley Ivan Hurt, Jan 22 2014

Table[n^10-1, {n,40}] (* Wesley Ivan Hurt, Jan 22 2014 *)

vector(25, n, n^10 -1) \\ G. C. Greubel, Aug 08 2019

[n^10 -1 for n in (1..25)] # G. C. Greubel, Aug 08 2019

From G. C. Greubel, Aug 08 2019: (Start)
G.f.: x^2*(1023 + 47795*x + 455312*x^2 + 1310144*x^3 + 1310606*x^4 + 454982*x^5 + 47960*x^6 + 968*x^7 + 11*x^8 + x^9)/(1-x)^11.
E.g.f.: 1 +(-1 + x + 511*x^2 + 9330*x^3 + 34105*x^4 + 42525*x^5 + 22827*x^6 + 5880*x^7 + 750*x^8 + 45*x^9 + x^10)*exp(x). (End)

A258807 a(n) = n^5 - 1.

Original entry on oeis.org

0, 31, 242, 1023, 3124, 7775, 16806, 32767, 59048, 99999, 161050, 248831, 371292, 537823, 759374, 1048575, 1419856, 1889567, 2476098, 3199999, 4084100, 5153631, 6436342, 7962623, 9765624, 11881375, 14348906, 17210367, 20511148, 24299999, 28629150, 33554431

Offset: 1

Vincenzo Librandi, Jun 11 2015

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Subsequence of A181124.
Cf. A024003, A024049, A300656.
Sequences of the type n^k-1: A132411 (k=2), A068601 (k=3), A123865 (k=4), this sequence (k=5), A123866 (k=6), A258808 (k=7), A258809 (k=8), A258810 (k=9), A123867 (k=10), A258812 (k=11), A123868 (k=12).

List([1..35],n->n^5-1); # Muniru A Asiru, Oct 28 2018

Magma
```
[n^5-1: n in [1..50]];
```

I:=[0,31,242,1023, 3124,7775]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+ 6*Self(n-5)-Self(n-6): n in [1..50]];

seq(n^5-1,n=1..35); # Muniru A Asiru, Oct 28 2018

Table[n^5 - 1, {n, 1, 50}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 31, 242, 1023, 3124, 7775}, 50]

a(n)=n^5-1 \\ Charles R Greathouse IV, Jun 11 2015

for n in range(1, 50): print(n**5 - 1, end=', ') # Stefano Spezia, Oct 28 2018

[n^5-1 for n in (1..50)] # Bruno Berselli, Jun 11 2015

G.f.: x^2*(31 + 56*x + 36*x^2 - 4*x^3 + x^4)/(1 - x)^6.
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).
a(n) = -A024003(n). - Bruno Berselli, Jun 11 2015
Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(5*n) - 1) = 0.0379539032... - Amiram Eldar, Nov 06 2020

A024002 a(n) = 1 - n^4.

Original entry on oeis.org

1, 0, -15, -80, -255, -624, -1295, -2400, -4095, -6560, -9999, -14640, -20735, -28560, -38415, -50624, -65535, -83520, -104975, -130320, -159999, -194480, -234255, -279840, -331775, -390624, -456975, -531440, -614655, -707280, -809999, -923520, -1048575, -1185920, -1336335, -1500624

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..630
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A123865.

[1-n^4: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[1 - n^4, {n, 0, 50}] (* Bruno Berselli, Jun 12 2015 *)
CoefficientList[Series[(1 - 5*x - 5*x^2 - 15*x^3)/(1 - x)^5, {x, 0, 50}], x] (* G. C. Greubel, May 11 2017 *)

x='x+O('x^50); Vec((1 - 5*x - 5*x^2 - 15*x^3)/(1 - x)^5) \\ G. C. Greubel, May 11 2017

a(n) = -A123865(n) for n>0.
From G. C. Greubel, May 11 2017: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1 - 5*x - 5*x^2 - 15*x^3)/(1 - x)^5.
E.g.f.: (1 - x - 7*x^2 - 6*x^3 - x^4)*exp(x). (End)
Sum_{k>=2} -1/a(k) = A256919 = 7/8 - Pi*coth(Pi)/4. - Vaclav Kotesovec, Dec 08 2020

Corrected by T. D. Noe, Nov 08 2006

A129292 Number of divisors of n^4 - 1 that are not greater than n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 5, 3, 8, 3, 10, 4, 6, 4, 12, 4, 13, 4, 11, 6, 14, 3, 10, 6, 12, 6, 17, 3, 16, 7, 10, 9, 13, 4, 18, 7, 11, 4, 22, 3, 26, 8, 9, 7, 23, 5, 18, 7, 13, 6, 25, 4, 24, 8, 21, 6, 18, 3, 18, 10, 12, 14, 16, 4, 26, 8, 17, 7, 31, 5, 30, 6, 11, 13, 26, 7, 25, 6, 16, 10, 35, 4, 18, 11

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

a(n) = #{d: d<=n and A123865(n) mod d = 0};

a(n)>1 for n>2, see A129293 for m such that a(m)=2: a(A129293(n))=2.

			a(100) = #{1,3,9,11,33,73,99} = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A129294, A129296.

f:= n -> nops(select(`<=`,numtheory:-divisors(n^4-1),n)):
1,seq(f(n), n=2..100); # Robert Israel, Sep 21 2014

Table[Count[Divisors[n^4-1],?(#<=n&)],{n,90}] (* _Harvey P. Dale, Aug 13 2014 *)

a(n) = if (n==1, 1, sumdiv(n^4-1, d, d <= n)); \\ Michel Marcus, Sep 21 2014

A260478 Cyclotomic polynomial value Phi(8,n!).

Original entry on oeis.org

2, 2, 17, 1297, 331777, 207360001, 268738560001, 645241282560001, 2642908293365760001, 17340121312772751360001, 173401213127727513600000001, 2538767161403058526617600000001, 52643875858853821607942553600000001, 1503561738404723998944447273369600000001

Offset: 0

Robert Price, Aug 28 2015

nonn

Robert Price, Table of n, a(n) for n = 0..100
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A123865, A200906, A259264, A259265.

Mathematica
```
Table[Cyclotomic[8, n!], {n, 0, 200}]
```

a(n)=polcyclo(8,n!) \\ Charles R Greathouse IV, Aug 29 2015

a(n) = A123865(n!) for n>0.

A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

Original entry on oeis.org

1, 16, 4, 81, 36, 15, 256, 144, 80, 32, 625, 400, 255, 140, 65, 1296, 900, 624, 396, 240, 108, 2401, 1764, 1295, 896, 609, 364, 175, 4096, 3136, 2400, 1760, 1280, 864, 544, 256, 6561, 5184, 4095, 3132, 2385, 1728, 1215, 756, 369, 10000, 8100, 6560, 5180, 4080, 3100, 2320, 1620, 1040, 500

Offset: 1

Stefano Spezia, Jul 17 2024

T(n, k) is the k-th super- and subdiagonal sum of the Hankel matrix M(n) whose permanent is A374668(n).

			n\k|    0    1    2    3    4    5
---+------------------------------
1  |    1
2  |   16    4
3  |   81   36   15
4  |  256  144   80   32
5  |  625  400  255  140   65
6  | 1296  900  624  396  240  108
      ...
For n = 3 the matrix M is
  [ 1,  4, 15]
  [ 4, 15, 32]
  [15, 32, 65]
and therefore T(3, 0) = 1 + 15 + 65 = 81, T(3, 1) = 4 + 32 = 36, and T(3, 2) = 15.

Cf. A317614 (diagonal), A374668.
Cf. A333119, A330613.
Cf. A000583 (k=0), A035287 (k=1), A123865, A374709 (row sums).

T[n_,k_]:=(n-k)*n*(4*n^2 - 4*n*k+2*k^2-1+(-1)^k)/4; Table[T[n,k],{n,10},{k,0,n-1}]//Flatten

O.g.f.: x*(1 - 4*x^8*y^5 + x*(11 + 2*y) - x^7*y^4*(7 + 16*y) - x^2*(-11 + 6*y - 6*y^2) - x^5*y^2*(2 - 46*y - 3*y^2) - x^6*y^3*(-2 - 27*y + 4*y^2) - x^3*(-1 + 18*y + 38*y^2 - 2*y^3) - x^4*y*(2 + 14*y + 2*y^2 - y^3))/((1 - x)^5*(1 - x*y)^4*(1 + x*y)^2).

T(n,2) = A123865(n-1) for n > 1.

A261535 Primes of the form Phi(8,n!), where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 17, 1297, 331777, 1503561738404723998944447273369600000001

Offset: 1

Robert Price, Aug 28 2015

nonn

a(6) is 730 digits.

OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A123865, A200906, A259264, A259265, A260478.

Select[Table[Cyclotomic[8, n!], {n, 0, 200}], PrimeQ]

for(n=1, 500, if(isprime(k=polcyclo(8, n!)), print1(k", "))) \\ Altug Alkan, Nov 16 2015

A336194 Table read by antidiagonals upwards: T(n,k) = (n - 1)*k^3 - 1, with n > 1 and k > 0.

Original entry on oeis.org

0, 1, 7, 2, 15, 26, 3, 23, 53, 63, 4, 31, 80, 127, 124, 5, 39, 107, 191, 249, 215, 6, 47, 134, 255, 374, 431, 342, 7, 55, 161, 319, 499, 647, 685, 511, 8, 63, 188, 383, 624, 863, 1028, 1023, 728, 9, 71, 215, 447, 749, 1079, 1371, 1535, 1457, 999, 10, 79, 242, 511, 874, 1295, 1714, 2047, 2186, 1999, 1330

Offset: 2

Stefano Spezia, Jul 11 2020

T(n, k) is a sharp upper bound of the tree width of a graph G that does not contain a clique on n vertices nor a minimal separator of size larger than k (see Theorem 2.1 in Pilipczuk et al.).

All the square matrices starting at top left of the table T are singular except for the 2 X 2 submatrix: det([0, 7; 1, 15]) = -7.

			The table starts at row n = 2 and column k = 1 as:
0   7   26   63  124   215 ...
1  15   53  127  249   431 ...
2  23   80  191  374   647 ...
3  31  107  255  499   863 ...
4  39  134  319  624  1079 ...
5  47  161  383  749  1295 ...
...

Marcin Pilipczuk, Ni Luh Dewi Sintiari, Stéphan Thomassé and Nicolas Trotignon, (Theta, triangle)-free and (even hole, K4)-free graphs. Part 2 : bounds on treewidth, arXiv:2001.01607 [cs.DM], 2020. See p. 7.
Index entries for sequences related to trees.

Cf. A000578, A001093, A001477 (k = 1), A004771 (k = 2), A068601 (n = 2), A085537, A109129, A123865 (main diagonal), A325543, A325612.

T[n_,k_]:=(n-1)*k^3-1; Flatten[Table[T[n+1-k,k],{n,2,12},{k,1,n-1}]]

PARI
```
T(n, k) = (n - 1)*k^3 - 1
```

O.g.f.: x^2*y*(y*(7 - 2*y + y^2) + x*(1 - y)^3)/((1 - x)^2*(1 - y)^4).
E.g.f.: -1 + exp(x) - x + exp(y)*x + exp(y)*(1 + y + 3*y^2 + y^3) + exp(x + y)*(-1 +(-1 + x)*y*(1 + 3*y + y^2)).
T(n, k) = n*A000578(k) - A001093(k).
T(n, n) = A085537(n) - 1 for n > 1.
T(n, k) = T(n+1, 1)*T(2, k) + T(n, 1).

cp's OEIS Frontend

A123868 a(n) = n^12 - 1.

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A123867 a(n) = n^10 - 1.

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A258807 a(n) = n^5 - 1.

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A024002 a(n) = 1 - n^4.

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A129292 Number of divisors of n^4 - 1 that are not greater than n.

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A260478 Cyclotomic polynomial value Phi(8,n!).

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A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.