A016169 -id:A016169 - cp's OEIS frontend

A047969 Square array of nexus numbers a(n,k) = (n+1)^(k+1) - n^(k+1) (n >= 0, k >= 0) read by upwards antidiagonals.

1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 19, 15, 1, 1, 9, 37, 65, 31, 1, 1, 11, 61, 175, 211, 63, 1, 1, 13, 91, 369, 781, 665, 127, 1, 1, 15, 127, 671, 2101, 3367, 2059, 255, 1, 1, 17, 169, 1105, 4651, 11529, 14197, 6305, 511, 1, 1, 19, 217, 1695, 9031

Offset: 0

N. J. A. Sloane

If each row started with an initial 0 (i.e., a(n,k) = (n+1)^k - n^k) then each row would be the binomial transform of the preceding row. - Henry Bottomley, May 31 2001
a(n-1, k-1) is the number of ordered k-tuples of positive integers such that the largest of these integers is n. - Alford Arnold, Sep 07 2005
From Alford Arnold, Jul 21 2006: (Start)
The sequences in A047969 can also be calculated using the Eulerian Array (A008292) and Pascal's Triangle (A007318) as illustrated below: (cf. A101095).
  1       1       1       1       1       1
  1       1       1       1       1       1
  -----------------------------------------
  1       2       3       4       5       6
          1       2       3       4       5
  1       3       5       7       9      11
  -----------------------------------------
  1       3       6      10      15      21
          4      12      24      40      60
                  1       3       6      10
  1       7      19      37      61      91
  -----------------------------------------
  1       4      10      20      35      56
         11      44     110     220     385
                 11      44     110     220
                          1       4      10
  1      15      65     175     369     671
  -----------------------------------------  (End)
From Peter Bala, Oct 26 2008: (Start)
The above remarks of Alford Arnold may be summarized by saying that (the transpose of) this array is the Hilbert transform of the triangle of Eulerian numbers A008292 (see A145905 for the definition of the Hilbert transform). In this context, A008292 is best viewed as the array of h-vectors of permutohedra of type A. See A108553 for the Hilbert transform of the array of h-vectors of type D permutohedra. Compare this array with A009998.
The polynomials n^k - (n-1)^k, k = 1,2,3,..., which give the nonzero entries in the columns of this array, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re s = 1/2 in the complex plane. See A019538 for the connection between the polynomials n^k - (n-1)^k and the Stirling polynomials of the simplicial complexes dual to the type A permutohedra.
(End)
Empirical: (n+1)^(k+1) - n^(k+1) is the number of first differences of length k+1 arrays of numbers in 0..n, k > 0. - R. H. Hardin, Jun 30 2013
a(n-1, k-1) is the number of bargraphs of width k and height n. Examples: a(1,2) = 7 because we have [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1], and [2,2,2]; a(2,1) = 5 because we have [1,3], [2,3], [3,1], [3,2], and [3,3] (bargraphs are given as compositions). This comment is equivalent to A. Arnold's Sep 2005 comment. - Emeric Deutsch, Jan 30 2017

			Array a begins:
  [n\k][0  1   2    3    4   5  6  ...
  [0]   1  1   1    1    1   1  1  ...
  [1]   1  3   7   15   31  63  ...
  [2]   1  5  19   65  211  ...
  [3]   1  7  37  175  ...
  ...
Triangle T begins:
  n\m   0   1    2     3     4      5      6      7      8     9  10 ...
  0:    1
  1:    1   1
  2:    1   3    1
  3:    1   5    7     1
  4:    1   7   19    15     1
  5:    1   9   37    65    31      1
  6:    1  11   61   175   211     63      1
  7:    1  13   91   369   781    665    127      1
  8:    1  15  127   671  2101   3367   2059    255      1
  9:    1  17  169  1105  4651  11529  14197   6305    511     1
  10:   1  19  217  1695  9031  31031  61741  58975  19171  1023   1
  ...  - _Wolfdieter Lang_, May 07 2021

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

T. D. Noe, Rows n = 0..100 of triangle, flattened
A. Blecher, C. Brennan, A. Knopfmacher and Helmut Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.
Eric Weisstein's World of Mathematics, Nexus Number

Cf. A047970.
Cf. A009998, A108553 (Hilbert transform of array of h-vectors of type D permutohedra), A145904, A145905.
Row n sequences of array a: A000012, A000225(k+1), A001047(k+1), A005061(k+1), A005060(k+1), A005062(k+1), A016169(k+1), A016177(k+1), A016185(k+1), A016189(k+1), A016195(k+1), A016197(k+1).
Column k sequences of array a: (nexus numbers): A000012, A005408, A003215, A005917(n+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
Cf. A343237 (row reversed triangle).

Flatten[Table[n = d - e; k = e; (n + 1)^(k + 1) - n^(k + 1), {d, 0, 100}, {e, 0, d}]] (* T. D. Noe, Feb 22 2012 *)

T(n,m):=if m=0 then 1 else sum(k!*(-1)^(m+k)*stirling2(m,k)*binomial(n+k-1,n),k,0,m); /* Vladimir Kruchinin, Jan 28 2018 */

From Vladimir Kruchinin: (Start)
O.g.f. of e.g.f of rows of array: ((1-x)*exp(y))/(1-x*exp(y))^2.
T(n,m) = Sum_{k=0..m} k!*(-1)^(m+k)*Stirling2(m,k)*C(n+k-1,n), T(n,0)=1.(End)
From Wolfdieter Lang, May 07 2021: (Start)
T(n,m) = a(n-m,m) = (n-m+1)^(m+1) - (n-m)^(m+1), n >= 0, m = 0, 1,..., n.
O.g.f. column k of the array: polylog(-(k+1), x)*(1-x)/x. See the Peter Bala comment above, and the Eulerian triangle A008292 formula by Vladeta Jovovic, Sep 02 2002.
E.g.f. of e.g.f. of row of the array: exp(y)*(1 + x*(exp(y) - 1))*exp(x*exp(y)).
O.g.f. of triangle's exponential row polynomials R(n, y) = Sum_{m=0} T(n, m)*(y^m)/m!: G(x, y) = exp(x*y)*(1 - x)/(1 - x*exp(x*y))^2. (End)

A016129 Expansion of 1/((1-2x)(1-6*x)).

Original entry on oeis.org

1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, 90698752, 544194560, 3265171456, 19591036928, 117546237952, 705277460480, 4231664828416, 25389989101568, 152339934871552, 914039609753600, 5484237659570176, 32905425959518208, 197432555761303552

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Row sums of A100851.
Cf. A071951, A089278, A089500, A129467, A144843.
Sequences with gf 1/((1-n*x)*(1-6*x)): A000400 (n=0), A003464 (n=1), this sequence (n=2), A016137 (n=3), A016149 (n=4), A005062 (n=5), A053469 (n=6), A016169 (n=7), A016170 (n=8), A016172 (n=9), A016173 (n=10), A016174 (n=11), A016175 (n=12).

[(6^(n+1)-2^(n+1))/4 : n in [0..30]]; // Vincenzo Librandi, Oct 09 2011

Table[(6^(n+1) -2^(n+1))/4, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-12},{1,8},30] (* Harvey P. Dale, Jan 15 2015 *)

Vec(1/(1-2*x)/(1-6*x)+O(x^30)) \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,8,12) for n in range(1, 31)] # Zerinvary Lajos, Apr 23 2009

[(6^n - 2^n)/4 for n in range(1,31)] # Zerinvary Lajos, Jun 04 2009

a(n) = A071951(n+2, 2) = 9*(2*3)^(n-1) - (2*1)^(n-1) = (2^(n-1))*(3^(n+1)-1), n>=0. - Wolfdieter Lang, Nov 07 2003
From Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005: (Start)
G.f.: 1/((1-2*x)*(1-6*x)).
E.g.f.: (-exp(2*x) + 3*exp(6*x))/2.
a(n) = (6^(n+1) - 2^(n+1))/4. (End)
a(n)^2 = A144843(n+1). - Philippe Deléham, Nov 26 2008
a(n) = 8*a(n-1) - 12*a(n-2). - Philippe Deléham, Jan 01 2009
a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013

A016189 a(n) = 10^n - 9^n.

Original entry on oeis.org

0, 1, 19, 271, 3439, 40951, 468559, 5217031, 56953279, 612579511, 6513215599, 68618940391, 717570463519, 7458134171671, 77123207545039, 794108867905351, 8146979811148159, 83322818300333431, 849905364703000879, 8649148282327007911, 87842334540943071199, 890581010868487640791

Offset: 0

N. J. A. Sloane

Almost all numbers contain any given sequence of digits (in any base) [Theorem 143 of Hardy and Wright]. a(7) = 5217031, more than 52% of the numbers < 10^7 contain any given nonzero decimal digit. - Frank Ellermann, May 30 2001
a(n) gives the number of integers from 0 to 10^n-1 which contain (at least) any one given decimal digit except 0. - Michael Taktikos, Aug 24 2004
These are the numerators of a(n)=(integral{x=0 to 0.2} (1-0.5*x)^n dx). E.g., a(3)=3439/20000. The denominators are b(n)=5*(n+1)*10^n. E.g., b(3)=20000. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004
Binomial transforms of sequences defined by a(n)=(C+1)^n-C^n are the sequences (C+2)^n-(C+1)^n. The binomial transform of this here is in A016195, for example. - R. J. Mathar, Nov 27 2008
First differences are given in A088924. - M. F. Hasler, May 04 2015

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 143

Vincenzo Librandi, Table of n, a(n) for n = 0..130
Alexander Bogomolny, Almost every integer has a digit 3 in it
John Elias, Illustration of Initial Terms
James Grime, 3 is everywhere, Numberphile video
Index entries for linear recurrences with constant coefficients, signature (19, -90).

Base 2: A000225, 3: A001047, 4: A005061, 5: A005060, 6: A005062, base 7: A016169, 8: A016177, 9: A016185 11: A016195 12: A016197.
Equals A155671 - 1.
Cf. A011557, A011533.

a016189 n = 10 ^ n - 9 ^ n
a016189_list = 0 : zipWith (+) (map (* 9) a016189_list) a011557_list
-- Reinhard Zumkeller, Apr 03 2015

[10^n - 9^n: n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=10^n-9^n;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

a(n)=10^n-9^n \\ M. F. Hasler, May 04 2015

G.f.: x/((1-9x)(1-10x)).
a(0) = 0, a(1) = 1, then a(n+1) = 9*a(n) + 10^n.
a(n) = 19*a(n-1) - 90*a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: e^(10*x) - e^(9*x). - Mohammad K. Azarian, Jan 14 2009

A081201 7th binomial transform of (0,1,0,1,0,1,....), A000035.

Original entry on oeis.org

0, 1, 14, 148, 1400, 12496, 107744, 908608, 7548800, 62070016, 506637824, 4113568768, 33271347200, 268347559936, 2159841173504, 17357093552128, 139326933401600, 1117436577120256, 8956419276406784, 71752914167922688, 574632673083392000, 4600717543107198976

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A081200.
Conjecture (verified up to a(8)): Number of collinear 6-tuples of points in a 6 X 6 X 6 X... n-dimensional cubic grid. [R. H. Hardin, May 23 2010]
From Wolfdieter Lang, Jul 17 2017: (Start)
For a combinatorial interpretation of a(n) with special 8-letter words of length n see the comment in A081200 on the 7-letter analog.
The binomial transform of {a(n)}_{n >= 0} is A081202, the 9-letter analog.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-48).

Cf. A000035, A016169, A081200, A081202.

Apart from offset same as A016170.

[(8^n-6^n)/2: n in [0..30]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x/((1 - 6 x) (1 - 8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{14,-48},{0,1},30] (* Harvey P. Dale, Oct 24 2022 *)

A081201=BinaryRecurrenceSequence(14,-48,0,1)
[A081201(n) for n in range(41)] # G. C. Greubel, Nov 10 2024

a(n) = 14*a(n-1) - 48*a(n-2) with n>1, a(0)=0, a(1)=1.
G.f.: x/((1-6*x)*(1-8*x)).
a(n) = (1/2)*(8^n - 6^n).
E.g.f.: exp(7*x)*sinh(x). - G. C. Greubel, Nov 10 2024

Name clarified by Pontus von Brömssen, Nov 11 2020

A016197 a(n) = 12^n - 11^n.

Original entry on oeis.org

0, 1, 23, 397, 6095, 87781, 1214423, 16344637, 215622815, 2801832661, 35979939623, 457696700077, 5777672071535, 72470493235141, 904168630965623, 11229773405170717, 138934529031464255, 1713164078241143221

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (23, -132).

Cf. k^n-(k-1)^n: A000225 (k=2), A001047 (k=3), A005061 (k=4), A005060 (k=5), A005062 (k=6), A016169 (k=7), A016177 (k=8), A016185 (k=9), A016189 (k=10), A016195 (k=11), this sequence (k=12).

[12^n-11^n: n in [0..40]]; // Vincenzo Librandi, Aug 03 2014

f[n_]:= 12^n - 11^n; f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
CoefficientList[Series[x/((1 - 11 x) (1 - 12 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 03 2014 *)

a(n)=12^n-11^n \\ Charles R Greathouse IV, Aug 03 2014

G.f.: x/((1-11x)(1-12x)).
E.g.f.: e^(12*x)-e^(11*x). - Mohammad K. Azarian, Jan 14 2009
a(0)=0, a(n)=12*a(n-1)+11^(n-1). - _Vincenzo Librandi-, Feb 09 2011
a(0)=0, a(1)=1, a(n)=23*a(n-1)-132*a(n-2). - Vincenzo Librandi, Feb 09 2011

A087759 Primes p such that 7^p - 6^p is composite.

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 31, 37, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Cino Hilliard, Oct 26 2003

nonn

Cf. A016169.

Primes p such that k^p - (k-1)^p is composite: A087489 (k=3), A087490 (k=4), A087685 (k=5), A087749 (k=6), this sequence (k=7), A087763 (k=8), A087894 (k=9), A087895 (k=10).

Select[Prime[Range[70]],CompositeQ[7^#-6^#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2019 *)

apmb(a,b,n) = { forprime(x=2,n, y=a^x-b^x; if(!ispseudoprime(y), print1(x","); ) ) }

Offset corrected by Mohammed Yaseen, Jul 18 2022

A257287 a(n) = 67^n - 56^n.

Original entry on oeis.org

1, 12, 114, 978, 7926, 61962, 472614, 3541578, 26190726, 191733162, 1392520614, 10049975178, 72163811526, 516030592362, 3677517616614, 26134444136778, 185292033880326, 1311149786699562, 9262681804120614, 65346572412186378

Offset: 0

M. F. Hasler, May 03 2015

First differences of 7^n - 6^n = A016169.
a(n-1) is the number of numbers with n digits having the largest digit equal to 6. Note that this is independent of the base b > 6.
Equivalently, number of n-letter words over a 7-letter alphabet {a,b,c,d,e,f,g}, which must not start with the first letter of the alphabet, and in which the last letter of the alphabet must appear.

Index entries for linear recurrences with constant coefficients, signature (13,-42).

Cf. A016169.

See also A000225, A027649, A255463, A257285, A257286, A257287, A257288, A257289, and A088924.

[6*7^n-5*6^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Table[6 7^n - 5 6^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)
LinearRecurrence[{13,-42},{1,12},20] (* Harvey P. Dale, Dec 10 2023 *)

PARI
```
a(n)=6*7^n-5*6^n
```

From Vincenzo Librandi, May 04 2015: (Start)
G.f.: (1-x)/((1-6*x)*(1-7*x)).
a(n) = 13*a(n-1) - 42*a(n-2). (End)
E.g.f.: exp(6*x)*(6*exp(x) - 5). - Stefano Spezia, Nov 15 2023

A190540 a(n) = 7^n - 2^n.

Original entry on oeis.org

0, 5, 45, 335, 2385, 16775, 117585, 823415, 5764545, 40353095, 282474225, 1977324695, 13841283105, 96889002215, 678223056465, 4747561477175, 33232930504065, 232630513856135, 1628413597648305, 11398895184848855, 79792266296563425, 558545864081186855, 3909821048578793745

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter >2. [Joerg Arndt, Jun 02 2011]

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-14).

Cf. A016130, A016169, A121213.

Magma
```
[7^n -2^n: n in [0..30]];
```

CoefficientList[Series[5 x/((1 - 2 x) (1 - 7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)

a(n)=7^n-1<Charles R Greathouse IV, Jun 08 2011

a(n) = 9*a(n-1) - 14*a(n-2).
G.f.: 5*x/((1-2*x)*(1-7*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = 5*A016130(n-1). - R. J. Mathar, Mar 10 2022
E.g.f.: exp(2*x)*(exp(5*x) - 1). - Elmo R. Oliveira, Sep 10 2024

A190541 a(n) = 7^n - 3^n.

Original entry on oeis.org

0, 4, 40, 316, 2320, 16564, 116920, 821356, 5758240, 40333924, 282416200, 1977149596, 13840755760, 96887416084, 678218289880, 4747547161036, 33232887522880, 232630384847044, 1628413210489960, 11398894023111676, 79792262810827600, 558545853622930804, 3909821017201928440

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter greater than 3. - Joerg Arndt, Jun 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Cf. A000244, A000420.

Similar sequences: A121213, A016169.

Magma
```
[7^n - 3^n: n in [0..30]];
```

A190541:=n->7^n-3^n: seq(A190541(n), n=0..25); # Wesley Ivan Hurt, Oct 04 2014

Table[7^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[4 x /((1 - 3 x) (1 - 7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)
LinearRecurrence[{10,-21},{0,4},20] (* Harvey P. Dale, Mar 30 2015 *)

a(n)=7^n-3^n \\ Charles R Greathouse IV, Jun 02 2011

a(n) = 10*a(n-1) - 21*a(n-2).
G.f.: 4*x/((1-3*x)*(1-7*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = A000420(n) - A000244(n). - Wesley Ivan Hurt, Oct 04 2014
E.g.f.: 2*exp(5*x)*sinh(2*x). - Elmo R. Oliveira, Mar 31 2025
a(n) = 4*A016138(n-1). - R. J. Mathar, Jun 07 2025

A204768 7^p - 6^p - 1, with p = prime(n).

Original entry on oeis.org

12, 126, 9030, 543606, 1614529686, 83828316390, 215703854542470, 10789535445362646, 26579017117027313526, 3183060102526390833854310, 156448938516521406467644086, 18500229372226631089176131976870

Offset: 1

Vincenzo Librandi, Mar 07 2012

nonn

For n>2 all terms are divisible by 1806 = 2*3*7*43.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A016169.

Magma
```
[(7^p-6^p-1): p in PrimesUpTo(100)];
```

Table[(7^Prime[n] - 6^Prime[n] - 1), {n, 1, 20}] (* Vincenzo Librandi, May 04 2014 *)
7^#-6^#-1&/@Prime[Range[20]] (* Harvey P. Dale, Aug 17 2024 *)

cp's OEIS Frontend

A047969 Square array of nexus numbers a(n,k) = (n+1)^(k+1) - n^(k+1) (n >= 0, k >= 0) read by upwards antidiagonals.

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A016129 Expansion of 1/((1-2*x)*(1-6*x)).

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A016189 a(n) = 10^n - 9^n.

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A081201 7th binomial transform of (0,1,0,1,0,1,....), A000035.

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A016197 a(n) = 12^n - 11^n.

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A087759 Primes p such that 7^p - 6^p is composite.

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A257287 a(n) = 6*7^n - 5*6^n.

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A016129 Expansion of 1/((1-2x)(1-6*x)).

A257287 a(n) = 67^n - 56^n.