A016169 -id:A016169 - cp's OEIS frontend

A190542 a(n) = 7^n - 4^n.

0, 3, 33, 279, 2145, 15783, 113553, 807159, 5699265, 40091463, 281426673, 1973132439, 13824509985, 96821901543, 677954637393, 4746487768119, 33228635602305, 232613334118023, 1628344878433713, 11398620307466199, 79791166785984225, 558541466036772903, 3909803456396943633, 27368676971336738679, 191580949905589703745

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter greater than 4. - 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 (11,-28).

Cf. A000302, A000420, A016169, A121213.

Magma
```
[7^n -4^n: n in [0..30]];
```

A190542:=n->7^n - 4^n; seq(A190542(n), n=0..30); # Wesley Ivan Hurt, Feb 26 2014

Table[7^n - 4^n, {n, 0, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)
LinearRecurrence[{11,-28},{0,3},30] (* Harvey P. Dale, Dec 21 2019 *)

a(n)=7^n-4^n \\ Charles R Greathouse IV, Jun 02 2011

def A190542(n): return pow(7,n) - pow(4,n)
print([A190542(n) for n in range(31)]) # G. C. Greubel, Nov 13 2024

a(n) = 11*a(n-1) - 28*a(n-2).
a(n) = A000420(n) - A000302(n). - Michel Marcus, Feb 26 2014
From G. C. Greubel, Nov 13 2024: (Start)
G.f.: 3*x/((1-4*x)*(1-7*x)).
E.g.f.: 2*exp(11*x/2)*sinh(3*x/2). (End)

A280307 Numbers m such that 7^m - 6^m is not squarefree, but 7^d - 6^d is squarefree for every proper divisor d of m.

Original entry on oeis.org

20, 26, 55, 68, 171, 258, 310

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2016

Numbers m such that 7^m - 6^m is not squarefree not divisible by any smaller number of the same form.
7^m - 6^m is nonsquarefree if and only if m is divisible by a term of this sequence. - Jon E. Schoenfield, Jan 01 2017
The smallest squares of 7^m - 6^m as defined above are 25, 169, 121, 289, 361, 1849, 961. - Robert Price, Mar 07 2017
a(8) >= 323. - Jinyuan Wang, May 15 2020
a(8) <= 381. 381, 406, 506, 610, 689, 979, 1027, 1081, 1332 are terms. - Chai Wah Wu, Jul 20 2020

			20 is in this sequence because 7^20 - 6^20 = 43242508113549025 is not squarefree but 7^d - 6^d is squarefree for every proper divisor d of 20 (i.e., for d = 1, 2, 4, 5, and 10): 7^1 - 6^1 = 1, 7^2 - 6^2 = 13, 7^4 - 6^4 = 1105, 7^5 - 6^5 = 13682, 7^10 - 6^10 = 222009013 are all squarefree.

Cf. A016169, A237043, A280203, A280208, A280209, A280302.

a(5)-a(7) from Jinyuan Wang, May 15 2020

A343237 Triangle T obtained from the array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0, by reading antidiagonals upwards.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, May 10 2021

This is the row reversed version of the triangle A047969(n, m). The corresponding array A047969 is a(n, k) = A(k, n) (transposed of array A).

A(n-1, k-1) = k^n - (k-1)^n gives the number of n-digit numbers with digits from K = {1, 2, 3, ..., k} such that any digit from K, say k, appears at least once. Motivated by a comment in A005061 by Enrique Navarrete, the instance k=4 for n >= 1 (the column 3 in array A), and the d = 3 (sub)-diagonal sequence of T for m >= 0.

			The array A begins:
n\k  0  1   2    3     4     5     6      7      8      9 ...
-------------------------------------------------------------
0:   1  1   1    1     1     1     1      1      1      1 ...
1:   1  3   5    7     9    11    13     15     17     19 ...
2:   1  7  19   37    61    91   127    169    217    271 ...
3:   1 15  65  175   369   671  1105   1695   2465   3439 ...
4:   1 31 211  781  2101  4651  9031  15961  26281  40951 ...
5:   1 63 665 3367 11529 31031 70993 144495 269297 468559 ...
...
The 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    7     5     1
4:    1   15    19     7     1
5:    1   31    65    37     9     1
6:    1   63   211   175    61    11    1
7:    1  127   665   781   369    91   13    1
8:    1  255  2059  3367  2101   671  127   15   1
9:    1  511  6305 14197 11529  4651 1105  169  17  1
10:   1 1023 19171 58975 61741 31031 9031 1695 217 19  1
...
Combinatorial interpretation (cf. A005061 by _Enrique Navarrete_)
The three digits numbers with digits from K ={1, 2, 3, 4} having at least one 4 are:
j=1 (one 4): 114, 141, 411; 224, 242, 422; 334, 343, 433; 124, 214, 142, 241, 412, 421; 134, 314, 143, 341, 413, 431; 234, 243, 423. That is,  3*3 + 3!*3 = 27 = binomial(3, 1)*(4-1)^(3-1) = 3*3^2;
j=2 (twice 4):  144, 414, 441;  244, 424, 442;  344, 434, 443; 3*3 = 9 = binomial(3, 2)*(4-1)^(3-2) = 3*3;
j=3 (thrice 4) 444; 1 = binomial(3, 3)*(4-1)^(3-3).
Together: 27 + 9 + 1 = 37 = A(2, 3) = T(5, 3).

Cf. A005061, A008292, A047969 (reversed), A045531 (central diagonal), A047970 (row sums of triangle).
Row sequences of array A (nexus numbers): A000012, A005408, A003215, A005917(k+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
Column sequences of array A: A000012, A000225(n+1), A001047(n+1), A005061(n+1), A005060(n+1), A005062(n+1), A016169(n+1), A016177(n+1), A016185(n+1), A016189(n+1), A016195(n+1), A016197(n+1).

egf := exp(exp(x)*y + x)*(exp(x)*y - y + 1): ser := series(egf, x, 12):
cx := n -> series(n!*coeff(ser, x, n), y, 12):
Arow := n -> seq(k!*coeff(cx(n), y, k), k=0..9):
for n from 0 to 5 do Arow(n) od; # Peter Luschny, May 10 2021

A[n_, k_] := (k + 1)^(n + 1) - k^(n + 1); Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 10 2021 *)

Array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0.
A(n-1, k-1) = Sum_{j=1} binomial(n, j)*(k-1)^(n-j) = Sum_{j=0} binomial(n, j)*(k-1)^(n-j) - (k-1)^n = (1+(k-1))^n - (k-1)^n = k^n - (k-1)^n (from the combinatorial comment on A(n-1, k-1) above).
O.g.f. row n of array A: RA(n, x) = P(n, x)/(1 - x)^n, with P(n, x) = Sum_{m=0..n} A008292(n+1, m+1)*x^m, (the Eulerian number triangle  A008292 has offset 1) for n >= 0. (See the Oct 26 2008 comment in A047969 by Peter Bala). RA(n, x) = polylog(-(n+1), x)*(1-x)/x. (For P(n, x) see the formula by Vladeta Jovovic, Sep 02 2002.)
E.g.f. of e.g.f.s of the rows of array A: EE(x, y) = exp(x)*(1 + y*(exp(x) - 1))*exp(y*exp(x)), that is A(n, k) = [y^k/k!][x^n/n!] EE(x, y).
Triangle T(n, m) = A(n-m, m) = (m+1)^(n-m+1) - m^(n-m+1), n >= 0, m = 0, 1, ..., n.
E.g.f.: -(exp(x)-1)/(x*exp(x)*y-x). - Vladimir Kruchinin, Nov 02 2022

A020570 Expansion of g.f. 1/((1-6x)(1-7x)(1-8*x)).

Original entry on oeis.org

1, 21, 295, 3465, 36751, 365001, 3463615, 31794105, 284628751, 2499039081, 21606842335, 184519243545, 1559982264751, 13079717026761, 108915112739455, 901732722577785, 7429565635164751, 60963378722560041, 498496565225842975, 4064108629664292825, 33049477950757248751

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (21,-146,336).

Cf. A016169, A016170.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-6*x)*(1-7*x)*(1-8*x)))); // Vincenzo Librandi, Jul 04 2013

I:=[1, 21, 295]; [n le 3 select I[n] else 21*Self(n-1)-146*Self(n-2)+336*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 04 2013

CoefficientList[Series[1/((1-6*x)*(1-7*x)*(1-8*x)), {x, 0, 20}], x]  (* Harvey P. Dale, Feb 24 2011 *)

my(x='x+O('x^30)); Vec(1/((1-6*x)*(1-7*x)*(1-8*x))) \\ G. C. Greubel, Feb 07 2018

If we define f(m,j,x) = Sum_{k=j..m} (binomial(m,k)*Stirling2(k,j)*x^(m-k)) then a(n-2) = f(n,2,6), (n>=2). - Milan Janjic, Apr 26 2009
a(n) = 18*6^n - 49*7^n + 32*8^n. - R. J. Mathar, Jun 30 2013
From Vincenzo Librandi, Jul 04 2013: (Start)
a(0)=1, a(1)=21, a(2)=295; for n>2, a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3).
a(n) = 15*a(n-1) - 56*a(n-2) + 6^n. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(6*x)*(18 - 49*exp(x) + 32*exp(2*x)).
a(n) = A016170(n+1) - A016169(n+2). (End)

A209901 7^p - 6^p - 2 with p = prime(n).

Original entry on oeis.org

11, 125, 9029, 543605, 1614529685, 83828316389, 215703854542469, 10789535445362645, 26579017117027313525, 3183060102526390833854309, 156448938516521406467644085, 18500229372226631089176131976869, 44487435359130133495783012898708549

Offset: 1

Jonathan Vos Post, Mar 14 2012

After 11 and 9029, there are no prime values of a(n) through 7^109 - 6^109 - 2.

			543605 is in the sequence because 543605 = 7^7 - 6^7 - 2, and 7 is prime.

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

Cf. A000040, A000400, A000420, A016169, A204768.

Table[7^p - 6^p - 2, {p, Prime[Range[20]]}] (* T. D. Noe, Mar 15 2012 *)

forprime(p=2,100,print1(7^p-6^p-2", ")) \\ Charles R Greathouse IV, Mar 15 2012

a(n) = A016169(A000040(n)) - 2 = A204768(n) - 1 = A000420(A000040(n)) - A000400(A000040(n)) - 2.

cp's OEIS Frontend

A190542 a(n) = 7^n - 4^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A280307 Numbers m such that 7^m - 6^m is not squarefree, but 7^d - 6^d is squarefree for every proper divisor d of m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A343237 Triangle T obtained from the array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0, by reading antidiagonals upwards.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A020570 Expansion of g.f. 1/((1-6*x)*(1-7*x)*(1-8*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

A209901 7^p - 6^p - 2 with p = prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A020570 Expansion of g.f. 1/((1-6x)(1-7x)(1-8*x)).