A016185 -id:A016185 - cp's OEIS frontend

A257289 a(n) = 89^n - 78^n.

1, 16, 200, 2248, 23816, 243016, 2416520, 23583688, 226933256, 2159839816, 20378082440, 190918934728, 1778399954696, 16486635929416, 152228014061960, 1400838452135368, 12853836673840136, 117654854901535816, 1074656292809619080, 9798007424852945608

Offset: 0

M. F. Hasler, May 03 2015

First differences of 9^n - 8^n = A016185.
a(n-1) is the number of numbers with n digits having the largest digit equal to 8. Note that this is independent of the base b > 8.
Equivalently, number of n-letter words over a 9-letter alphabet, which must not start with the last letter of the alphabet, and in which the first letter of the alphabet must appear.

Index entries for linear recurrences with constant coefficients, signature (17,-72).

Cf. A016185.

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

[8*9^n-7*8^n: n in [0..20]]; // Vincenzo Librandi, May 04 2015

Table[8 9^n - 7 8^n, {n, 0, 20}] (* Vincenzo Librandi, May 04 2015 *)
LinearRecurrence[{17,-72},{1,16},30] (* Harvey P. Dale, May 26 2019 *)

PARI
```
a(n)=8*9^n-7*8^n
```

[8*9^n-7*8^n for n in (0..20)] # Bruno Berselli, May 04 2015

G.f.: (1-x)/((1-8*x)*(1-9*x)). - Vincenzo Librandi, May 04 2015

E.g.f.: exp(8*x)*(8*exp(x) - 7). - Stefano Spezia, Nov 15 2023

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 19683, 13122, 8748, 5832, 3888, 2592, 1728, 1152, 768, 512

Offset: 0

Johannes W. Meijer, Sep 21 2010, Jul 13 2011, Jun 03 2012

Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.

This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.

			1;
3, 2;
9, 6, 4;
27, 18, 12, 8;
81, 54, 36, 24, 16;
243, 162, 108, 72, 48, 32;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J. O'Connor and E.F. Robertson, Nicomachus of Gerasa, The MacTutor History of Mathematics archive, 2010.
Jay Kappraff, The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion, Nexus Network Journal, vol. 2, no. 4 (October 2000).
Hendrik Lenstra, Aeternitatem Cogita, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
Ivars Peterson, Medieval Harmony, Math Trek, Mathematical Association of America, 1998.

Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).

Cf. A000079, A000244, A000400, A003586.

a175840 n k = a175840_tabf !! n !! k
a175840_row n = a175840_tabf !! n
a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);

Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 08 2013 *)

T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.

A191465 a(n) = 9^n - 2^n.

Original entry on oeis.org

0, 7, 77, 721, 6545, 59017, 531377, 4782841, 43046465, 387419977, 3486783377, 31381057561, 282429532385, 2541865820137, 22876792438577, 205891132061881, 1853020188786305, 16677181699535497, 150094635296736977, 1350851717672467801, 12157665459055880225, 109418989131510262057

Offset: 0

Vincenzo Librandi, Jun 03 2011

a(n) is the number of words of length n over the alphabet {1,2,...,9} where at least one letter >= 3 appears. - Joerg Arndt, Jan 18 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-18).

Cf. A016133, A016185, A118004.

Table[9^n-2^n,{n,0,20}] (* Harvey P. Dale, Apr 16 2014 *)

a(n)=9^n-1<Charles R Greathouse IV, Jun 08 2011

a(n) = 11*a(n-1) - 18*a(n-2).
G.f.: 7*x/((1-2*x)*(1-9*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = 7*A016133(n-1). - R. J. Mathar, Mar 10 2022
E.g.f.: 2*exp(11*x/2)*sinh(7*x/2). - Elmo R. Oliveira, Mar 31 2025

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

A191466 a(n) = 9^n - 5^n.

Original entry on oeis.org

0, 4, 56, 604, 5936, 55924, 515816, 4704844, 42656096, 385467364, 3477018776, 31332231484, 282185395856, 2540645125204, 22870688939336, 205860614516524, 1852867600961216, 16676418760213444, 150090820599733496, 1350832644186663964, 12157570091625288176, 109418512294354156084

Offset: 0

Vincenzo Librandi, Jun 03 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (14,-45).

Cf. A016163, A016185, A059410, A118004.

Magma
```
[9^n-5^n: n in [0..20]];
```

Table[9^n - 5^n, {n, 0, 25}] (* or *) CoefficientList[Series[4 x/((1 - 5 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 05 2014 *)
LinearRecurrence[{14,-45},{0,4},20] (* Harvey P. Dale, Jun 26 2019 *)

a(n)=9^n-5^n \\ Charles R Greathouse IV, Jun 08 2011

a(n) = 14*a(n-1) - 45*a(n-2).
From Vincenzo Librandi, Oct 05 2014: (Start)
G.f.: 4*x/((1-5*x)*(1-9*x)).
a(n+1) = 4*A016163(n). (End)
E.g.f.: 2*exp(14*x/2)*sinh(2*x). - Elmo R. Oliveira, Mar 31 2025

A191467 9^n - 7^n.

Original entry on oeis.org

0, 2, 32, 386, 4160, 42242, 413792, 3959426, 37281920, 347066882, 3204309152, 29403732866, 268588249280, 2444976817922, 22198569382112, 201143570584706, 1819787258282240, 16444551185679362, 148466221699088672, 1339452822487618946

Offset: 0

Vincenzo Librandi, Jun 03 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (16,-63).

Cf. A118004, A016178, A016185, A059410, A139745, A191467.

Magma
```
[9^n - 7^n: n in [0..20]]:
```

Table[9^n-7^n,{n,0,20}] (* or *) LinearRecurrence[{16,-63},{0,2},20] (* Harvey P. Dale, Jun 21 2014 *)
CoefficientList[Series[2 x/((1 - 7 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 05 2014 *)

a(n)=9^n-7^n \\ Charles R Greathouse IV, Jun 08 2011

a(n) = 16*a(n-1) - 63*a(n-2).
G.f.: 2*x/((1-7*x)*(1-9*x)). - Vincenzo Librandi, Oct 05 2014
a(n+1) = 2*A016178(n). - Vincenzo Librandi, Oct 05 2014

A147671 Primes of the form 9^k-8^k.

Original entry on oeis.org

17, 2685817, 4555386192335572300559213161, 371616904162662789429456905017, 8591830681082909151487632391785043782074371844781522582861081817, 15210163976423790740121668878903464496715921225994590375394324312407011207180253342612172354203245908479382157462960757179523559095554271755561

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 10 2008

nonn

Cf. A016185, A059803.

lst={};Do[p=9^n-8^n;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst
Select[Table[9^n-8^n,{n,150}],PrimeQ] (* Harvey P. Dale, May 13 2018 *)

A248343 a(n) = 10^n - 8^n.

Original entry on oeis.org

0, 2, 36, 488, 5904, 67232, 737856, 7902848, 83222784, 865782272, 8926258176, 91410065408, 931280523264, 9450244186112, 95601953488896, 964815627911168, 9718525023289344, 97748200186314752, 981985601490518016, 9855884811924144128

Offset: 0

Vincenzo Librandi, Oct 05 2014

Index entries for linear recurrences with constant coefficients, signature (18,-80).

Cf. A016185, A016186, A139746.

Cf. A000079, A001018, A005060, A011557.

Magma
```
[10^n-8^n: n in [0..30]];
```

Table[10^n - 8^n, {n, 0, 25}] (* or *) CoefficientList[Series[2 x/((1 - 8 x) (1 - 10 x)), {x, 0, 30}], x]
LinearRecurrence[{18,-80},{0,2},30] (* Harvey P. Dale, May 21 2018 *)

G.f.: 2*x/((1-8*x)*(1-10*x)).
a(n) = 18*a(n-1) - 80*a(n-2).
a(n) = 2^n*(5^n - 4^n) = A000079(n)*A005060(n) = A011557(n) - A001018(n).
a(n+1) = 2*A016186(n).
E.g.f.: 2*exp(9*x)*sinh(x). - Elmo R. Oliveira, Mar 31 2025

cp's OEIS Frontend

A257289 a(n) = 8*9^n - 7*8^n.

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A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

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A191465 a(n) = 9^n - 2^n.

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

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A191466 a(n) = 9^n - 5^n.

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A191467 9^n - 7^n.

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A147671 Primes of the form 9^k-8^k.

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A248343 a(n) = 10^n - 8^n.

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A257289 a(n) = 89^n - 78^n.