A016145 -id:A016145 - cp's OEIS frontend

A248226 a(n) = 10^n - 3^n.

0, 7, 91, 973, 9919, 99757, 999271, 9997813, 99993439, 999980317, 9999940951, 99999822853, 999999468559, 9999998405677, 99999995217031, 999999985651093, 9999999956953279, 99999999870859837, 999999999612579511, 9999999998837738533, 99999999996513215599

Offset: 0

Vincenzo Librandi, Oct 05 2014

Index entries for linear recurrences with constant coefficients, signature (13,-30).

Cf. similar sequences listed in A248225.

Cf. A016145, A016189.

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

Table[10^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[(7 x)/((1 - 3 x) (1 - 10 x)), {x, 0, 30}], x]
LinearRecurrence[{13,-30},{0,7},30] (* Harvey P. Dale, Jul 10 2021 *)

a(n) = 10^n - 3^n \\ Michel Marcus, Oct 05 2014

G.f.: 7*x/((1 - 3*x)*(1 - 10*x)).
a(n) = 13*a(n-1) - 30*a(n-2).
a(n) = A011557(n) - A000244(n).
a(2*n) mod 10^n = A016189(n). - Michel Marcus, Oct 05 2014
a(n+1) = 7*A016145(n). - Bruno Berselli, Oct 05 2014
E.g.f.: exp(3*x)*(exp(7*x) - 1). - Stefano Spezia, Mar 09 2025

A021724 Expansion of 1/((1-x)(1-3x)(1-10x)(1-12x)).

Original entry on oeis.org

1, 26, 465, 7150, 101621, 1378026, 18123145, 233349350, 2958918141, 37094306626, 461004657425, 5690785933950, 69876732453061, 854393804284826, 10411455807073305, 126524771262956950, 1534170271000826381

Offset: 0

N. J. A. Sloane

From Bruno Berselli, May 08 2013: (Start)
Naturally, the sequence is related to:
A018207, 1/((1-3x)(1-10x)(1-12x)):  A018207(n) = a(n)-a(n-1), n>0;
A016267, 1/((1-x)(1-10x)(1-12x)):  A016267(n) = a(n)-3*a(n-1), n>0;
A016217, 1/((1-x)(1-3x)(1-12x)):  A016217(n) = a(n)-10*a(n-1), n>0;
A016215, 1/((1-x)(1-3x)(1-10x)):  A016215(n) = a(n)-12*a(n-1), n>0;
A016196, 1/((1-10x)(1-12x)):  A016196(n) = a(n)-4*a(n-1)+3*a(n-2), n>1;
A016147, 1/((1-3x)(1-12x)):  A016147(n) = a(n)-11*a(n-1)+10*a(n-2), n>1;
A016145, 1/((1-3x)(1-10x)):  A016145(n) = a(n)-13*a(n-1)+12*a(n-2), n>1;
A016125, 1/((1-x)(1-12x)):  A016125(n) = a(n)-13*a(n-1)+30*a(n-2), n>1;
A002275, x/((1-x)(1-10x)):  A002275(n) = a(n-1)-15*a(n-2)+36*a(n-3), n>2;
A003462, x/((1-x)(1-3x)):  A003462(n) = a(n-1)-22*a(n-2)+120*a(n-3), n>2;
A000012, 1/(1-x):  A000012(n) = a(n)-25*a(n-1)+186*a(n-2)-360*a(n-3), n>2;
A000244, 1/(1-3x):  A000244(n) = a(n)-23*a(n-1)+142*a(n-2)-120*a(n-3), n>2;
A011557, 1/(1-10x):  A011557(n) = a(n)-16*a(n-1)+51*a(n-2)-36*a(n-3), n>2;
A001021, 1/(1-12x):  A001021(n) = a(n)-14*a(n-1)+43*a(n-2)-30*a(n-3), n>2. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (26,-211,546,-360).

Cf. A000012, A000244, A001021, A002275, A003462, A011557, A016125, A016145, A016147, A016196, A016215, A016217, A016267, A018207.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)))); // Bruno Berselli, May 07 2013

CoefficientList[Series[1/((1 - x) (1 - 3 x) (1 - 10 x) (1 - 12 x)), {x, 0, 20}], x] (* Bruno Berselli, May 07 2013 *)
LinearRecurrence[{26,-211,546,-360},{1,26,465,7150},120] (* Harvey P. Dale, Jul 06 2019 *)

Vec(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x))+O(x^20)) \\ Bruno Berselli, May 07 2013

G.f.: 1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)).

a(n) = -1/198 +3^(n+1)/14 -2^(n+2)*5^(n+3)/63 +2^(2n+5)*3^(n+1)/11. [Bruno Berselli, May 07 2013]

cp's OEIS Frontend

A248226 a(n) = 10^n - 3^n.

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