A070365 -id:A070365 - cp's OEIS frontend

A005060 a(n) = 5^n - 4^n.

0, 1, 9, 61, 369, 2101, 11529, 61741, 325089, 1690981, 8717049, 44633821, 227363409, 1153594261, 5835080169, 29443836301, 148292923329, 745759583941, 3745977788889, 18798608421181, 94267920012849

Offset: 0

N. J. A. Sloane

Also, the number of numbers with at most n digits whose largest digit equals 4. - M. F. Hasler, May 03 2015
a(n) is divisible by 7 iff n is divisible by 6; for example: a(6) = 11529 = 7 * 1647 (see 'Les cahier du bac' or subtract A070365 and A153727 and locate zeros). - Bernard Schott, Oct 02 2020
a(n) is the number of n-digit numbers whose smallest decimal digit is 5. - Stefano Spezia, Nov 15 2023

Les Cahiers du Bac, Terminales C & E, Tome 1, 1985, Exercice 109, p. 18; Bac Rouen, Série C, 1978.

Muniru A Asiru, Table of n, a(n) for n = 0..200
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (9,-20).

Cf. A070365, A153727.

List([0..20],n->5^n - 4^n); # Muniru A Asiru, Mar 04 2018

a:=n->sum(4^(n-j)*binomial(n,j),j=1..n): seq(a(n), n=0..18); # Zerinvary Lajos, Jan 04 2007

a[n_]:=5^n-4^n; a[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
LinearRecurrence[{9,-20},{0,1},30] (* Harvey P. Dale, Oct 01 2016 *)

a(n)=5^n-4^n \\ M. F. Hasler, May 03 2015

[lucas_number1(n, 9, 20) for n in range(21)]  # Zerinvary Lajos, Apr 23 2009

a(n) = 5*a(n-1) + 4^(n-1). - Xavier Acloque, Oct 20 2003
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-5*x) - 1/(1-4*x).
E.g.f.: e^(5*x) - e^(4*x). (End)
a(n) = 9*a(n-1) - 20*a(n-2), a(0)=0, a(1)=1. - Vincenzo Librandi, Jan 28 2011

A201910 Irregular triangle of 5^k mod prime(n).

Original entry on oeis.org

1, 1, 2, 0, 1, 5, 4, 6, 2, 3, 1, 5, 3, 4, 9, 1, 5, 12, 8, 1, 5, 8, 6, 13, 14, 2, 10, 16, 12, 9, 11, 4, 3, 15, 7, 1, 5, 6, 11, 17, 9, 7, 16, 4, 1, 5, 2, 10, 4, 20, 8, 17, 16, 11, 9, 22, 18, 21, 13, 19, 3, 15, 6, 7, 12, 14, 1, 5, 25, 9, 16, 22, 23, 28, 24, 4

Offset: 1

T. D. Noe, Dec 07 2011

Except for the third row, the first term of each row is 1. Many sequences are in this one: starting at A036121 (mod 23) and A070365 (mod 7).

			The first 9 rows are:
1
1, 2
0
1, 5, 4, 6, 2, 3
1, 5, 3, 4, 9
1, 5, 12, 8
1, 5, 8, 6, 13, 14, 2, 10, 16, 12, 9, 11, 4, 3, 15, 7
1, 5, 6, 11, 17, 9, 7, 16, 4
1, 5, 2, 10, 4, 20, 8, 17, 16, 11, 9, 22, 18, 21, 13, 19, 3, 15, 6, 7, 12, 14

T. D. Noe, Rows n = 1..60, flattened

Cf. A201908 (2^k), A201909 (3^k), A201911 (7^k).

Cf. A070365 (7), A070367 (11), A070368 (13), A070371 (17), A070373 (19), A036121 (23), A070379 (29), A070384 (37), A070387 (41), A070389 (43), A036127 (47), A036133 (73), A036137 (97), A036139 (103), A036149 (157), A036151 (167), A036156 (193).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(5,P[n]));;
Flat(Concatenation([1,1,2,0],List([3..10],n->List([0..R[n]-1],k->PowerMod(5,k,P[n]))))); # Muniru A Asiru, Feb 02 2019

nn = 10; p = 5; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

A271378 a(n) = 5^n mod 31.

Original entry on oeis.org

1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25

Offset: 0

Vincenzo Librandi, Apr 06 2016

Period 3: repeat [1, 5, 25].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. similar sequences of the type 5^n mod p, where p is a prime: A070365 (p=7), A070367 (p=11), A070368 (p=13), A070371 (p=17), A070373 (p=19), A036121 (p=23), A070379 (p=29), this sequence (p=31), A070384 (p=37), A070387 (p=41), A070389 (p=43), A036127 (p=47), A036133 (p=73), A036137 (p=97), A271379 (p=101), A036139 (p=103), A036149 (p=157), A271380 (p=163) A036151 (p=167), A036156 (p=193).

Magma
```
[Modexp(5, n, 31): n in [0..100]];
```

&cat [[1,5,25]^^30]; // Bruno Berselli, Apr 07 2016

seq(op([1, 5, 25]), n=0..50); # Wesley Ivan Hurt, Jun 30 2016

Mathematica
```
PowerMod[5, Range[0, 100], 31]
```

x='x+O('x^99); Vec((1+5*x+25*x^2)/(1-x^3)) \\ Altug Alkan, Apr 06 2016

G.f.: (1+5*x+25*x^2)/(1-x^3).
a(n) = a(n-3) for n>2.
a(n) = 5^(n mod 3).
a(n) = (31 - 28*cos(2*n*Pi/3) - 20*sqrt(3)*sin(2*n*Pi/3))/3. - Wesley Ivan Hurt, Jun 30 2016

Edited by Bruno Berselli, Apr 07 2016

A178141 Period 6: repeat [4, -1, 2, -4, 1, 2].

Original entry on oeis.org

4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2

Offset: 0

Paul Curtz, May 21 2010

Differences of the period 6: repeat [1, 5, 4, 6, 2, 3] (A070365).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A069705, A070365, A132954, A153727.

&cat [[4, -1, 2, -4, 1, 2]^^20]; // Wesley Ivan Hurt, Jun 23 2016

A178141:=n->[4, -1, 2, -4, 1, 2][(n mod 6)+1]: seq(A178141(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

PadRight[{}, 100, {4, -1, 2, -4, 1, 2}] (* Wesley Ivan Hurt, Jun 23 2016 *)

a(n)=[1,5,4,6,2,3][n%5+1] \\ Charles R Greathouse IV, Jun 02 2011

Mix A153727(n+1) with -A153727(n).
From Wesley Ivan Hurt, Jun 23 2016: (Start)
G.f.: (4-x+2*x^2-4*x^3+x^4+2*x^5)/(1-x^6).
a(n) = a(n-6) for n>5.
a(n) = (2 + 5*cos(n*Pi) + 7*cos(n*Pi/3) - 2*cos(2*n*Pi/3) - sqrt(3)*sin(n*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3))/3. (End)

New name from Wesley Ivan Hurt, Jun 23 2016

A178229 Decimal expansion of (221+11*sqrt(1086))/490.

Original entry on oeis.org

1, 1, 9, 0, 8, 1, 5, 6, 2, 2, 8, 0, 0, 5, 0, 9, 7, 6, 4, 4, 5, 5, 1, 1, 9, 9, 8, 3, 0, 7, 4, 3, 7, 9, 5, 7, 5, 5, 7, 9, 9, 0, 8, 8, 6, 7, 8, 0, 8, 8, 1, 1, 0, 3, 3, 1, 8, 6, 0, 1, 0, 1, 1, 9, 0, 3, 6, 9, 8, 4, 0, 1, 3, 4, 5, 4, 0, 8, 5, 4, 0, 0, 6, 1, 6, 7, 8, 5, 9, 6, 3, 7, 9, 5, 8, 5, 8, 3, 3, 5, 1, 0, 2, 3, 0

Offset: 1

Klaus Brockhaus, May 23 2010

Continued fraction expansion of (221+11*sqrt(1086))/490 is A070365.

			(221+11*sqrt(1086))/490 = 1.19081562280050976445...

Cf. A178230 (decimal expansion of sqrt(1086)), A070365 (repeat 1, 5, 4, 6, 2, 3).

cp's OEIS Frontend

A005060 a(n) = 5^n - 4^n.

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A201910 Irregular triangle of 5^k mod prime(n).

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A271378 a(n) = 5^n mod 31.

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A178141 Period 6: repeat [4, -1, 2, -4, 1, 2].

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Magma

Maple

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A178229 Decimal expansion of (221+11*sqrt(1086))/490.

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