A070366 -id:A070366 - cp's OEIS frontend

A173598 Period 6: repeat [1, 8, 7, 2, 4, 5].

1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8

Offset: 0

Paul Curtz, Nov 23 2010

For A141425 = 1,2,4,5,7,8 permutations, see A153130. a(n) is based on A022998. Successive differences are linked to A070366.

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

Cf. A022998, A070366, A141425, A153130, A166138.

&cat [[1, 8, 7, 2, 4, 5]^^20]; // Wesley Ivan Hurt, Jun 23 2016

A173598:=n->[1, 8, 7, 2, 4, 5][(n mod 6)+1]: seq(A173598(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

PadRight[{}, 100, {1, 8, 7, 2, 4, 5}] (* Wesley Ivan Hurt, Jun 23 2016 *)

a(n)=[1,8,7,2,4,5][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n) = A166138(n) mod 9.
a(2n+1) + a(2n+2) = 9.
G.f.: (1+8*x+7*x^2+2*x^3+4*x^4+5*x^5) / ((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)). - R. J. Mathar, Mar 08 2011
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (9 - cos(n*Pi) - 6*cos(2*n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/2. (End)

A234038 Smallest positive integer solution x of 9x - 2^ny = 1.

Original entry on oeis.org

1, 1, 1, 1, 9, 25, 57, 57, 57, 57, 569, 1593, 3641, 3641, 3641, 3641, 36409, 101945, 233017, 233017, 233017, 233017, 2330169, 6524473, 14913081, 14913081, 14913081, 14913081, 149130809, 417566265, 954437177, 954437177, 954437177

Offset: 0

Wolfdieter Lang, Feb 17 2014

The solution of the linear Diophantine equation 9*x - 2^n*y = 1 with smallest positive x is x=a(n), n>= 0, and the corresponding y is given by y(n) = 5^(n+3) (mod 9) = A070366(n+3) with o.g.f. (8-4*x-2*x^2+7*x^3)/((1-x+x^2)*(1-x)*(1+x)) (derived from the one given in A070366). This is the periodic sequence with period [8, 4, 2, 1, 5, 7].

			n = 0:  9*1 - 1*8 = 1; n = 3:  9*1 - 8*1  = 1.
a(4) = (1 + 2^4*5)/9 = 9.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710, 2014 and J. Int. Seq. 17 (2014) # 14.11.7.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-8,24,-16)

Cf. A070366, A007583 (see the Feb 15 2013 comment).

A234038[n_] := Ceiling[(1 + 2^n*Mod[5^(n - 3), 9])/9]; Array[A234038, 50, 0] (* or *)
LinearRecurrence[{3, -2, -8, 24, -16}, {1, 1, 1, 1, 9}, 50] (* Paolo Xausa, Nov 05 2024 *)

a(n) = (1 + 2^n*(5^(n-3)(mod 9)))/3^2, n >= 3.
O.g.f.: (1-2*x+8*x^3-8*x^4)/((1-x)*(1-4*x^2)*(1-2*x+4*x^2)) (derived from the one for y(n) given above in a comment).
a(n) = 2*(a(n-1) - 4*a(n-3) + 8*a(n-4)) - 1, n >= 4, a(0)=a(1)=a(2)=a(3) = 1 (from the y(n) recurrence given in A070366).

A253298 Digital root for the following sequences, F(4n)/F(4); F(12n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.

Original entry on oeis.org

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

Offset: 1

Peter M. Chema, Dec 30 2014

Cyclical and palindromic in two parts with periodicity 18: {1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8, 9}.
Digital root of the period is 9, its mean and median is 5, and its product is (9!)^2.
See A253368 for the initial motivation for this sequence.
From Peter M. Chema, Jul 04 2016: (Start)
A composite of three respective digital root sequences in alternation: a "halving sequence" of 1, 5, 7, 8, 4, 2, a "doubling sequence" of 7, 5, 1, 2, 4, 8, and a three-six-nine circuit of 3, 3, 9, 6, 6, 9.
Also the digital root of A000045(4n)/3 or A004187(n).  In general terms, sequences defined by Fib(x*n)/ Fib(x) where x=(8*a-4) all share the same digital root (e.g., F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20); F(28*n)/F(28); F(36*n)/F(36), etc.) (End)

Tom Barnett, Phi VBM Tori Array, YouTube video (see first two minutes).
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1).

Cf. A010888, A253368, A070366 and A001370.

f[n_] := Mod[ Fibonacci[ 12n]/144, 9]; Array[f, 5*18] (* Robert G. Wilson v, Jan 23 2015 *)
LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1},{1, 7, 3, 5, 5, 3, 7, 1, 9, 8, 2, 6, 4, 4, 6, 2, 8},72] (* Ray Chandler, Aug 12 2015 *)

a(n) = A010888(A253368(n)).

G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3 + 5*x^4 + 3*x^5 + 7*x^6 + x^7 + 9*x^8 + 8*x^9 + 2*x^10 + 6*x^11 + 4*x^12 + 4*x^13 + 6*x^14 + 2*x^15 + 8*x^16 + 9*x^17)/(1 - x^18). - Vincenzo Librandi, Mar 28 2016

Edited. Numbers and name changed to fit A253368. Formula adapted. Cross reference added. - Wolfdieter Lang, Jan 28 2015

Name generalized by Peter M. Chema, Jul 04 2016

A321643 a(n) = 5*2^n - (-1)^n.

Original entry on oeis.org

4, 11, 19, 41, 79, 161, 319, 641, 1279, 2561, 5119, 10241, 20479, 40961, 81919, 163841, 327679, 655361, 1310719, 2621441, 5242879, 10485761, 20971519, 41943041, 83886079, 167772161, 335544319, 671088641, 1342177279, 2684354561, 5368709119, 10737418241, 21474836479

Offset: 0

Paul Curtz, Dec 03 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A010690, A010701, A020714, A051049, A070366, A110286.

Cf. A001045, A081808, A321483.

List([0..30],n->5*2^n-(-1)^n); # Muniru A Asiru, Dec 05 2018

[5*2^n-(-1)^n$n=0..30]; # Muniru A Asiru, Dec 05 2018

a[n_] := 5*2^n - (-1)^n; Array[a, 30, 0] (* Amiram Eldar, Dec 03 2018 *)

Vec((4 + 7*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 04 2018

for n in range(0,30): print(5*2**n - (-1)**n) # Stefano Spezia, Dec 05 2018

a(n+2) - a(n) = a(n+1) + a(n) = 15*2^n, n >= 0.
a(n) - 2*a(n-1) = period 2: repeat [3, -3], n > 0, a(0)=4, a(1)=11.
a(n+1) = 10*A051049(n) + period 2: repeat [1, 9].
a(n) = 12*2^n - A321483(n), n >= 0.
a(n) = 2^(n+2) + 3*A001045(n), n >= 0.
a(n) == A070366(n+4) (mod 9).
From Colin Barker, Dec 04 2018: (Start)
G.f.: (4 + 7*x) / ((1 + x)*(1 - 2*x)).
a(n) = a(n-1) + 2*a(n-2) for n > 1. (End)
E.g.f.: exp(-x)*(5*exp(3*x) - 1). - Elmo R. Oliveira, Aug 17 2024

A132708 Period 6: repeat [4, 2, 1, -4, -2, -1].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Nov 16 2007

Old definition was: "First differences of 5^n mod 9".

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

Cf. A070366 (5^n mod 9).

&cat [[4,2,1,-4,-2,-1]^^30]; // Wesley Ivan Hurt, Jun 28 2016

A132708:= n -> [4, 2, 1, -4, -2, -1][(n mod 6)+1]: seq(A132708(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

PadRight[{}, 120, {4, 2, 1, -4, -2, -1}] (* Harvey P. Dale, Nov 21 2015 *)
Differences[PowerMod[5, Range[0, 90], 9]] (* Alonso del Arte, Jun 04 2016 *)

G.f.: (x^2 + 2*x + 4)/(x^3 + 1). - Chai Wah Wu, Jun 04 2016
From Wesley Ivan Hurt, Jun 28 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = a(n-6) for n>5.
a(n) = cos(n*Pi) + 3*cos(n*Pi/3) + sqrt(3)*sin(n*Pi/3). (End)

Name changed by Wesley Ivan Hurt, Jun 28 2016

cp's OEIS Frontend

A173598 Period 6: repeat [1, 8, 7, 2, 4, 5].

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A234038 Smallest positive integer solution x of 9*x - 2^n*y = 1.

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A253298 Digital root for the following sequences, F(4*n)/F(4); F(12*n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.

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A321643 a(n) = 5*2^n - (-1)^n.

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A132708 Period 6: repeat [4, 2, 1, -4, -2, -1].

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A234038 Smallest positive integer solution x of 9x - 2^ny = 1.

A253298 Digital root for the following sequences, F(4n)/F(4); F(12n)/F(12); F(20*n)/F(20), where the pattern increases by 8, ad infinitum, with the Fibonacci numbers F = A000045.