A141425 -id:A141425 - cp's OEIS frontend

A141430 a(n) = A000111(n) mod 9.

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

Offset: 0

Paul Curtz, Aug 06 2008

After the initial 1,1, the sequence is periodic with period 12.

This sequence's periodic part is a shuffled version of the two period-6 sequences A070366 and A010697. The sequence contains only the digits 1, 2, 4, 5, 7 and 8 (those of A141425).

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

Cf. A000111, A004099, A010686, A014021.

def A141430(n): return (2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2)[n%12] if n>1 else 1 # Chai Wah Wu, Apr 17 2023

a(n) = A000111(n) mod 9 = A004099(n) mod 9.
a(n+12) = a(n), n > 1.
a(n) + a(n+6) = 9, n > 1.
a(n+11-p) - a(n+p) = 6 (p=0 or 5), 0 (p=1 or 4), -3 (p=2 or 3), any n > 1.
G.f.: (6x^8-5x^7+x^6+2x^5+3x^4+x^3+1) / ((1-x)(x^2+1)(x^4-x^2+1)). - R. J. Mathar, Dec 05 2008
a(n) = 9/2 +(-1)^floor(n/2)*A010686(n)/2 - 3*A014021(n), n > 1. - R. J. Mathar, Dec 05 2008
a(n) = 9/2 - (3/2)*cos(Pi*n/6) + (1/2)*3^(1/2)*sin(Pi*n/6) - (1/2)*cos(Pi*n/2) - (5/2)*sin(Pi*n/2) - (3/2)*cos(5*Pi*n/6) - (1/2)*3^(1/2)*sin(5*Pi*n/6). - Richard Choulet, Dec 12 2008

Edited by R. J. Mathar, Dec 05 2008

A153990 Period 6: repeat [1, 2, 5, 4, 7, 8].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Jan 04 2009

Shares digits with other 6-periodic sequences, see the list in A153130.

Also the decimal expansion of the constant 13942/111111. [R. J. Mathar, Jan 23 2009]

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

Cf. A131533, A141425, A153130, A154811.

&cat[[1, 2, 5, 4, 7, 8]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

A153990:=n->(27-cos(n*Pi)-8*sqrt(3)*cos((1-4*n)*Pi/6)-16*sin((1+2*n)*Pi/6))/6: seq(A153990(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016

Flatten[Table[{1, 2, 5, 4, 7, 8}, {20}]] (* Wesley Ivan Hurt, Jun 17 2016 *)
PadRight[{},120,{1,2,5,4,7,8}] (* Harvey P. Dale, Nov 08 2017 *)

a(n) - A141425(n) = A131533(n+2).
a(6n+0) + a(6n+5) = a(6n+1) + a(6n+4) = a(6n+2) + a(6n+3) = 9.
G.f.: (1+2*x+5*x^2+4*x^3+7*x^4+8*x^5)/((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)). [R. J. Mathar, Jan 23 2009]
From Wesley Ivan Hurt, Jun 17 2016: (Start)
a(n) = (27-cos(n*Pi)-8*sqrt(3)*cos((1-4*n)*Pi/6)-16*sin((1+2*n)*Pi/6))/6.
a(n) = a(n-6) for n>5. (End)

Edited by R. J. Mathar, Jan 23 2009

A144453 a(n) = A061039(8*n+5).

Original entry on oeis.org

16, 160, 16, 832, 1360, 224, 2800, 3712, 176, 5920, 7216, 320, 10192, 11872, 1520, 15616, 17680, 736, 22192, 24640, 336, 29920, 32752, 3968, 38800, 42016, 560, 48832, 52432, 2080, 60016, 64000, 7568, 72352, 76720, 3008, 85840, 90592, 3536, 100480, 105616

Offset: 0

Paul Curtz, Oct 07 2008

Numerators of 16*(n+1)*(4*n+1)/(9*(8*n+5)^2), so all numbers are multiples of 16 because the denominator is always odd.

Interpreted modulo 9, all numbers from 1 to 8 appear: a(20) is the first entry = 3 (mod 9), a(26) is the first entry = 2 (mod 9), a(80) is the first entry = 6 (mod 9).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A020806, A141425, A146537.

Numerator[1/9 - 1/(8*Range[0,100] +5)^2] (* G. C. Greubel, Mar 07 2022 *)

[numerator(1/9 - 1/(8*n+5)^2) for n in (0..100)] # G. C. Greubel, Mar 07 2022

a(n) = A061039(8*n+5).

a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81) for n>83. - Colin Barker, Oct 10 2016

Edited and extended by R. J. Mathar, Oct 24 2008

A156283 Period 6: repeat [1, 2, 4, -4, -2, -1].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Feb 07 2009

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

Cf. A141425.

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

A156283:=n->[1, 2, 4, -4, -2, -1][(n mod 6)+1]: seq(A156283(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

PadRight[{}, 80, {1,2,4,-4,-2,-1}] (* or *) LinearRecurrence[{-1,-1,-1,-1,-1}, {1,2,4,-4,-2}, 80] (* Harvey P. Dale, May 29 2013 *)

a(n) == A141425(n) (mod 9). - Paul Curtz, Feb 08 2009
a(n) = ( (2*A141425(n)) mod 9) - A141425(n). - Paul Curtz, Feb 08 2009
G.f.: (1+x^4+3*x^3+7*x^2+3*x)/( (x+1)*(x^2-x+1)*(x^2+x+1) ). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) = 0 for n>4.
a(n) = cos(n*Pi) + 2*sqrt(3)*cos(n*Pi/6)*sin(n*Pi/6) - sqrt(3)*cos(n*Pi/2)*sin(n*Pi/6) + 3*sin(n*Pi/6)*sin(n*Pi/2). (End)

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

Original entry on oeis.org

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)

A177883 Period 6: repeat [4, 5, 7, 2, 1, 8].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Dec 14 2010

Represents also the decimal expansion of 16934/37037 and the continued fractions of 0.23839... = (sqrt(496555)-667)/158 or of 4.194699... = (667+sqrt(496555))/327. - R. J. Mathar, Dec 20 2010

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

Cf. A173598, A141425, A153130 (permutations).

&cat[[4, 5, 7, 2, 1, 8]: n in [0..20]]; // Wesley Ivan Hurt, Jun 18 2016

A177883:=n->[4, 5, 7, 2, 1, 8][(n mod 6)+1]: seq(A177883(n), n=0..100); # Wesley Ivan Hurt, Jun 18 2016

PadRight[{}, 120, {4,5,7,2,1,8}] (* Harvey P. Dale, Feb 11 2016 *)

a(n)=[4, 5, 7, 2, 1, 8][n%6+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = A166304(n) mod 9 = A022998(3n+2) mod 9.
a(2n) + a(2n+1) = 9.
G.f.: (4+5*x+7*x^2+2*x^3+x^4+8*x^5) / ( (1-x)*(1+x)*(1+x+x^2)*(x^2-x+1) ). - R. J. Mathar, Dec 20 2010
From Wesley Ivan Hurt, Jun 18 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (9 -cos(n*Pi) + 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) + sqrt(3)*sin(n*Pi/3) - 3*sqrt(3)*sin(2*n*Pi/3))/2. (End)

A141446 A102055(n) mod 9.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Aug 07 2008

sign

We compute the positive remainder modulo 9 and subtract 9 if A102055(n) is negative.

Appears to be periodic with period length 18 after the transitional first 3 elements. (This would imply only the same 6 digits appear as found in A141425.)

Cf. A141430.

A102055 := proc(n) local k; if n = 0 then 1; else 1-add(A001469(k),k=1..n) ; end if; end proc:
A141446 := proc(n) local a; a := A102055(n) ; if a > 0 then a mod  9; else (a mod  9)-9; end if; end proc; # R. J. Mathar, Jul 07 2011

a(3n) + a(3n+1) + a(3n+2) = 4, 7, -2, -2, -2, 5 ever same six digits?

A144471 Inverse binomial transform of A020806.

Original entry on oeis.org

1, 3, -5, 13, -30, 61, -119, 234, -467, 937, -1878, 3757, -7511, 15018, -30035, 60073, -120150, 240301, -480599, 961194, -1922387, 3844777, -7689558, 15379117, -30758231, 61516458, -123032915, 246065833, -492131670, 984263341, -1968526679, 3937053354, -7874106707

Offset: 0

Paul Curtz, Oct 10 2008

sign

Index entries for linear recurrences with constant coefficients, signature (-3,-3,-2)

Cf. A141425, A141532.

Digits := 200 ; read("transforms") ; read("transforms3") ; x := 1/7 ; L := CONSTTOLIST(x) ; BINOMIALi(L) ; # R. J. Mathar, Sep 07 2009

LinearRecurrence[{-3,-3,-2},{1,3,-5,13},40] (* Harvey P. Dale, Nov 11 2017 *)

|a(n+1)| - 2*|a(n)| = -A117378(n-1) = A117378(n+2), n>0.
a(n) = -3*a(n-1) - 3*a(n-2) - 2*a(n-3), n > 3.
G.f.: (6*x+7*x^2+9*x^3+1) / ((2*x+1) * (1+x+x^2)). - R. J. Mathar, Sep 07 2009

Edited and extended by R. J. Mathar, Sep 07 2009

A167280 Period length 12: 0,0,1,2,4,7,4,8,7,4,8,5 (and repeat).

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7, 4, 8, 5, 0, 0, 1, 2, 4, 7, 4, 8, 7

Offset: 0

Paul Curtz, Nov 01 2009

The sum of the terms in the period is 50, so the partial sums of the sequence are also 12-periodic if reduced modulo 50 or modulo 10.
The weighted partial sums b(n) = sum_{i=0..n} a(i)*2^i obey b(n) = b(n+12) (mod 10).
Third column is A000689. (Which table or array is this referring to? R. J. Mathar, Nov 01 2009)
The set of digits in the period is the same as in A141425.
A derived sequence with terms a(n)+a(n+6) has period length 6: 4, 8, 8, 6, 12, 12 (repeat).

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

a(n) = A113405(n+1) mod 10.

G.f.: x^2*(1+2*x+4*x^2+7*x^3+4*x^4+8*x^5+7*x^6+4*x^7+8*x^8+5*x^9)/( (1-x)*(1+x+x^2)*(1+x)*(1-x+x^2)*(1+x^2)*(x^4-x^2+1)) [R. J. Mathar, Nov 03 2009]

Edited by R. J. Mathar, Nov 05 2009

A171677 Period 9:repeat 7,5,7,4,2,4,1,8,1.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Dec 15 2009

Represents also the decimal expansion of 252474727/333333333.

Contains the same set of numbers as A141425.

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

Cf. A165568.

PadRight[{},80,{7,5,7,4,2,4,1,8,1}] (* Harvey P. Dale, Aug 27 2019 *)

a(n) = A165563(n+1) mod 9.

G.f.: ( -7-5*x-7*x^2-4*x^3-2*x^4-4*x^5-x^6-8*x^7-x^8 ) / ( (x-1) *(1+x+x^2) *(x^6+x^3+1) ). - R. J. Mathar, Mar 08 2011

More terms from Jinyuan Wang, Feb 26 2020

cp's OEIS Frontend

A141430 a(n) = A000111(n) mod 9.

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A153990 Period 6: repeat [1, 2, 5, 4, 7, 8].

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A144453 a(n) = A061039(8*n+5).

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A156283 Period 6: repeat [1, 2, 4, -4, -2, -1].

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

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A177883 Period 6: repeat [4, 5, 7, 2, 1, 8].

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A141446 A102055(n) mod 9.

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