A131711 -id:A131711 - cp's OEIS frontend

A131725 Partial sums of A131711.

0, 1, 3, 8, 10, 19, 19, 28, 36, 41, 49, 50, 50, 51, 53, 58, 60, 69, 69, 78, 86, 91, 99, 100, 100, 101, 103, 108, 110, 119, 119, 128, 136, 141, 149, 150, 150, 151, 153, 158, 160, 169, 169, 178, 186, 191, 199, 200

Offset: 0

Paul Curtz, Sep 16 2007

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

Cf. A131711.

O.g.f.: x*(x^8+8x^7+4x^6+5x^4+4x^2+2x+1)/((x-1)^2*(1+x)*(x^2+x+1)*(x^2-x+1)*(x^4-x^2+1)). - R. J. Mathar, Jul 04 2008

a(n) = a(n-1) + a(n-2) - a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-7) - a(n-8) + a(n-9) + a(n-10) - a(n-11) for n > 10. - Chai Wah Wu, Jan 09 2021

Edited by R. J. Mathar, Jul 04 2008

A131036 First differences of A131711.

Original entry on oeis.org

1, 1, 3, -3, 7, -9, 9, -1, -3, 3, -7, -1, 1, 1, 3, -3, 7, -9, 9, -1, -3, 3, -7, -1, 1, 1, 3, -3, 7, -9, 9, -1, -3, 3, -7, -1, 1, 1, 3, -3, 7, -9, 9, -1, -3, 3, -7, -1

Offset: 0

Paul Curtz, Sep 23 2007

sign

The equivalent operation on A001333 yields last digits A131707 and their first differences A131715.

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

Cf. A131711.

Differences[PadRight[{},120,{0,1,2,5,2,9,0,9,8,5,8,1}]] (* Harvey P. Dale, Dec 29 2014 *)

a(n) = a(n-12).
From Chai Wah Wu, Jan 09 2021: (Start)
a(n) = - a(n-1) - a(n-4) - a(n-5) - a(n-8) - a(n-9) for n > 8.
G.f.: (x^8 + 8*x^7 + 4*x^6 + 5*x^4 + 4*x^2 + 2*x + 1)/(x^9 + x^8 + x^5 + x^4 + x + 1). (End)

Edited by R. J. Mathar, Jul 07 2008

A131707 Period 12: repeat 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9 .

Original entry on oeis.org

1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9, 1, 1, 3, 7, 7, 1, 9, 9, 7

Offset: 0

Paul Curtz, Sep 14 2007

Also the decimal expansion of 1023949/9000009. [R. J. Mathar, Feb 07 2009]

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

Cf. A131711.

PadRight[{},120,{1,1,3,7,7,1,9,9,7,3,3,9}] (* Harvey P. Dale, May 02 2012 *)

a(n) = A001333(n) mod 10. - Paul Curtz, Apr 08 2008
G.f.: (1+2x^2+4x^3-6x^5+9x^6)/((1-x)(1+x^2)(1-x^2+x^4)). a(n)=a(n-1)-a(n-6)+a(n-7). [R. J. Mathar, Feb 07 2009]
a(n) = 5-2*cos(Pi*n/6) -2*sin(Pi*n/6)/3 -10*sin(Pi*n/2)/3 -2*cos(5*Pi*n/6) -2*sin(5*Pi*n/6)/3. - R. J. Mathar, Oct 08 2011

More terms from Tracy Poff (tracy.poff(AT)gmail.com), Dec 21 2008

Even more periods from R. J. Mathar, Feb 07 2009

A131727 Pell numbers A000129 without last digit.

Original entry on oeis.org

1, 2, 7, 16, 40, 98, 237, 574, 1386, 3346, 8078, 19502, 47083, 113668, 274421, 662510, 1599442, 3861396, 9322235, 22505868, 54333972, 131173812, 316681596, 764537004, 1845755605, 4456048214, 10757852035, 25971752284, 62701356604

Offset: 4

Paul Curtz, Sep 16 2007, Sep 27 2007

A000129 := proc(n) option remember ; if n <= 2 then n ; else 2*A000129(n-1)+A000129(n-2) ; fi ; end: A131727 := proc(n) floor(A000129(n)/10) ; end: seq(A131727(n),n=4..40) ; # R. J. Mathar, Oct 24 2007

a(n)=(A000129(n)-A131711(n))/10. - R. J. Mathar, Jul 17 2009

More terms from R. J. Mathar, Oct 24 2007

A131716 Period 6: repeat [0, 1, 2, 5, 8, 9].

Original entry on oeis.org

0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1, 2, 5, 8, 9, 0, 1

Offset: 0

Paul Curtz, Sep 14 2007

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

Cf. A131711.

&cat[[0, 1, 2, 5, 8, 9]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A131716:=n->[0, 1, 2, 5, 8, 9][(n mod 6)+1]: seq(A131716(n), n=0..100); # Wesley Ivan Hurt, Jun 19 2016

PadRight[{}, 200, {0, 1, 2, 5, 8, 9}] (* Wesley Ivan Hurt, Jun 19 2016 *)

From Wesley Ivan Hurt, Jun 19 2016: (Start)
G.f.: x*(1+2*x+5*x^2+8*x^3+9*x^4)/(1-x^6).
a(n) = a(n-6) for n>5.
a(n) = (25 - 5*cos(n*Pi) - 10*cos(n*Pi/3) - 10*cos(2*n*Pi/3) - 14*sqrt(3)*sin(n*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3))/6. (End)

cp's OEIS Frontend

A131725 Partial sums of A131711.

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A131036 First differences of A131711.

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A131707 Period 12: repeat 1, 1, 3, 7, 7, 1, 9, 9, 7, 3, 3, 9 .

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A131727 Pell numbers A000129 without last digit.

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A131716 Period 6: repeat [0, 1, 2, 5, 8, 9].

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