A131707 -id:A131707 - cp's OEIS frontend

A131091 Partial sums of A131707.

1, 2, 5, 12, 19, 20, 29, 38, 45, 48, 51, 60, 61, 62, 65, 72, 79, 80, 89, 98, 105, 108, 111, 120, 121, 122, 125, 132, 139, 140, 149, 158, 165, 168, 171, 180, 181, 182, 185, 192, 199, 200, 209, 218, 225, 228, 231, 240, 241, 242, 245, 252, 259, 260, 269, 278, 285

Offset: 0

Paul Curtz, Sep 24 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,-1,2,-1).

Cf. A001333, A131707, A131725.

Accumulate[PadRight[{},60,{1,1,3,7,7,1,9,9,7,3,3,9}]] (* Harvey P. Dale, Jul 30 2025 *)

a(n) = Sum_{i=0..n} A131707(i), where A131707(n) = A001333(n) mod 10.
O.g.f.: (9x^6-6x^5+4x^3+2x^2+1)/((x-1)^2*(x^2+1)*(x^4-x^2+1)). - R. J. Mathar, Jul 16 2008
a(n) = 2*a(n-1) - a(n-2) - a(n-6) + 2*a(n-7) - a(n-8) for n > 7. - Chai Wah Wu, Jan 09 2021

Edited by R. J. Mathar, Jul 16 2008

A001903 Final digit of 7^n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Period 4: repeat [1, 7, 9, 3]. - Joerg Arndt, Aug 12 2014

Derek Orr, Table of n, a(n) for n = 0..1000
Edward Omey and Stefan Van Gulck, What are the last digits of ...?, International Journal of Mathematical Education in Science and Technology, (2015) 46:1, 147-155.
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000420, A001148, A010879, A131707, A159966.

[7^n mod 10: n in [0..57]]; // Vincenzo Librandi, Feb 08 2011

A001903:=n->7^n mod 10: seq(A001903(n), n=0..100); # Wesley Ivan Hurt, Aug 12 2014

Table[PowerMod[7, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{1,-1,1},{1,7,9},100] (* or *) PadRight[{},100,{1,7,9,3}] (* Harvey P. Dale, May 21 2025 *)

a(n)=lift(Mod(7,10)^n) \\ Charles R Greathouse IV, Dec 28 2012

[power_mod(7,n,10)for n in range(0,81)] # Zerinvary Lajos, Nov 03 2009

a(n) = 7^n mod 10. - Zerinvary Lajos, Nov 03 2009
From R. J. Mathar, Apr 20 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 2.
G.f.: ( 1+6*x+3*x^2 ) / ( (1-x)*(1+x^2) ). (End)
a(n) = 10 - a(n-2) for n > 1. - Vincenzo Librandi, Feb 08 2011
From Bruno Berselli, Feb 08 2011: (Start)
a(n) = 5 - (2-i)*(-i)^n - (2+i)*i^n, where i=sqrt(-1).
a(n) = A001148(A159966(n)). (End)
a(n) = A010879(A000420(n)). - Michel Marcus, Jul 06 2016
E.g.f.: 2*sin(x) - 4*cos(x) + 5*exp(x). - Ilya Gutkovskiy, Jul 06 2016

A131711 Period 12: repeat 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Sep 14 2007

Final digits of Pell numbers. First differences: 1, 1, 3, -3, 7, -9, 9, -1, -3, 3 -7, -1, 1 (cf. A131707).

Can be though of as 2 interlocking sequences, each of the form a(n) = a(n - 1) - a(n - 2) + a(n - 3) - a(n - 4) + a(n - 5).

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

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

[0,1,2,5,2,9,0,9,8,5,8,1][n%12+1] \\ Charles R Greathouse IV, Jun 02 2011

G.f.: (x^8+8x^7+4x^6+5x^4+4x^2+2x+1)x/((1-x) (1+x) (x^2+x+1) (x^2-x+1) (x^4-x^2+1)). a(n) = |A131201(n)| = A000129(n) mod 10 = A000129(n)-10*A131727(n). [From R. J. Mathar, Sep 20 2008]

a(n) = 25/6 -4*cos(Pi*n/6)/sqrt(3) -sqrt(3)*sin(Pi*n/6) -5*cos(Pi*n/3)/3 -5*cos(2*Pi*n/3)/3 +4*cos(5*Pi*n/6)/sqrt(3) +sqrt(3)*sin(5*Pi*n/6) -5*(-1)^n/6. - R. J. Mathar, Oct 08 2011

A131712 Period 4: repeat [1, 3, 7, 9].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Sep 14 2007

Decimal expansion of 1379/9999. - Klaus Brockhaus, May 21 2010

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

Cf. A072845, A131707.

Cf. A178148 (decimal expansion of (243+17*sqrt(285))/402). - Klaus Brockhaus, May 21 2010

&cat[[1, 3, 7, 9]: k in [1..30]]; // Vincenzo Librandi, Nov 23 2010

seq(op([1, 3, 7, 9]), n=0..40); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 100, {1, 3, 7, 9}] (* Wesley Ivan Hurt, Jul 09 2016 *)

a(n)=1+2*(n%4)+2*(n%4\2) \\ Jaume Oliver Lafont, Aug 28 2009

G.f.: (1+3*x+7*x^2+9*x^3)/((1-x)*(x+1)*(1+x^2)). - R. J. Mathar, Nov 14 2007
From Wesley Ivan Hurt, Jul 09 2016: (Start)
a(n) = a(n-4) for n>3.
a(n) = 5 - 3*cos(n*Pi/2) - cos(n*Pi) - 3*sin(n*Pi/2) - I*sin(n*Pi). (End)

More terms from Klaus Brockhaus, May 21 2010

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

A131037 Sequence A001333 with last digits set to zero.

Original entry on oeis.org

0, 0, 0, 0, 10, 40, 90, 230, 570, 1390, 3360, 8110, 19600, 47320, 114240, 275800, 665850, 1607520, 3880890, 9369310, 22619530, 54608390, 131836320, 318281030, 768398400, 1855077840, 4478554080, 10812186000, 26102926090, 63018038200

Offset: 0

Paul Curtz, Sep 23 2007

a(n)=A001333(n)-A131707(n) = 10*A131607(n).

Edited by R. J. Mathar, Jul 07 2008

cp's OEIS Frontend

A131091 Partial sums of A131707.

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A001903 Final digit of 7^n.

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A131711 Period 12: repeat 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1.

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A131712 Period 4: repeat [1, 3, 7, 9].

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A131036 First differences of A131711.

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A131037 Sequence A001333 with last digits set to zero.

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