A001148 -id:A001148 - cp's OEIS frontend

A000689 Final decimal digit of 2^n.

1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6

Offset: 0

N. J. A. Sloane

These are the analogs of the powers of 2 in carryless arithmetic mod 10.
Let G = {2,4,8,6}. Let o be defined as XoY = least significant digit in XY. Then (G,o) is an Abelian group wherein 2 is a generator (also see the first comment under A001148). - K.V.Iyer, Mar 12 2010
This is also the decimal expansion of 227/1818. - Kritsada Moomuang, Dec 21 2021

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 6*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 6*x^8 + ...

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000079, A000855, A008952, A034887, A173635.

a000689 n = a000689_list !! n
a000689_list = 1 : cycle [2,4,8,6]  -- Reinhard Zumkeller, Sep 15 2011

[2^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011

Table[PowerMod[2, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

for(n=0,80, if(n,{x=(n+3)%4+1; print1(10-(4*x^3+47*x-27*x^2)/3,", ")},{print1("1, ")}))

[power_mod(2,n,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 03 2009

Periodic with period 4.
a(n) = 2^n mod 10.
a(n) = A002081(n) - A002081(n-1), for n > 0.
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3), n > 3.
G.f.: (x+3*x^2+5*x^3+1)/((1-x) * (1+x^2)). (End)
For n >= 1, a(n) = 10 - (4x^3 + 47x - 27x^2)/3, where x = (n+3) mod 4 + 1.
For n >= 1, a(n) = A070402(n) + 5*floor( ((n-1) mod 4)/2 ).
G.f.: 1 / (1 - 2*x / (1 + 5*x^3 / (1 + x / (1 - 3*x / (1 + 3*x))))). - Michael Somos, May 12 2012
a(n) = 5 + cos((n*Pi)/2) - 3*sin((n*Pi)/2) for n >= 1. - Kritsada Moomuang, Dec 21 2021

A216099 Period of powers of 3 mod 10^n.

Original entry on oeis.org

4, 20, 100, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000

Offset: 1

V. Raman, Sep 01 2012

Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A000244, A001148, A093143, A001218.

Table[If[n<5,(4*5^n)/5,10^n/20],{n,20}] (* or *) Join[{4,20,100},NestList[ 10#&,500,20]] (* Harvey P. Dale, May 31 2017 *)

a(n)=if(n<5,4*5^n/5,10^n/20) \\ Charles R Greathouse IV, Mar 26 2016

a(n) = 4*5^(n-1) for n <= 4.

a(n) = 5*10^(n-2) for n >= 5.

A281181 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^3 dx ).

Original entry on oeis.org

1, 1, 13, 493, 37369, 4732249, 901188997, 240798388357, 85948640603761, 39504564917358001, 22726779729476308093, 15998009117983994065693, 13526765851190230940840809, 13528070218935445806530640649, 15795819619923464298050697616117, 21294937666865806704402646632389557, 32828500597549179599563478551377297121, 57385924456400269824204023290894357442401, 112904615348383588847189789579363784912180973

Offset: 0

Paul D. Hanna, Jan 16 2017

nonn

From Paul Curtz, Jan 20 2017: (Start)
a(n) mod 10 = periodic sequence of length 8: repeat [1, 1, 3, 3, 9, 9, 7, 7] = duplicated A001148(n).
a(n) mod 9 = 1, followed by period 3: repeat [1, 4, 7]. See A100402. See also A281280, A281182, A281183, A281184 (1, followed by 3's).
a(n+p) - a(n) is a multiple of 12. (End)

			E.g.f.: C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! + 39504564917358001*x^18/18! + 22726779729476308093*x^20/20! +...
such that
(1) C(x) = cosh( Integral C(x)^3 dx ),
(2) C(x)^2 - S(x)^2 = 1, and
(3) C(x) = 1 + Integral C(x)^3*S(x) dx,
where S(x) begins:
S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! + 1000749157519458304*x^19/19! + 635146072839001735168*x^21/21! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...
RELATED SERIES.
As power series with reduced fractional coefficients, S(x) and C(x) begin:
S(x) = x + 2/3*x^3 + 11/15*x^5 + 292/315*x^7 + 3548/2835*x^9 + 273766/155925*x^11 + 15360178/6081075*x^13 + 214706776/58046625*x^15 +...
C(x) = 1 + 1/2*x^2 + 13/24*x^4 + 493/720*x^6 + 37369/40320*x^8 + 4732249/3628800*x^10 + 901188997/479001600*x^12 + 240798388357/87178291200*x^14 +...
Related powers of series C(x) are given as follows.
C(x)^2 = 1 + 2*x^2/2! + 32*x^4/4! + 1376*x^6/6! + 114176*x^8/8! + 15519488*x^10/10! + 3132551168*x^12/12! + 879422726144*x^14/14! + 327670676455424*x^16/16! + 156439068819587072*x^18/18! +...+ A281183(n)*x^(2*n)/(2*n)! +...
where C(x)^2 = 1 + S(x)^2.
C(x)^3 = 1 + 3*x^2/2! + 57*x^4/4! + 2739*x^6/6! + 246801*x^8/8! + 35822307*x^10/10! + 7636142793*x^12/12! + 2246286827091*x^14/14! + 871869519033249*x^16/16! + 431649452286233283*x^18/18! +...+ A281184(n)*x^(2*n)/(2*n)! +...
where C(x)^3 = d/dx log( C(x) + S(x) ).
Also, C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
C(x)^4 = 1 + 4*x^2/2! + 88*x^4/4! + 4672*x^6/6! + 454144*x^8/8! + 70084096*x^10/10! + 15728822272*x^12/12! + 4836914249728*x^14/14! + 1952137912385536*x^16/16! + 1000749157519458304*x^18/18! +...+ A281180(n+1)*x^(2*n)/(2*n)! +...
where C(x)^4 = d/dx S(x).

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A281180 (S), A281182 (C+S), A281183 (C^2), A281184 (C^3), A001148, A100402, A122553.

nMax = 30; m = maxExponent = 2*nMax; a[n_] := Module[{S = x, C = 1}, For[i = 1, i <= n, i++, S = Integrate[C^4 + x*O[x]^m // Normal, x] + O[x]^m // Normal; C = 1 + Integrate[S*C^3 + O[x]^m // Normal, x]] + O[x]^m // Normal; (2*n)!*Coefficient[C, x, 2*n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, nMax}] (* Jean-François Alcover, Jan 20 2017, adapted from PARI *)
nmax = 20; Table[(CoefficientList[Sqrt[D[InverseSeries[Series[(2*x + Sin[2*x])/4, {x, 0, 2*nmax - 1}], x], x]], x] * Range[0, 2*nmax - 2]!)[[2*n - 1]], {n, 1, nmax}] (* Vaclav Kotesovec, Sep 02 2017 *)

{a(n) = my(S=x,C=1); for(i=0,n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n)!*polcoeff(C,2*n)}
for(n=0,30,print1(a(n),", "))

E.g.f. C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
E.g.f. C(x) = d/dx Series_Reversion( ( x*sqrt(1 - x^2) + asin(x) )/2 ).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral cos(x)^2 dx ) )^(1/2).
E.g.f. C(x) = ( d/dx Series_Reversion( (2*x + sin(2*x))/4 ) )^(1/2).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ) )^(1/3).
E.g.f. C(x) = ( d/dx Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ) )^(1/3).
E.g.f. C(x) and related series S(x) (e.g.f. of A281180) satisfy:
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
Integrals.
(2.a) S(x) = Integral C(x)^4 dx.
(2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
Exponential.
(3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
(3.b) C(x) = cosh( Integral C(x)^3 dx ).
(3.c) S(x) = sinh( Integral C(x)^3 dx ).
Derivatives.
(4.a) S'(x) = C(x)^4.
(4.b) C'(x) = C(x)^3*S(x).
(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
(4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
Explicit Solutions.
(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
(5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
(5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
(5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
(5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
(5.f) C(x)^4 = d/dx Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
(5.g) C(x)^5 = d/dx Series_Reversion( Integral C(i*x)^5 dx ).

Name simplified by Paul D. Hanna, Jan 22 2017

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

A001218 a(n) = 3^n mod 100.

Original entry on oeis.org

1, 3, 9, 27, 81, 43, 29, 87, 61, 83, 49, 47, 41, 23, 69, 7, 21, 63, 89, 67, 1, 3, 9, 27, 81, 43, 29, 87, 61, 83, 49, 47, 41, 23, 69, 7, 21, 63, 89, 67, 1, 3, 9, 27, 81, 43, 29, 87, 61, 83, 49, 47, 41, 23, 69, 7, 21, 63, 89, 67

Offset: 0

N. J. A. Sloane

Period is 20.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
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,-1,1).

Cf. A001148, A216096, A216097. - Zak Seidov, Jul 27 2014

[Modexp(3, n, 100): n in [0..100]]; // Vincenzo Librandi, Aug 15 2016

PowerMod[3, Range[0, 60], 100] (* Vincenzo Librandi, Aug 15 2016 *)

a(n)=lift(Mod(3,100)^n) \\ Charles R Greathouse IV, Dec 27 2012

Definition corrected by Zak Seidov, Jul 27 2014

A216096 a(n) = 3^n mod 1000.

Original entry on oeis.org

1, 3, 9, 27, 81, 243, 729, 187, 561, 683, 49, 147, 441, 323, 969, 907, 721, 163, 489, 467, 401, 203, 609, 827, 481, 443, 329, 987, 961, 883, 649, 947, 841, 523, 569, 707, 121, 363, 89, 267, 801, 403, 209, 627, 881, 643, 929, 787, 361, 83, 249, 747, 241, 723, 169, 507, 521, 563, 689

Offset: 0

V. Raman, Sep 01 2012

Period = 100.

V. Raman and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 100 terms from V. Raman)
Index entries for linear recurrences with constant coefficients, order 100.

Cf. A001148, A001218, A216097.

[Modexp(3, n, 1000): n in [0..110]]; // Vincenzo Librandi, Aug 16 2016

PowerMod[3, Range[0, 100], 1000] (* Vincenzo Librandi, Aug 16 2016 *)

a(n) = lift(Mod(3, 1000)^n); \\ Michel Marcus, Aug 16 2016

Definition corrected by Zak Seidov, Jul 27 2014

Offset changed and a(0) = 1 prepended by Vincenzo Librandi, Aug 16 2016

A216097 3^n mod 10000.

Original entry on oeis.org

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 9683, 9049, 7147, 1441, 4323, 2969, 8907, 6721, 163, 489, 1467, 4401, 3203, 9609, 8827, 6481, 9443, 8329, 4987, 4961, 4883, 4649, 3947, 1841, 5523, 6569, 9707, 9121, 7363, 2089, 6267, 8801, 6403, 9209, 7627, 2881, 8643, 5929, 7787, 3361, 83, 249, 747

Offset: 0

V. Raman, Sep 01 2012

Period = 500.

V. Raman and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 500 terms from V. Raman)

Cf. A001148, A001218, A216096.

[Modexp(3, n, 10000): n in [0..110]]; // Vincenzo Librandi, Aug 16 2016

PowerMod[3,Range[0,60],10000] (* Harvey P. Dale, Oct 18 2015 *)

PARI
```
for(i=0, 100, print(3^i%10000" "))
```

Definition corrected by Zak Seidov, Jul 27 2014

a(0) = 1, offset changed by Vincenzo Librandi, Aug 16 2016

A168427 a(n) = 3^n mod 30.

Original entry on oeis.org

1, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27, 21, 3, 9, 27

Offset: 0

Zerinvary Lajos, Nov 25 2009

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

Cf. A001148.

PowerMod[3,Range[0,90],30] (* Harvey P. Dale, Nov 04 2011 *)

a(n)=lift(Mod(3,30)^n) \\ Charles R Greathouse IV, Mar 22 2016

def A168427(n): return (21,3,9,27)[n&3] if n else 1 # Chai Wah Wu, Jan 22 2023

[power_mod(3,n,30) for n in range(0, 88)] #

From Chai Wah Wu, Jan 22 2023: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 3.
G.f.: (-20*x^3 - 7*x^2 - 2*x - 1)/((x - 1)*(x^2 + 1)). (End)

A361390 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 6, 7, 6, 5, 1, 0, 1, 2, 1, 4, 5, 6, 1, 0, 1, 4, 3, 6, 5, 6, 7, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 0, 1, 6, 7, 6, 5, 6, 3, 4, 9, 1, 0, 1, 2, 1, 4, 5, 6, 1, 2, 1, 10, 1, 0, 1, 4, 3, 6, 5, 6, 7, 6, 9, 100, 11, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 1000, 121, 12, 1

Offset: 0

Seiichi Manyama, Mar 10 2023

			4 * 4 = 16, so T(4,2) = 6. 6 * 4 = 24, so T(4,3) = 4.
Square array begins:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 4, 8, 6, 2, 4, 8, ...
  1, 3, 9, 7, 1, 3, 9, 7, ...
  1, 4, 6, 4, 6, 4, 6, 4, ...
  1, 5, 5, 5, 5, 5, 5, 5, ...
  1, 6, 6, 6, 6, 6, 6, 6, ...
  1, 7, 9, 3, 1, 7, 9, 3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Columns k=0..4 give A000012, A001477, A059729, A169885, A169886.
Rows n=0..4 give A000007, A000012, A000689, A001148, A168428.
T(11,k) gives A059734.
Main diagonal gives A361351.
Cf. A004248, A359697.

T(n, k) = fromdigits(Vec(Pol(digits(n))^k)%10);

cp's OEIS Frontend

A000689 Final decimal digit of 2^n.

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A216099 Period of powers of 3 mod 10^n.

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A281181 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^3 dx ).

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A001903 Final digit of 7^n.

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A001218 a(n) = 3^n mod 100.

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A216096 a(n) = 3^n mod 1000.

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A216097 3^n mod 10000.

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