A000689 -id:A000689 - cp's OEIS frontend

A005054 a(0) = 1; a(n) = 4*5^(n-1) for n >= 1.

1, 4, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500, 1907348632812500, 9536743164062500

Offset: 0

N. J. A. Sloane

Consider the sequence formed by the final n decimal digits of {2^k: k >= 0}. For n=1 this is 1, 2, 4, 8, 6, 2, 4, ... (A000689) with period 4. For any n this is periodic with period a(n). Cf. A000855 (n=2), A126605 (n=3, also n=4). - N. J. A. Sloane, Jul 08 2022
First differences of A000351.
Length of repeating cycle of the final n+1 digits in Fermat numbers. - Lekraj Beedassy, Robert G. Wilson v and Eric W. Weisstein, Jul 05 2004
Number of n-digit endings for a power of 2 whose exponent is greater than or equal to n. - J. Lowell
For n>=1, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
Equals INVERT transform of A033887: (1, 3, 13, 55, 233, ...) and INVERTi transform of A001653: (1, 5, 29, 169, 985, 5741, ...). - Gary W. Adamson, Jul 22 2010
a(n) = (n+1) terms in the sequence (1, 3, 4, 4, 4, ...) dot (n+1) terms in the sequence (1, 1, 4, 20, 100, ...). Example: a(4) = 500 = (1, 3, 4, 4, 4) dot (1, 1, 4, 20, 100) = (1 + 3 + 16, + 80 + 400), where (1, 3, 16, 80, 400, ...) = A055842, finite differences of A005054 terms. - Gary W. Adamson, Aug 03 2010
a(n) is the number of compositions of n when there are 4 types of each natural number. - Milan Janjic, Aug 13 2010
Apart from the first term, number of monic squarefree polynomials over F_5 of degree n. - Charles R Greathouse IV, Feb 07 2012
For positive integers that can be either of two colors (designated by ' or ''), a(n) is the number of compositions of 2n that are cardinal palindromes; that is, palindromes that only take into account the cardinality of the numbers and not their colors. Example: 3', 2'', 1', 1, 2', 3'' would count as a cardinal palindrome. - Gregory L. Simay, Mar 01 2020
a(n) is the length of the period of the sequence Fibonacci(k) (mod 5^(n-1)) (for n>1) and the length of the period of the sequence Lucas(k) (mod 5^n) (Kramer and Hoggatt, 1972). - Amiram Eldar, Feb 02 2022

T. Koshy, "The Ends Of A Fermat Number", pp. 183-4 Journal Recreational Mathematics, vol. 31(3) 2002-3 Baywood NY.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 458.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Judy Kramer and V. E. Hoggatt, Jr., Special Cases of Fibonacci Periodicity (Part 1, Part 2), The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 519-522, 530.
Eric Weisstein's World of Mathematics, Fermat Number.
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000010, A000032, A000045, A000351, A000215, A055842.
Cf. A001653, A033887. - Gary W. Adamson, Jul 22 2010
See also A000689, A000855, A126605, A216099, A216164. - N. J. A. Sloane, Jul 08 2022

[(4*5^n+0^n)/5: n in [0..30]]; // Vincenzo Librandi, Jun 08 2013

a:= n-> ceil(4*5^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 08 2022

CoefficientList[Series[(1 - x) / (1 - 5 x), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

Vec((1-x)/(1-5*x) + O(x^100)) \\ Altug Alkan, Dec 07 2015

a(n) = (4*5^n + 0^n) / 5. - R. J. Mathar, May 13 2008
G.f.: (1-x)/(1-5*x). - Philippe Deléham, Nov 02 2009
G.f.: 1/(1 - 4*Sum_{k>=1} x^k).
a(n) = 5*a(n-1) for n>=2. - Vincenzo Librandi, Dec 31 2010
a(n) = phi(5^n) = A000010(A000351(n)).
E.g.f.: (4*exp(5*x)+1)/5. - Paul Barry, Apr 20 2003
a(n + 1) = (((1 + sqrt(-19))/2)^n + ((1 - sqrt(-19))/2)^n)^2 - (((1 + sqrt(-19))/2)^n - ((1 - sqrt(-19))/2)^n)^2. - Raphie Frank, Dec 07 2015

Better definition from R. J. Mathar, May 13 2008

Edited by N. J. A. Sloane, Jul 08 2022

A000855 Final two digits of 2^n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4, 8, 16, 32, 64, 28

Offset: 0

N. J. A. Sloane

Has period 20 (starting with a(2)=4).

T. D. Noe, 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 (1,0,0,0,0,0,0,0,0,-1,1).

Cf. A000689, A126605.

PowerMod[2,Range[0,60],100] (* Harvey P. Dale, Jan 05 2012 *)

a(n)=lift(Mod(2,100)^n) \\ Charles R Greathouse IV, Dec 28 2012

G.f.: -(50*x^12 +25*x^11 +13*x^10 -44*x^9 +28*x^8 -36*x^7 +32*x^6 +16*x^5 +8*x^4 +4*x^3 +2*x^2 +x +1) / ((x -1)*(x^2 +1)*(x^8 -x^6 +x^4 -x^2 +1)). - Colin Barker, Dec 01 2014

For n > 13: a(n) = a(n-1) - a(n-10) + a(n-11). - Ray Chandler, Aug 09 2025

A070402 a(n) = 2^n mod 5.

Original entry on oeis.org

1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1

Offset: 0

N. J. A. Sloane, May 12 2002

Periodic with period 4: [1, 2, 4, 3]. - Washington Bomfim, Nov 23 2010

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

Cf. A173635.

List([0..83],n->PowerMod(2,n,5)); # Muniru A Asiru, Feb 01 2019

[Modexp(2, n, 5): n in [0..100]]; // Bruno Berselli, Mar 23 2016

A070402 := proc(n) op((n mod 4)+1,[1,2,4,3]) ; end proc: # R. J. Mathar, Feb 05 2011

PadRight[{}, 100, {1, 2, 4, 3}] (* or *) CoefficientList[Series[(1 + 2 x + 4 x^2 + 3 x^3) / (1 - x^4), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 25 2016 *)
PowerMod[2,Range[0,120],5] (* Harvey P. Dale, Sep 16 2020 *)

for(n=0, 80, x=n%4; print1(1 + (15*x^2 -5*x -4*x^3)/6, ", ")) \\ Washington Bomfim, Nov 23 2010

[power_mod(2,n,5) for n in range(0, 105)] # Zerinvary Lajos, Jun 08 2009

From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3).
G.f.: (1 + x + 3*x^2) / ((1-x)*(1+x^2)). (End)
From Washington Bomfim, Nov 23 2010: (Start)
a(n) = 1 + (15*r^2 - 5*r - 4*r^3)/6, where r = n mod 4.
a(n) = A000689(n) - 5*floor(((n-1) mod 4)/2) for n>0. (End)
E.g.f.: (1/2)*(5*exp(x) - 3*cos(x) - sin(x)). - G. C. Greubel, Mar 19 2016

A126605 Final three digits of 2^n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 24, 48, 96, 192, 384, 768, 536, 72, 144, 288, 576, 152, 304, 608, 216, 432, 864, 728, 456, 912, 824, 648, 296, 592, 184, 368, 736, 472, 944, 888, 776, 552, 104, 208, 416, 832, 664, 328, 656, 312, 624, 248, 496, 992, 984, 968

Offset: 0

Zak Seidov, Mar 13 2007

Has period 100. Sequence of last four digits has period 500. Cf. A000855 Final two digits of 2^n, has period 20.

V. Raman and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 103 terms from V. Raman)
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Cf. A000689, A000855.

[Modexp(2, n, 1000): n in [0..110]]; // Vincenzo Librandi, Aug 16 2016

Mathematica
```
Table[PowerMod[2,n,1000],{n,0,1000}]
```

for(i=0,103,print(i" "(2^i)%1000)) \\ V. Raman, Sep 01 2012

For n > 54: a(n) = a(n-1) - a(n-50) + a(n-51). - Ray Chandler, Aug 09 2025

A002081 Numbers congruent to {2, 4, 8, 16} (mod 20).

Original entry on oeis.org

2, 4, 8, 16, 22, 24, 28, 36, 42, 44, 48, 56, 62, 64, 68, 76, 82, 84, 88, 96, 102, 104, 108, 116, 122, 124, 128, 136, 142, 144, 148, 156, 162, 164, 168, 176, 182, 184, 188, 196, 202, 204, 208, 216, 222, 224, 228, 236, 242, 244, 248, 256, 262, 264, 268, 276, 282

Offset: 0

N. J. A. Sloane

First differences are periodic, cf. A000689.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
C. Babbage, On the Determination of the General Term of a New Class of Infinite Series, Trans. Camb. Phil. Soc., 2 (1827), 217-225 (see p. 220).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A002082, A008587.

a002081 n = a002081_list
a002081_list = filter ((`elem` [2,4,8,16]) . (`mod` 20)) [1..]
-- Reinhard Zumkeller, Sep 15 2011

A002081:=2*(1+2*z**2+2*z**3)/(z**2+1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Flatten[Table[20n + {2, 4, 8, 16}, {n, 0, 14}]] (* Alonso del Arte, Nov 30 2011 *)
LinearRecurrence[{2, -2, 2, -1},{2, 4, 8, 16},57] (* Ray Chandler, Aug 25 2015 *)
Select[Range[300],MemberQ[{2,4,8,16},Mod[#,20]]&] (* Harvey P. Dale, Jul 20 2021 *)

a(n) = 5*n + [2,-1,-2,1][(n%4)+1] \\ Ralf Stephan, Jun 08 2005

is(n) = n > 0 && setsearch([2,4,8,16], n%20) > 0 \\ Rick L. Shepherd, Aug 17 2016

G.f.: 2*(1+2*x^2+2*x^3)/((1-x)^2*(1+x^2)). - Simon Plouffe
a(n+4) = a(n) + 20 for n > 3. - Reinhard Zumkeller, Sep 15 2011
a(n) = 5*n + (1/2)*(3 + (-1)^n)*(-1)^(n(n+1)/2). - Bruno Berselli, Sep 15 2011
E.g.f.: 2*cos(x) - sin(x) + 5*x*exp(x). - Ilya Gutkovskiy, Aug 17 2016

More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000

A160590 Next-to-least significant digit of 2^n.

Original entry on oeis.org

1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, 6, 3, 7, 4, 8, 7, 5, 0, 0, 1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, 6, 3, 7, 4, 8, 7, 5, 0, 0, 1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, 6, 3, 7, 4, 8, 7, 5, 0, 0, 1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, 6, 3, 7, 4, 8, 7, 5, 0, 0, 1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, 6, 3, 7, 4, 8, 7, 5, 0, 0, 1, 3, 6, 2, 5

Offset: 4

Brian Tristam Williams (briantw(AT)briantw.com), May 20 2009

In the sequence [1, 2, 4, 8,] 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, (A000079) the least-significant digit recurs as [..], 6,2,4,8.. (A000689).

The second-least-significant digit repeats in the sequence [..],1,3,6,2,5,1,2,4,9,9,8,6,3,7,4,8,7,5,0,0...

Paolo Xausa, Table of n, a(n) for n = 4..10000
M. Haapanen, What is this and how to prove?, newgsroups sci.math, Mar 28 1996
K. Makella, Mika luku tulee seuraavana?, sfnet.tiede.matematiikka, Jan 2008
Lars Radvall, Multiplar (PDF).
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,-1,1).

Cf. A000079, A000689.

for n from 4 to 130 do 2 &^ n mod 100 ; printf("%d,", floor(%/10)) ; od:

PadRight[{},100,{1,3,6,2,5,1,2,4,9,9,8,6,3,7,4,8,7,5,0,0}] (* Paolo Xausa, Oct 22 2023 *)

Periodic: a(n+20) = a(n).

Edited by R. J. Mathar, May 22 2009

A367361 Comma transform of powers of 2.

Original entry on oeis.org

12, 24, 48, 81, 63, 26, 41, 82, 65, 21, 42, 84, 68, 21, 43, 86, 61, 22, 45, 81, 62, 24, 48, 81, 63, 26, 41, 82, 65, 21, 42, 84, 68, 21, 43, 86, 61, 22, 45, 81, 62, 24, 48, 81, 63, 27, 41, 82, 65, 21, 42, 84, 69, 21, 43, 87, 61, 22, 45, 81, 62, 24, 49, 81, 63, 27, 41, 82, 65, 21, 42, 84, 69, 21, 43, 87, 61, 23, 46, 81

Offset: 0

N. J. A. Sloane, Nov 22 2023

See A367360 for further information.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000079, A000689, A008952, A166499, A367360.

FromDigits /@ Partition[Rest@ Flatten[{First[#], Last[#]} & /@ IntegerDigits[2^Range[0, 120]]], 2, 2] (* Michael De Vlieger, Nov 22 2023 *)

from itertools import count, islice, pairwise
def S(): yield from (str(2**i) for i in count(0))
def agen(): yield from (int(t[-1]+u[0]) for t, u in pairwise(S()))
print(list(islice(agen(), 80))) # Michael S. Branicky, Nov 22 2023

def A367361(n): return (60,20,40,80)[n&3]+int(str(1<Chai Wah Wu, Dec 22 2023

a(n) = 10 * A000689(n) + A008952(n+1). - Alois P. Heinz, Nov 22 2023

A220755 Numbers n such that n^2 + n(n+1)/2 is an oblong number (A002378).

Original entry on oeis.org

0, 1, 28, 117, 2760, 11481, 270468, 1125037, 26503120, 110242161, 2597035308, 10802606757, 254482957080, 1058545220041, 24936732758548, 103726628957277, 2443545327380640, 10164151092593121, 239442505350544188, 995983080445168597, 23462921979025949800

Offset: 1

Alex Ratushnyak, Apr 13 2013

Numbers n such that 6*n^2 + 2*n + 1 is a square. - Joerg Arndt, Apr 14 2013
a(n+4) - a(n) is divisible by 40. (a(n+2) - a(n)) mod 10 = period 4: repeat 8, 6, 2, 4. See A000689. - Paul Curtz, Apr 15 2013
For this 5 consecutive terms recurrence,the main (or principal) sequence is: CRR(n)= 0, 0, 0, 0, 1, 1, 99, 99, 9702, 9702,... . - Paul Curtz, Apr 16 2013
Also numbers n such that the sum of the octagonal numbers N(n) and N(n+1) is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).

Cf. A000217, A005449 (n^2 + n(n+1)/2).
Cf. A011916 (numbers n>=0 such that n^2 + n(n+1)/2 is a triangular number).
Cf. A220186 (numbers n>=0 such that n^2 + n(n+1)/2 is a square).
Cf. A220185 (numbers n>=0 such that n^2 + n(n+1) is an oblong number).
(Example of a family of main sequences: A131577, A024495, A000749, A139761. )
Cf. A251793.

#include 
typedef unsigned long long U64;
U64 rootPronic(U64 a) {
    U64 sr = 1L<<31, s, b;
    while (a < sr*(sr+1))  sr>>=1;
    for (b = sr>>1; b; b>>=1) {
            s = sr+b;
            if (a >= s*(s+1))  sr = s;
    }
    return sr;
}
int main() {
  U64 a, n, r, t;
  for (n=0; n < 3L<<30; n++) {
    a = n*(n+1)/2 + n*n;
    t = rootPronic(a);
    if (a == t*(t+1)) {
        printf("%llu\n", n);
    }
  }
}

LinearRecurrence[{1, 98, -98, -1, 1}, {0, 1, 28, 117, 2760}, 30] (* Giovanni Resta, Apr 14 2013 *)
CoefficientList[Series[x (1 + 27 x - 9 x^2 - 3 x^3)/((1 - x) (1 - 10 x + x^2) (1 + 10 x + x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 13 2014 *)

makelist(expand(((-(-1)^n+sqrt(6))*(5+2*sqrt(6))^(n-1)-((-1)^n+sqrt(6))*(5-2*sqrt(6))^(n-1)-2)/12), n, 1, 25); /* Bruno Berselli, Apr 14 2013 */

concat([0], Vec( x * (1+27*x-9*x^2-3*x^3) / ( (1-x)*(1-10*x+x^2)*(1+10*x+x^2) ) + O(x^66) ) )  /* Joerg Arndt, Apr 14 2013 */

G.f.: x^2 * (1+27*x-9*x^2-3*x^3) / ( (1-x)*(1-10*x+x^2)*(1+10*x+x^2) ). - Giovanni Resta, Apr 14 2013, adapted by Vincenzo Librandi Aug 13 2014
a(n) = ((-(-1)^n+sqrt(6))*(5+2*sqrt(6))^(n-1)-((-1)^n+sqrt(6))*(5-2*sqrt(6))^(n-1)-2)/12. - Bruno Berselli, Apr 14 2013
a(n) = a(n-1) + 98*a(n-2) - 98*a(n-3) - a(n-4) + a(n-5).

a(11)-a(21) from Giovanni Resta, Apr 14 2013

A216095 a(n) = 2^n mod 10000.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 6384, 2768, 5536, 1072, 2144, 4288, 8576, 7152, 4304, 8608, 7216, 4432, 8864, 7728, 5456, 912, 1824, 3648, 7296, 4592, 9184, 8368, 6736, 3472, 6944, 3888, 7776, 5552, 1104, 2208, 4416, 8832, 7664, 5328, 656, 1312, 2624, 5248, 496, 992, 1984, 3968, 7936, 5872, 1744, 3488, 6976, 3952, 7904, 5808

Offset: 0

V. Raman, Sep 01 2012

Period = 500.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 504 terms from V. Raman)
Index entries for linear recurrences with constant coefficients, order 251.

Cf. A000855, A126605, A000689.

[Modexp(2, n, 10000): n in [0..110]]; // Vincenzo Librandi, Aug 16 2016

PowerMod[2, Range[0, 100], 10000] (* Vincenzo Librandi, Aug 16 2016 *)

PARI
```
for(i=0,500,print(2^i%10000" "))
```

For n > 255: a(n) = a(n-1) - a(n-250) + a(n-251). - Ray Chandler, Aug 09 2025

A267317 a(n) = final digit of 2^n-1.

Original entry on oeis.org

0, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

Decimal expansion of 25/1818.

Period 4: repeat [1, 3, 7, 5] for n > 0.

Eric Weisstein's World of Mathematics, Mersenne Number
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000225, A000689, A010688, A010703, A010879, A080172.

[0] cat &cat[[1, 3, 7, 5]^^25]; // Bruno Berselli, Jan 13 2016

A267317:=n->(2^n-1) mod 10: seq(A267317(n), n=0..150); # Wesley Ivan Hurt, Jun 15 2016

Mathematica
```
Table[Mod[2^n - 1, 10], {n, 0, 120}]
```

a(n) = if(n==0, 0, if(n%4==0, 5, if(n%4==1, 1, if(n%4==2, 3, if(n%4==3, 7))))) \\ Felix Fröhlich, Jan 19 2016

a(n) = lift(Mod(2^n-1, 10)) \\ Felix Fröhlich, Jan 19 2016

G.f.: x*(1 + 2*x + 5*x^2)/(1 - x + x^2 - x^3).
a(n) = A010879(A000225(n)).
a(n) = A000689(n) - 1.
a(n) = (1+(-1)^n)*(-1)^(n*(n-1)/2)/2 + 3*(1-(-1)^n)*(-1)^(n*(n+1)/2)/2 + 4 for n > 0, a(0) = 0. [Bruno Berselli, Jan 13 2016]
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = a(n-4) for n>4.
a(2k+2) = A010703(k), a(2k+1) = A010688(k). (End)
From Wesley Ivan Hurt, Jul 06 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 3.
a(n) = 4 + cos(n*Pi/2) - 3*sin(n*Pi/2) for n > 0. (End)
E.g.f.: -5 + cos(x) - 3*sin(x) + 4*exp(x). - Ilya Gutkovskiy, Jul 06 2016

cp's OEIS Frontend

A005054 a(0) = 1; a(n) = 4*5^(n-1) for n >= 1.

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A000855 Final two digits of 2^n.

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A070402 a(n) = 2^n mod 5.

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A126605 Final three digits of 2^n.

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A002081 Numbers congruent to {2, 4, 8, 16} (mod 20).

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Haskell

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PARI

PARI

Formula

Extensions

A160590 Next-to-least significant digit of 2^n.

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Maple

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