A160700 -id:A160700 - cp's OEIS frontend

A161450 Numbers n such that A160700(n) = 10.

10, 27, 40, 57, 78, 95, 108, 125, 130, 147, 160, 177, 198, 215, 228, 245, 267, 282, 297, 312, 335, 350, 365, 380, 387, 402, 417, 432, 455, 470, 485, 500, 520, 537, 554, 571, 588, 605, 622, 639, 640, 657, 674, 691, 708, 725, 742, 759, 777, 792, 811, 826, 845

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161451, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==10, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161451 Numbers n such that A160700(n) = 11.

Original entry on oeis.org

11, 26, 41, 56, 79, 94, 109, 124, 131, 146, 161, 176, 199, 214, 229, 244, 266, 283, 296, 313, 334, 351, 364, 381, 386, 403, 416, 433, 454, 471, 484, 501, 521, 536, 555, 570, 589, 604, 623, 638, 641, 656, 675, 690, 709, 724, 743, 758, 776, 793, 810, 827, 844

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161452, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==11, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161452 Numbers m such that A160700(m) = 12.

Original entry on oeis.org

12, 29, 46, 63, 72, 89, 106, 123, 132, 149, 166, 183, 192, 209, 226, 243, 269, 284, 303, 318, 329, 344, 363, 378, 389, 404, 423, 438, 449, 464, 483, 498, 526, 543, 556, 573, 586, 603, 616, 633, 646, 663, 676, 693, 706, 723, 736, 753, 783, 798, 813, 828, 843

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161453, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==12, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161453 Numbers n such that A160700(n) = 13.

Original entry on oeis.org

13, 28, 47, 62, 73, 88, 107, 122, 133, 148, 167, 182, 193, 208, 227, 242, 268, 285, 302, 319, 328, 345, 362, 379, 388, 405, 422, 439, 448, 465, 482, 499, 527, 542, 557, 572, 587, 602, 617, 632, 647, 662, 677, 692, 707, 722, 737, 752, 782, 799, 812, 829, 842

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161454, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==13, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161454 Numbers n such that A160700(n) = 14.

Original entry on oeis.org

14, 31, 44, 61, 74, 91, 104, 121, 134, 151, 164, 181, 194, 211, 224, 241, 271, 286, 301, 316, 331, 346, 361, 376, 391, 406, 421, 436, 451, 466, 481, 496, 524, 541, 558, 575, 584, 601, 618, 635, 644, 661, 678, 695, 704, 721, 738, 755, 781, 796, 815, 830, 841

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161455.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==14, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A161455 Numbers n such that A160700(n) = 15.

Original entry on oeis.org

15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 270, 287, 300, 317, 330, 347, 360, 377, 390, 407, 420, 437, 450, 467, 480, 497, 525, 540, 559, 574, 585, 600, 619, 634, 645, 660, 679, 694, 705, 720, 739, 754, 780, 797, 814, 831, 840

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A161440, A161441, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454.

A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==15, return(k))) \\ Charles R Greathouse IV, Jan 25 2018

16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018

A000302 Powers of 4: a(n) = 4^n.

Original entry on oeis.org

1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664, 281474976710656

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 4), L(1, 4), P(1, 4), T(1, 4). Essentially same as Pisot sequences E(4, 16), L(4, 16), P(4, 16), T(4, 16). See A008776 for definitions of Pisot sequences.
The convolution square root of this sequence is A000984, the central binomial coefficients: C(2n,n). - T. D. Noe, Jun 11 2002
With P(n) being the number of integer partitions of n, p(i) as the number of parts of the i-th partition of n, d(i) as the number of different parts of the i-th partition of n, m(i, j) the multiplicity of the j-th part of the i-th partition of n, one has a(n) = Sum_{i = 1..P(n)} p(i)!/(Product_{j = 1..d(i)} m(i, j)!) * 2^(n-1). - Thomas Wieder, May 18 2005
Sums of rows of the triangle in A122366. - Reinhard Zumkeller, Aug 30 2006
Hankel transform of A076035. - Philippe Deléham, Feb 28 2009
Equals the Catalan sequence: (1, 1, 2, 5, 14, ...), convolved with A032443: (1, 3, 11, 42, ...). - Gary W. Adamson, May 15 2009
Sum of coefficients of expansion of (1 + x + x^2 + x^3)^n.
a(n) is number of compositions of natural numbers into n parts less than 4. For example, a(2) = 16 since there are 16 compositions of natural numbers into 2 parts less than 4.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 4-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Squares in A002984. - Reinhard Zumkeller, Dec 28 2011
Row sums of Pascal's triangle using the rule that going left increases the value by a factor of k = 3. For example, the first three rows are {1}, {3, 1}, and {9, 6, 1}. Using this rule gives row sums as (k+1)^n. - Jon Perry, Oct 11 2012
First differences of A002450. - Omar E. Pol, Feb 20 2013
Sum of all peak heights in Dyck paths of semilength n+1. - David Scambler, Apr 22 2013
Powers of 4 exceed powers of 2 by A020522 which is the m-th oblong number A002378(m), m being the n-th Mersenne number A000225(n); hence, we may write, a(n) = A000079(n) + A002378(A000225(n)). - Lekraj Beedassy, Jan 17 2014
a(n) is equal to 1 plus the sum for 0 < k < 2^n of the numerators and denominators of the reduced fractions k/2^n. - J. M. Bergot, Jul 13 2015
Binomial transform of A000244. - Tony Foster III, Oct 01 2016
From Ilya Gutkovskiy, Oct 01 2016: (Start)
Number of nodes at level n regular 4-ary tree.
Partial sums of A002001. (End)
Satisfies Benford's law [Berger-Hill, 2011]. - N. J. A. Sloane, Feb 08 2017
Also the number of connected dominating sets in the (n+1)-barbell graph. - Eric W. Weisstein, Jun 29 2017
Side length of the cells at level n in a pyramid scheme where a square grid is decomposed into overlapping 2 X 2 blocks (cf. Kropatsch, 1985). - Felix Fröhlich, Jul 04 2019
a(n-1) is the number of 3-compositions of n; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020

H. W. Gould, Combinatorial Identities, 1972, eq. (1.93), p. 12.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, eq. (5.39), p. 187.
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

T. D. Noe, Table of n, a(n) for n = 0..100
Arno Berger and Theodore P. Hill, Benford's law strikes back: no simple explanation in sight for mathematical gem, The Mathematical Intelligencer 33.1 (2011): 85-91.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
R. Duarte and A. G. de Oliveira, Short note on the convolution of binomial coefficients, arXiv preprint arXiv:1302.2100 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.7.6.
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See p. 5.
R. K. Guy, Letter to N. J. A. Sloane
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 8
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 269
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Tanya Khovanova, Recursive Sequences
Craig Knecht, Number of tilings for a 6 sphinx tile repetitive unit.
Walter G. Kropatsch, A pyramid that grows by powers of 2, Pattern Recognition Letters, Vol. 3, No. 5 (1985), 315-322 [Subscription required].
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
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
Robert Price, Comments on A000302 concerning Elementary Cellular Automata, Feb 26 2016.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9., 2016.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv preprint arXiv:1504.04404 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Barbell Graph
Eric Weisstein's World of Mathematics, Cantor Dust
Eric Weisstein's World of Mathematics, Connected Dominating Set
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for "core" sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4).
Index entries for sequences related to Benford's law

Cf. A024036, A052539, A032443, A000351 (Binomial transform).
Cf. A249307.
Cf. A083420.

a000302 = (4 ^)
a000302_list = iterate (* 4) 1  -- Reinhard Zumkeller, Apr 04 2012

A000302 := n->4^n;
for n from 0 to 10 do sum(2^(n-j)*binomial(n+j,j),j=0..n); od; # Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
A000302:=-1/(-1+4*z); # Simon Plouffe in his 1992 dissertation.

Table[4^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
CoefficientList[Series[1/(1 - 4 x), {x, 0, 50}], x]  (* Vincenzo Librandi, May 29 2014 *)
NestList[4 # &, 1, 30] (* Harvey P. Dale, Mar 26 2015 *)
4^Range[0, 30] (* Eric W. Weisstein, Jun 29 2017 *)
LinearRecurrence[{4}, {1}, 31] (* Robert A. Russell, Nov 08 2018 *)

A000302(n):=4^n$ makelist(A000302(n),n,0,30); /* Martin Ettl, Oct 24 2012 */

A000302(n)=4^n \\ Michael B. Porter, Nov 06 2009

print([4**n for n in range(25)]) # Michael S. Branicky, Jan 04 2021

is_A000302 = lambda n: n.bit_count()==1 and n.bit_length()&1 # M. F. Hasler, Nov 25 2024

[4**n for n in range(0,25)] # Stefano Spezia, Jul 23 2025

(List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, Jun 22 2019

a(n) = 4^n.
a(0) = 1; a(n) = 4*a(n-1).
G.f.: 1/(1-4*x).
E.g.f.: exp(4*x).
a(n) = Sum_{k = 0..n} binomial(2k, k) * binomial(2(n - k), n - k). - Benoit Cloitre, Jan 26 2003 [See Graham et al., eq. (5.39), p. 187. - Wolfdieter Lang, Aug 16 2019]
1 = Sum_{n >= 1} 3/a(n) = 3/4 + 3/16 + 3/64 + 3/256 + 3/1024, ...; with partial sums: 3/4, 15/16, 63/64, 255/256, 1023/1024, ... - Gary W. Adamson, Jun 16 2003
a(n) = A001045(2*n) + A001045(2*n+1). - Paul Barry, Apr 27 2004
A000005(a(n)) = A005408(n+1). - Reinhard Zumkeller, Mar 04 2007
a(n) = Sum_{j = 0..n} 2^(n - j)*binomial(n + j, j). - Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
Hankel transform of A115967. - Philippe Deléham, Jun 22 2007
a(n) = 6*Stirling2(n+1, 4) + 6*Stirling2(n+1, 3) + 3*Stirling2(n+1, 2) + 1 = 2*Stirling2(2^n, 2^n - 1) + Stirling2(n+1, 2) + 1. - Ross La Haye, Jun 26 2008
a(n) = A159991(n)/A001024(n) = A047653(n) + A181765(n). A160700(a(n)) = A010685(n). - Reinhard Zumkeller, May 02 2009
a(n) = A188915(A006127(n)). - Reinhard Zumkeller, Apr 14 2011
a(n) = Sum_{k = 0..n} binomial(2*n+1, k). - Mircea Merca, Jun 25 2011
Sum_{n >= 1} Mobius(n)/a(n) = 0.1710822479183... - R. J. Mathar, Aug 12 2012
a(n) = Sum_{k = 0..n} binomial(2*k + x, k)*binomial(2*(n - k) - x, n - k) for every real number x. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
a(n) = 5*a(n - 1) - 4*a(n - 2). - Jean-Bernard François, Sep 12 2013
a(n) = (2*n+1) * binomial(2*n,n) * Sum_{j=0..n} (-1)^j/(2*j+1)*binomial(n,j). - Vaclav Kotesovec, Sep 15 2013
a(n) = A000217(2^n - 1) + A000217(2^n). - J. M. Bergot, Dec 28 2014
a(n) = (2^n)^2 = A000079(n)^2. - Doug Bell, Jun 23 2015
a(n) = A002063(n)/3 - A004171(n). - Zhandos Mambetaliyev, Nov 19 2016
a(n) = (1/2) * Product_{k = 0..n} (1 + (2*n + 1)/(2*k + 1)). - Peter Bala, Mar 06 2018
a(n) = A001045(n+1)*A001045(n+2) + A001045(n)^2. - Ezhilarasu Velayutham, Aug 30 2019
a(n) = denominator(zeta_star({2}(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - _Roudy El Haddad, Feb 22 2022
a(n) = 1 + 3*Sum_{k=0..n} binomial(2*n, n+k)*(k|9), where (k|9) is the Jacobi symbol. - Greg Dresden, Oct 11 2022
a(n) = Sum_{k = 0..n} binomial(2*n+1, 2*k) = Sum_{k = 0..n} binomial(2*n+1, 2*k+1). - Sela Fried, Mar 23 2023

Partially edited by Joerg Arndt, Mar 11 2010

A001025 Powers of 16: a(n) = 16^n.

Original entry on oeis.org

1, 16, 256, 4096, 65536, 1048576, 16777216, 268435456, 4294967296, 68719476736, 1099511627776, 17592186044416, 281474976710656, 4503599627370496, 72057594037927936, 1152921504606846976, 18446744073709551616, 295147905179352825856, 4722366482869645213696, 75557863725914323419136, 1208925819614629174706176

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 16), L(1, 16), P(1, 16), T(1, 16). Essentially same as Pisot sequences E(16, 256), L(16, 256), P(16, 256), T(16, 256). See A008776 for definitions of Pisot sequences.
Convolution-square (auto-convolution) of A098430. - R. J. Mathar, May 22 2009
Subsequence of A161441: A160700(a(n)) = 1. - Reinhard Zumkeller, Jun 10 2009
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 16-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

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).

Muniru A Asiru, Table of n, a(n) for n = 0..820 (terms n = 0..100 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 280
Tanya Khovanova, Recursive Sequences
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
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (16).

Partial sums give A131865.

Cf. A000079, A000302, A001018, A005843, A008586.

List([0..20],n->16^n); # Muniru A Asiru, Nov 07 2018

a001025 = (16 ^)
a001025_list = iterate (* 16) 1  -- Reinhard Zumkeller, Nov 07 2012

A001025:=-1/(-1+16*z); # Simon Plouffe in his 1992 dissertation

Table[4^(2*n), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2009 *)

A001025(n):=16^n$
makelist(A001025(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=1<<(4*n) \\ Charles R Greathouse IV, Feb 01 2012

print([16**n for n in range(20)]) # Stefano Spezia, Nov 10 2018

[lucas_number1(n,16,0) for n in range(1, 18)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-16*x).
E.g.f.: exp(16*x).
From Muniru A Asiru, Nov 07 2018: (Start)
a(n) = 16^n.
a(0) = 1, a(n) = 16*a(n-1). (End)
a(n) = 4^A005843(n) = 2^A008586(n) = A000302(n)^2 = A000079(n)*A001018(n). - Muniru A Asiru, Nov 10 2018
a(n) = ( Sum_{k = 0..n} (2*k + 1)*binomial(2*n + 1, n - k) ) * ( Sum_{k = 0..n} (-1)^k/(2*k + 1)*binomial(2*n + 1, n - k) ). - Peter Bala, Feb 12 2019
a(n) = Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k). - Peter Bala, Aug 23 2025

A010685 Period 2: repeat (1,4).

Original entry on oeis.org

1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1

Offset: 0

N. J. A. Sloane

Continued fraction of (1 + sqrt(2))/2. - R. J. Mathar, Nov 21 2011

This sequence can be generated by an infinite number of formulas all having the form a^(b*n) mod c subject to the following conditions. The number a can be congruent to either 2,3, or 4 mod 5 (A047202). If a is congruent to 2 or 3 mod 5, then b can be any number of the form 4k+2 and c = 5 or 15. If a is congruent to 4 mod 5, then b can be any number of the form 2k+1 and c = 5. For example: a(n) = 29^(13*n) mod 5, a(n) = 24^(11*n) mod 5, and a(n) = 22^(10*n) mod 15. - Gary Detlefs, May 19 2014

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

Cf. sequences listed in Comments section of A283393.

Cf. A047202.

[Modexp(4,n,5): n in [0..100]]; // G. C. Greubel, Nov 22 2021

A010685 := proc(n)
    if type(n,'even') then
        1 ;
    else
        4;
    end if;
end proc: # R. J. Mathar, Aug 03 2015

Table[(5-3(-1)^n)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 26 2014 *)
PadRight[{},120,{1,4}] (* Harvey P. Dale, Aug 08 2022 *)

values(m)=my(v=[]);for(i=1,m,v=concat([1,4],v));v; /* Anders Hellström, Aug 03 2015 */

[power_mod(4,n,5)for n in range(0,81)] # Zerinvary Lajos, Nov 26 2009

a(2n) = 1, a(2n+1) = 4.
From Paul Barry, Jun 03 2003: (Start)
G.f.: (1+4*x)/((1-x)*(1+x)).
E.g.f.: (5*exp(x) - 3*exp(-x))/2.
a(n) = (5 - 3*(-1)^n)/2.
a(n) = 4^((1-(-1)^n)/2) = 2^(1-(-1)^n) = 2/(2^((-1)^n)).
a(n) = 4^(ceiling(n/2) - floor(n/2)). (End)
a(n) = gcd((n-1)^2, (n+1)^2). - Paul Barry, Sep 16 2004
a(n) = A160700(A000302(n)). - Reinhard Zumkeller, Jun 10 2009
a(n) = 4^n mod 5. - Zerinvary Lajos, Nov 26 2009
a(n) = 4^(n mod 2). - Wesley Ivan Hurt, Mar 29 2014

A133145 Period 4: repeat [1, 2, 4, 8].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Dec 16 2007

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

Cf. A069705. [Jaume Oliver Lafont, Mar 27 2009]

Cf. A000079, A160700.

&cat [[1, 2, 4, 8]^^30]; // Wesley Ivan Hurt, Jul 09 2016

seq(op([1, 2, 4, 8]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 100, {1, 2, 4, 8}] (* Wesley Ivan Hurt, Jul 09 2016 *)
Table[First@ IntegerDigits[2^n, 16], {n, 0, 120}] (* Michael De Vlieger, Jul 09 2016 *)

a(n)=2^(n%4) \\ Jaume Oliver Lafont, Mar 27 2009

[power_mod(2,n,15) for n in range(0,80)] # Zerinvary Lajos, Nov 03 2009

a(n) == 2*a(n-1) mod 15.
a(n) = 2^(n mod 4). - Jaume Oliver Lafont, Mar 27 2009
a(n) = A160700(A000079(n)). [Reinhard Zumkeller, Jun 10 2009]
a(n) = 2^n (mod 15). G.f.: (1+2*x)*(4*x^2+1)/ ((1-x)*(1+x)*(x^2+1)). [R. J. Mathar, Apr 13 2010]
From Wesley Ivan Hurt, Jul 09 2016: (Start)
a(n) = a(n-4) for n>3.
a(n) = (15-6*cos(n*Pi/2)-5*cos(n*Pi)-12*sin(n*Pi/2)-5*I*sin(n*Pi))/4. (End)

cp's OEIS Frontend

A161450 Numbers n such that A160700(n) = 10.

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A161451 Numbers n such that A160700(n) = 11.

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A161452 Numbers m such that A160700(m) = 12.

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A161453 Numbers n such that A160700(n) = 13.

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A161454 Numbers n such that A160700(n) = 14.

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A161455 Numbers n such that A160700(n) = 15.

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A000302 Powers of 4: a(n) = 4^n.

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A001025 Powers of 16: a(n) = 16^n.

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