A153643 -id:A153643 - cp's OEIS frontend

A010684 Period 2: repeat (1,3); offset 0.

1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1

Offset: 0

N. J. A. Sloane

Hankel transform is [1,-8,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Mar 29 2007
Binomial transform gives [1,4,8,16,32,64,...] (A151821(n+1)). - Philippe Deléham, Sep 17 2009
Continued fraction expansion of (3+sqrt(21))/6. - Klaus Brockhaus, May 04 2010
Positive sum of the coordinates from the image of the point (1,-2) after n 90-degree rotations about the origin. - Wesley Ivan Hurt, Jul 06 2013
This sequence can be generated by an infinite number of formulas having the form a^(b*n) mod c where a is congruent to 3 mod 4 and b is any odd number. If a is congruent to 3 mod 4 then c can be 4; if a is also congruent to 3 mod 8 then c can be 8. For example: a(n)= 15^(3*n) mod 4, a(n) = 19^(5*n) mod 4, a(n) = 19^(5*n) mod 8. - Gary Detlefs, May 19 2014
This sequence is also the unsigned periodic Schick sequence for p = 5. See the Schick reference, p. 158, for p = 5.- Wolfdieter Lang, Apr 03 2020
Digits following the decimal point when 1/3 is converted to base 5. - Jamie Robert Creasey, Oct 15 2021
Decimal expansion of 13/99. - Stefano Spezia, Feb 09 2025

			0.131313131313131313131313131313131313131313131...

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

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

Cf. A112030, A112033, A176014 (decimal expansion of (3+sqrt(21))/6).

[seq (modp((2*n+1),4),n=0..80)]; # Zerinvary Lajos, Nov 30 2006

Table[2-(-1)^n, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 24 2014 *)
PadRight[{},120,{1,3}] (* Harvey P. Dale, Mar 11 2025 *)

a(n)=1+n%2*2 \\ Charles R Greathouse IV, Dec 28 2011

def A010684(n): return 3 if n&1 else 1 # Chai Wah Wu, Jan 17 2023

[power_mod(3, n, 8)for n in range(0, 81)] # Zerinvary Lajos, Nov 24 2009

From Paul Barry, Apr 29 2003: (Start)
a(n) = 2-(-1)^n.
G.f.: (1+3x)/((1-x)(1+x)).
E.g.f.: 2*exp(x) - exp(-x). (End)
a(n) = 2*A153643(n) - A153643(n+1). - Paul Curtz, Dec 30 2008
a(n) = 3^(n mod 2). - Jaume Oliver Lafont, Mar 27 2009
a(n) = 7^n mod 4. - Vincenzo Librandi, Feb 07 2011
a(n) = 1 + 2*(n mod 2). - Wesley Ivan Hurt, Jul 06 2013
a(n) = A000034(n) + A000035(n). - James Spahlinger, Feb 14 2016

A128209 Jacobsthal numbers(A001045) + 1.

Original entry on oeis.org

1, 2, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532, 5726623062

Offset: 0

Paul Barry, Feb 19 2007

Row sums of A128208.
Essentially the same as A052953. - R. J. Mathar, Jun 14 2008
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 1, a(n+1) is the number of different representations of matrix P^(-1)+I+P by sum of permutation matrices. - Vladimir Shevelev, Apr 12 2010
a(n) is the rank of Fibonacci(n+2) in row n of A049456 (regarded as an irregular triangle read by rows). - N. J. A. Sloane, Nov 23 2016

V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A167030, A153643, A049456.

[1+2^n/3-(-1)^n/3: n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

CoefficientList[Series[(1-3*x^2)/(1-2*x-x^2+2*x^3),{x,0,40}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)

a(n)=2^n\/3+1 \\ Charles R Greathouse IV, Jan 31 2012

def A128209(n): return ((1<Chai Wah Wu, Apr 17 2025

a(n) = 1 + 2^n/3 - (-1)^n/3.

G.f.: (1-3*x^2)/(1 - 2*x - x^2 + 2*x^3).

A163834 a(n) = (4^n + 5)/3.

Original entry on oeis.org

2, 3, 7, 23, 87, 343, 1367, 5463, 21847, 87383, 349527, 1398103, 5592407, 22369623, 89478487, 357913943, 1431655767, 5726623063, 22906492247, 91625968983, 366503875927, 1466015503703, 5864062014807, 23456248059223, 93824992236887, 375299968947543

Offset: 0

Juri-Stepan Gerasimov, Aug 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000027, A135351, A140966, A153643.

Table[(4^n + 5)/3, {n, 0, 50}] (* G. C. Greubel, Aug 05 2017 *)
LinearRecurrence[{5,-4},{2,3},30] (* Harvey P. Dale, Jun 14 2023 *)

x='x+O('x^50); concat([0], Vec((2-7*x)/((4*x-1)*(x-1)))) \\ G. C. Greubel, Aug 05 2017

a(n) = (4^n + 5)/3 = A135351(2*n+1) = A140966(2*n) = A153643(2*n).
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: (2-7*x)/((4*x-1)*(x-1)).
a(n+1) - a(n) = A000302(n).
E.g.f.: (1/3)*(5*exp(x) + exp(4*x)). - G. C. Greubel, Aug 05 2017

Offset set to 0 by R. J. Mathar, Aug 06 2009

A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 5, 8, 9, 10, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 2, 5, 8, 9, 10, 17, 20, 21, 26, 29, 32, 33, 34, 33, 36, 37, 34, 37, 40, 41, 42, 1, 4, 5, 10, 13, 16, 17, 18, 17, 20, 21, 34, 37, 40, 41, 42, 49, 52, 53, 58, 61, 64, 65, 66, 65, 68

Offset: 0

Alex Ratushnyak, Jun 13 2012

Indices of 1's: A007583(n),
indices of 2's: A047849(n+1),
indices of 4's: A039301(n+2),
indices of 5's: A153643(n+3),
indices of 8's: A155701(n+2),
indices of 9's: A155701(n+2)+1 = A163868(n+2),
indices of 10's: A153643(n+4)+3^((n+1) mod 2),
indices of 13's: A039301(n+3)+3,
indices of 16's: A039301(n+3)+4,
indices of 17's: 17, 19, 27, 49, 51, 91, 177, 179, 347, 689, 691, 1371, 2737, 2739, 5467, 10929, 10931, 21851, 43697, 43699, 87387, 174769, 174771, 349531, 699057, 699059, 1398107, 2796209, 2796211, 5592411, 11184817, 11184819, 22369627, 44739249, 44739251, 89478491, ...
indices of 18's: A039301(n+3)+6,
n's such that a(n)<3: A005578, except the first term.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a:= proc(n) local i, k, m, r;
      k, m, r:= n, 3*n, 0;
      for i from 0 while (m>0 or k>0) do
        r:= r +2^i* irem(m, 2, 'm') *irem(k, 2, 'k')
      od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 22 2012

Table[BitAnd[n, 3*n], {n, 0, 68}] (* Arkadiusz Wesolowski, Jun 23 2012 *)

a(n)=bitand(n,3*n) \\ Charles R Greathouse IV, Feb 05 2013

for n in range(99):
    print(3*n & n, end=',')

A154890 Jacobsthal numbers A001045 alternatingly incremented by 3 and 5.

Original entry on oeis.org

3, 6, 4, 8, 8, 16, 24, 48, 88, 176, 344, 688, 1368, 2736, 5464, 10928, 21848, 43696, 87384, 174768, 349528, 699056, 1398104, 2796208, 5592408, 11184816, 22369624, 44739248, 89478488, 178956976, 357913944, 715827888, 1431655768, 2863311536, 5726623064

Offset: 0

Paul Curtz, Jan 17 2009

a(2n+1) = 2*a(2n).
a(n) = A153643(n)+A010684(n).
a(n+2) = 4*A128209(n).
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3). G.f.: (3-11x^2)/((1-x)(1+x)(1-2x)). [R. J. Mathar, Jan 23 2009]

Edited and extended by R. J. Mathar, Jan 23 2009

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A010684 Period 2: repeat (1,3); offset 0.

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A128209 Jacobsthal numbers(A001045) + 1.

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A163834 a(n) = (4^n + 5)/3.

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A213526 a(n) = 3*n AND n, where AND is the bitwise AND operator.

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A154890 Jacobsthal numbers A001045 alternatingly incremented by 3 and 5.

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