A007583 -id:A007583 - cp's OEIS frontend

A115716 A divide-and-conquer sequence.

1, 1, 3, 1, 3, 1, 11, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 683, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1

Offset: 0

Paul Barry, Jan 29 2006

			G.f. = 1 + x + 3*x^2 + x^3 + 3*x^4 + x^5 + 11*x^6 + x^7 + 3*x^8 + x^9 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..16381
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences , arXiv:math/0307027 [math.CO], 2003.

Partial sums are A032925.
Row sums of number triangle A115717.
Bisection: A276390.
Cf. A007583, A081294, A091090.
See A276391 for a closely related sequence.

a:= proc(n) option remember; `if`(n=0, 1,
      `if`(n::odd, 1, 4*a(n/2-1)-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 07 2016

a[n_] := a[n] = If[n == 0, 1, If[OddQ[n], 1, 4*a[n/2-1]-1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 25 2017, after Alois P. Heinz *)

{a(n) = if( n<1, n==0, n%2, 1, 4 * a(n/2-1) - 1)}; /* Michael Somos, Sep 07 2016 */

The g.f. G(x) satisfies G(x)-4*x^2*G(x^2)=(1+2*x)/(1+x). - Argument and offset corrected by Bill Gosper, Sep 07 2016
G.f.: 1/(1-x) + Sum_{k>=0} ((4^k-0^k)/2) *x^(2^(k+1)-2) /(1-x^(2^k)). - corrected by R. J. Mathar, Sep 07 2016
a(n)=A007583(A091090(n+1)-1). - Adapted to new offset by R. J. Mathar, Sep 07 2016
a(0) = 1, a(2*n + 1) = 1 for n>=0. a(2*n + 2) = 4*a(n) - 1 for n>=0. - Michael Somos, Sep 07 2016

A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

Original entry on oeis.org

0, 2, 0, 3, 2, 7, 10, 23, 42, 87, 170, 343, 682, 1367, 2730, 5463, 10922, 21847, 43690, 87383, 174762, 349527, 699050, 1398103, 2796202, 5592407, 11184810, 22369623, 44739242, 89478487, 178956970, 357913943, 715827882, 1431655767, 2863311530, 5726623063, 11453246122, 22906492247, 45812984490

Offset: 0

Miklos Kristof, Dec 07 2007

Partial sums of A155980 for n > 2. - Klaus Purath, Jan 30 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A005578, A099754.
Cf. A001045, A327767, A328284.
Cf. A007583, A062092, A087289, A020988 (even bisection), A163834 (odd bisection), A078008, A084247, A181565.

List([0..40], n-> (2^n+3-7*(-1)^n+3*0^n)/6); # G. C. Greubel, Sep 02 2019

a135351:=func< n | (2^n+3-7*(-1)^n+3*0^n)/6 >; [ a135351(n): n in [0..32] ]; // Klaus Brockhaus, Dec 05 2009

G(x):=x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)): f[0]:=G(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/n!,n=0..30);

Join[{0}, Table[(2^n +3 -7*(-1)^n)/6, {n,40}]] (* G. C. Greubel, Oct 11 2016 *)
LinearRecurrence[{2,1,-2},{0,2,0,3},40] (* Harvey P. Dale, Feb 13 2024 *)

a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; \\ Michel Marcus, Oct 11 2016

[(2^n+3-7*(-1)^n+3*0^n)/6 for n in (0..40)] # G. C. Greubel, Sep 02 2019

G.f.: x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)).
E.g.f.: (exp(2*x) + 3*exp(x) - 7*exp(-x) + 3)/6.
From Paul Curtz, Dec 20 2020: (Start)
a(n) + (period 2 sequence: repeat [1, -2]) = A328284(n+2).
a(n+1) + (period 2 sequence: repeat [-2, 1]) = A001045(n).
a(n+1) + (period 2 sequence: repeat [-1, 0]) = A078008(n).
a(n+1) + (period 2 sequence : repeat [-3, 2]) = -(-1)^n*A084247(n).
a(n+4) = a(n+1) + 7*A001045(n).
a(n+4) + a(n+1) = A181565(n).
a(2*n+2) + a(2*n+3) = A087289(n) = 3*A007583(n).
a(2*n+1) = A163834(n), a(2*n+2) = A020988(n). (End)

First part of definition corrected by Klaus Brockhaus, Dec 05 2009

A151917 a(0)=0, a(1)=1; for n>=2, a(n) = (2/3)*(Sum_{i=1..n-1} 3^wt(i)) + 1, where wt() = A000120().

Original entry on oeis.org

0, 1, 3, 5, 11, 13, 19, 25, 43, 45, 51, 57, 75, 81, 99, 117, 171, 173, 179, 185, 203, 209, 227, 245, 299, 305, 323, 341, 395, 413, 467, 521, 683, 685, 691, 697, 715, 721, 739, 757, 811, 817, 835, 853, 907, 925, 979, 1033, 1195, 1201, 1219

Offset: 0

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Also, total number of "ON" cells at n-th stage in two of the four wedges of the "Ulam-Warburton" two-dimensional cellular automaton of A147562, but including the central ON cell. It appears that this is very close to A139250, the toothpick sequence. - Omar E. Pol, Feb 22 2015

			n=3: (2/3)*(3^1+3^1+3^2+3^1) + 1 = (2/3)*18 + 1 = 13.

Michel Marcus, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000120, A007583, A139250, A147562, A151914, A151920.

Array[(2/3) Sum[3^(Total@ IntegerDigits[i, 2]), {i, # - 1}] + 1 &, 50] (* Michael De Vlieger, Nov 01 2022 *)

a(n) = if (n<2, n, 1 + 2*sum(i=1,n-1, 3^hammingweight(i))/3); \\ Michel Marcus, Feb 22 2015

a(n) = A151914(n)/4.
a(n) = A079315(2n)/4.
For n>=2, a(n) = 2*A151920(n-2) + 1.
For n>=1, a(n) = (1 + A147562(n))/2. - Omar E. Pol, Mar 13 2011
a(2^k) = A007583(k), if k >= 0. - Omar E. Pol, Feb 22 2015

A160128 a(n) = number of grid points that are covered after (2^n)th stage of A139250.

Original entry on oeis.org

3, 7, 19, 63, 235, 919, 3651, 14575, 58267, 233031, 932083, 3728287, 14913099, 59652343, 238609315, 954437199, 3817748731, 15270994855, 61083979347, 244335917311, 977343669163, 3909374676567, 15637498706179

Offset: 0

Omar E. Pol, May 09 2009

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A000079, A007583, A139250, A139251, A139252, A139560, A147614.

Cf. Same recurrence: A073724, A210985, A014825.

Vec((3 - 11*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)) + O(x^40)) \\ Colin Barker, May 13 2020

a(n) = A147614(A000079(n)).
a(n) = (1/9)*(2^(2*n+3) + 12*n + 19). [Nathaniel Johnston, Mar 29 2011]
It appears that a(n) = A139252(2^(n+1)). - Omar E. Pol, Sep 11 2012
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). - Paul Curtz, May 07 2020
G.f.: (3 - 11*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)). - Colin Barker, May 13 2020

Terms after a(10) from Nathaniel Johnston, Mar 29 2011

A199210 a(n) = (11*4^n + 1)/3.

Original entry on oeis.org

4, 15, 59, 235, 939, 3755, 15019, 60075, 240299, 961195, 3844779, 15379115, 61516459, 246065835, 984263339, 3937053355, 15748213419, 62992853675, 251971414699, 1007885658795, 4031542635179, 16126170540715, 64504682162859

Offset: 0

Vincenzo Librandi, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Sequences of the form (m*4^n + 1)/3: A007583 (m=2), A136412 (m=5), this sequence (m=11), A199210 (m=11), A206373 (m=14).

Magma
```
[(11*4^n+1)/3: n in [0..30]];
```

LinearRecurrence[{5,-4}, {4,15}, 31] (* G. C. Greubel, Jan 19 2023 *)

[(11*4^n+1)/3 for n in range(31)] # G. C. Greubel, Jan 19 2023

a(n) = 4*a(n-1) - 1.
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: (4-5*x)/((1-x)*(1-4*x)). - Bruno Berselli, Nov 04 2011
E.g.f.: (1/3)*(11*exp(4*x) + exp(x)). - G. C. Greubel, Jan 19 2023

A097072 Expansion of (1 - 2x + 2x^2)/((1 - x^2)(1 - 2x)).

Original entry on oeis.org

1, 0, 3, 4, 11, 20, 43, 84, 171, 340, 683, 1364, 2731, 5460, 10923, 21844, 43691, 87380, 174763, 349524, 699051, 1398100, 2796203, 5592404, 11184811, 22369620, 44739243, 89478484, 178956971, 357913940, 715827883, 1431655764, 2863311531, 5726623060, 11453246123

Offset: 0

Paul Barry, Jul 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2)

Cf. A000035, A000975, A001045, A007583, A080674.

[(4*2^n - 3 + 5*(-1)^n)/6: n in [0..50]]; // G. C. Greubel, Oct 10 2017

a:= n-> ceil(2*(2^n-1)/3)+(-1)^n:
seq(a(n), n=0..32);  # Alois P. Heinz, Jun 15 2023

CoefficientList[Series[(1-2x+2x^2)/((1-x^2)(1-2x)),{x,0,50}],x]  (* Harvey P. Dale, Mar 09 2011 *)
Table[2*2^n/3 - 1/2 + 5 (-1)^n/6, {n, 0, 32}] (* Michael De Vlieger, Feb 22 2017 *)

for(n=0,50, print1((4*2^n - 3 + 5*(-1)^n)/6, ", ")) \\ G. C. Greubel, Oct 10 2017

a(n) = (4*2^n - 3 + 5*(-1)^n)/6.
a(n) = Sum_{k=0..n} (2^k - 1 + 0^k)(-1)^(n-k).
a(n) = A001045(n+1) - A000035(n).
a(n) = a(n-1) + 2*a(n-2) + 1, n > 1. - Gary Detlefs, Jun 20 2010
a(2*n) = A007583(n), a(2*n+1) = A080674(n), n >= 0. - Yosu Yurramendi, Feb 21 2017
a(n) = A000975(n) + (-1)^n. - Alois P. Heinz, Jun 15 2023

A175625 Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).

Original entry on oeis.org

7, 11, 23, 31, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1123, 1187, 1283, 1291, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543

Offset: 1

Alzhekeyev Ascar M, Jul 28 2010, Jul 30 2010

nonn

All composites in this sequence are 2-pseudoprimes, A001567. That subsequence begins with 536870911, 46912496118443, 192153584101141163, with no other composites below 2^64 (the first two were found by 'venco' from the dxdy.ru forum), and contains the terms of A303448 that are not multiples of 3. Correspondingly, composite terms include those of the form A007583(m) = (2^(2m+1) + 1)/3 for m in A303009. The only known composite member not of this form is a(1018243) = 536870911.
Intended as a pseudoprimality test; note that many primes do not pass the third condition either.
Conjecture: The prime values belong to A039787. - Bill McEachen, Dec 27 2023

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

Cf. A001567, A007583, A303009, A303448.

Cf. A039787.

Select[Array[(6 # + (-1)^# - 3)/2 &, 3000], And[PowerMod[2, (# - 1), #] == 1, PowerMod[2, (# - 3), (# - 1)/2] == 1] &] (* Michael De Vlieger, Dec 27 2023 *)

isA175625(n) = gcd(n,6)==1 && Mod(2,n)^(n-1)==1 && Mod(2,n\2)^(n-3)==1

Partially edited by N. J. A. Sloane, Jul 29 2010
Entry rewritten by Charles R Greathouse IV, Aug 04 2010
Comment and b-file from Charles R Greathouse IV, Sep 06 2010
Edited by Max Alekseyev, May 28 2014, Apr 24 2018

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=',')

A255138 a(n) = (1 + 2^n(3 + 2(-1)^n))/3.

Original entry on oeis.org

2, 1, 7, 3, 27, 11, 107, 43, 427, 171, 1707, 683, 6827, 2731, 27307, 10923, 109227, 43691, 436907, 174763, 1747627, 699051, 6990507, 2796203, 27962027, 11184811, 111848107, 44739243, 447392427, 178956971

Offset: 0

L. Edson Jeffery, May 04 2015

Let N_1 be the set of odd natural numbers and v(y) the 2-adic valuation of y. Define the map F : N_1 -> N_1 by F(x) = (3*x+1)/2^v(3*x+1) (see A075677). Let F^(k)(x) denote k-fold iteration of F, with recurrence F^(k)(x) = F(F^(k-1)(x)), k > 0, and initial condition F^(0)(x) = x. Then, for n>0, a(n) is the least m such that F^(n)(4*m-3) == 1 (mod 4). Cf. A257499.

Let k == 1 mod 4, and k(r) be the r-th iteration at which k appears in a Collatz sequence. When n >= 2 and k(r) == [2^(n+1) - a(n)] mod 2^(n+1), then n is the number of halving steps following k(r+1). For instance, since a(5) = 11, there are 5 halving steps following k(r+1) when k(r) == 53 mod 64, because 2^(5+1) = 64 and 64-11 = 53; e.g., k(r) = 117: 117 -> 352 -> 176 -> 88 -> 44 -> 22 -> 11. - Bob Selcoe, Feb 09 2017

Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Cf. A007583, A075677, A096773, A136412, A257499.

[(1 + 2^n*(3 + 2*(-1)^n))/3: n in [0..50]]; // Wesley Ivan Hurt, Nov 05 2015

A255138:=n->(1 + 2^n*(3 + 2*(-1)^n))/3: seq(A255138(n), n=0..50); # Wesley Ivan Hurt, Nov 05 2015

a[n_] := (1 + 2^n*(3 + 2*(-1)^n))/3; Table[a[n], {n, 0, 29}]
LinearRecurrence[{1,4,-4},{2,1,7},30] (* Harvey P. Dale, Aug 03 2024 *)

vector(30, n, n--; (1 + 2^n*(3 + 2*(-1)^n))/3) \\ Altug Alkan, Nov 05 2015

a(2*n) = A136412(n); a(2*n+1) = A007583(n).
G.f.: (2-x-2*x^2)/((x-1)*(2*x-1)*(2*x+1)). - R. J. Mathar, Jul 25 2015
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 2. - Wesley Ivan Hurt, Nov 05 2015
a(n) = 4*a(n-2) - 1. - Bob Selcoe, Feb 09 2017
a(n) = 2^(n+1) - A096773(n+1). - Ruud H.G. van Tol, Sep 04 2023

A277955 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 3, 7, 11, 23, 43, 87, 171, 343, 683, 1367, 2731, 5463, 10923, 21847, 43691, 87383, 174763, 349527, 699051, 1398103, 2796203, 5592407, 11184811, 22369623, 44739243, 89478487, 178956971, 357913943, 715827883, 1431655767, 2863311531, 5726623063

Offset: 0

Robert Price, Nov 05 2016

Initialized with a single black (ON) cell at stage zero.

Essentially the same as A267052. - R. J. Mathar, Nov 09 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of the first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A277952, A277953, A277954.
Cf. A010673, A001045, A007583, A163834.
Cf. A078008.

I:=[1,3,3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Nov 06 2016

CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=14; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
Table[FromDigits[Part[ca[[i]][[i]],Range[i,2*i-1]],2], {i,1,stages-1}]
LinearRecurrence[{2, 1, -2}, {1, 3, 3}, 32] (* or *)
CoefficientList[ Series[(1 + x - 4x^2)/(1 - 2x - x^2 + 2x^3), {x, 0, 31}], x] (* Robert G. Wilson v, Nov 05 2016 *)

G.f.: (1 + x - 4*x^2)/(1 - 2*x - x^2 + 2*x^3). - Robert G. Wilson v, Nov 05 2016
From Colin Barker, Nov 06 2016: (Start)
a(n) = (3 - 2*(-1)^n + 2^(1+n))/3.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
From Paul Curtz, May 08 2024: (Start)
a(2*n) = A007583(n). a(2*n+1) = A163834(n+1).
a(n) = A001045(n+1) + A010673(n).
a(n) = a(n-1) + 2*A078008(n-1). (End)

cp's OEIS Frontend

A115716 A divide-and-conquer sequence.

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A135351 a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

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A151917 a(0)=0, a(1)=1; for n>=2, a(n) = (2/3)*(Sum_{i=1..n-1} 3^wt(i)) + 1, where wt() = A000120().

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A160128 a(n) = number of grid points that are covered after (2^n)th stage of A139250.

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A199210 a(n) = (11*4^n + 1)/3.

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A097072 Expansion of (1 - 2*x + 2*x^2)/((1 - x^2)*(1 - 2*x)).

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A175625 Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).

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A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

A097072 Expansion of (1 - 2x + 2x^2)/((1 - x^2)(1 - 2x)).

A255138 a(n) = (1 + 2^n(3 + 2(-1)^n))/3.