A024036 -id:A024036 - cp's OEIS frontend

A156760 5*4^n-1.

4, 19, 79, 319, 1279, 5119, 20479, 81919, 327679, 1310719, 5242879, 20971519, 83886079, 335544319, 1342177279, 5368709119, 21474836479, 85899345919, 343597383679, 1374389534719, 5497558138879, 21990232555519, 87960930222079, 351843720888319

Offset: 0

Paul Curtz, Feb 15 2009

Second column of the array A132207, or, if this array is flattened, a(n)=A132207(A007583(n)).

			Binary.......................................Decimal
100................................................4
10011.............................................19
1001111...........................................79
100111111........................................319
10011111111.....................................1279
1001111111111...................................5119
100111111111111................................20479
10011111111111111..............................81919
1001111111111111111...........................327679
100111111111111111111........................1310719
10011111111111111111111......................5242879
1001111111111111111111111...................20971519
100111111111111111111111111.................83886079
10011111111111111111111111111..............335544319
1001111111111111111111111111111...........1342177279
... - _Philippe Deléham_, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500

[5*4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Table[5*4^n - 1, {n, 0, 18}] (* Michael De Vlieger, Apr 20 2015 *)

a(n) mod 9 = A070403(n+2).
a(n+1) = 10*A083420(n)+9 .
a(n) = 5*A000302(n)-1.
a(n) = ( A024036(n+1)+A140529(n) )/2.
a(n) = 4a(n-1)+3, a(0)=4.
a(n) = A003947(n+1)-1 = 5*a(n-1)-4*a(n-2). G.f.: (4-x)/((1-x)(1-4x)). - R. J. Mathar, Feb 23 2009
a(n) = A198693(n) + 2^(2n+1). - Bob Selcoe, Apr 20 2015

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

A188529 Numbers which contain only the digit 3 in their base-4 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1 or 2, otherwise the exception must be the digit 2.

Original entry on oeis.org

1, 2, 3, 7, 11, 14, 15, 31, 47, 59, 62, 63, 127, 191, 239, 251, 254, 255, 511, 767, 959, 1007, 1019, 1022, 1023, 2047, 3071, 3839, 4031, 4079, 4091, 4094, 4095, 8191, 12287, 15359, 16127, 16319, 16367, 16379, 16382, 16383, 32767, 49151, 61439, 64511, 65279, 65471, 65519

Offset: 1

Vladimir Shevelev, Apr 03 2011

The sequence lists the positive binomial coefficient predictors in base 4. For definition, see paper in link.

A subsequence of A089633.

			(767)_10 = (23333)_4 contains only digits 3, with the exception the leading digit which is 2. Therefore 767 is in the sequence.
(4091)_10 = (333323)_4 contains only digits 3, with the exception a digit 2.

V. Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8

Cf. A024036, A089633, A188341, A188499.

isA188529 := proc(n) local dgs ,n3,p,d; dgs := convert(n,base,4); n3 := 0 ; for p from 1 to nops(dgs) do d := op(p,dgs) ; if d <> 3 then n3 := n3+1 ; if n3 >=2 then return false; end if; if p < nops(dgs) and d <> 2 then return false; end if; if p = nops(dgs) and d = 0 then return false; end if; end if; end do: return true; end proc:
A188529 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if isA188529(a) then return a; end if; end do; end if; end proc:
seq(A188529(n),n=1..50) ; # R. J. Mathar, Apr 03 2011

A248217 a(n) = 8^n - 2^n.

Original entry on oeis.org

0, 6, 60, 504, 4080, 32736, 262080, 2097024, 16776960, 134217216, 1073740800, 8589932544, 68719472640, 549755805696, 4398046494720, 35184372056064, 281474976645120, 2251799813554176, 18014398509219840, 144115188075331584, 1152921504605798400

Offset: 0

Vincenzo Librandi, Oct 04 2014

If 2^(n+1) is the length of the even leg of a primitive Pythagorean triangle (PPT) then it constrains the odd leg to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n+2^n, an area of 8^n-2^n and an inradius of 2^n-1. For n > 0, a(n) is the area of such triangles. - Frank M Jackson, Sep 07 2018

Maximum anomalous cancellation multiplicity of (2n+1)-digit integers: number of (2n+1)-digit integers which can be anomalously canceled with a fixed (2n+1)-digit integer. The maximum is obtained at 88...88911...11 containing n 8's and n 1's (see Example below). Anomalous cancellation is a "canceling" of digits of a and b in the numerator and denominator of a fraction a/b which results in a fraction equal to the original, and no 0 or digits that appear different times in a and b are canceled. For example, 49/98 = 4/8, 138/184 = 3/4, 1985/5955 = 185/555, 88911/43956 = 8811/4356, but 120/340 is not because canceling the 0's is not an anomalous cancellation. - Xiaohan Zhang, Nov 21 2019

			For n=1, there are 6 numbers with 3 digits that can be anomalously canceled with 891: 297, 396, 495, 594, 693, 792. For n=2 there are 60 numbers with 88911: 12987, 13986, 14985, 15984, 16983, 17982, 21978, 22977, 23976, 24975, 25974, 26973, 27972, 28971, 31968, 32967, 33966, 34965, 35964, 36963, 37962, 38961, 41958, 42957, 43956, 44955, 45954, 46953, 47952, 48951, 51948, 52947, 53946, 54945, 55944, 56943, 57942, 58941, 61938, 62937, 63936, 64935, 65934, 66933, 67932, 68931, 71928, 72927, 73926, 74925, 75924, 76923, 77922, 78921, 82917, 83916, 84915, 85914, 86913, 87912. For n=3 504 numbers with 8889111, and no other (2n+1)-digit number has greater multiplicity. There seems to be a pattern of integer partitions in these examples, because the sum of the digits of numbers above are all multiples of 9. - _Xiaohan Zhang_, Nov 21 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Anomalous Cancellation
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. similar sequences listed in A248216.

Cf. A000079, A024036.

Magma
```
[8^n-2^n: n in [0..25]];
```

Table[8^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 2 x) (1 - 8 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-16},{0,6},30] (* Harvey P. Dale, Mar 29 2015 *)

a(n) = 8^n-2^n; \\ Altug Alkan, Sep 07 2018

def A248217(n): return 6*binomial(pow(2,n) +1, 3)
print([A248217(n) for n in range(41)]) # G. C. Greubel, Dec 26 2024

G.f.: 6*x/((1-2*x)*(1-8*x)).
a(n) = 10*a(n-1) - 16*a(n-2).
a(n) = 2^n*(4^n-1) = A000079(n) * A024036(n) = A001018(n) - A000079(n).
E.g.f.: exp(2*x)*(-1 + exp(6*x)). - Stefano Spezia, Sep 07 2018
a(n) = 6*A016131(n-1). - R. J. Mathar, Mar 10 2022

A048329 Numbers that are repdigits in base 4.

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 15, 21, 42, 63, 85, 170, 255, 341, 682, 1023, 1365, 2730, 4095, 5461, 10922, 16383, 21845, 43690, 65535, 87381, 174762, 262143, 349525, 699050, 1048575, 1398101, 2796202, 4194303, 5592405, 11184810, 16777215, 22369621, 44739242, 67108863

Offset: 0

Patrick De Geest, Feb 15 1999

			10_10 = 22_4, 15_10 = 33_4, 5461_10 = 1111111_4.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-4).

Cf. A010785, A033017, A028987, A028988, A248721.

Base 4 repdigits 1,2,3 (trisections): A002450, A020988, A024036.

[0] cat  [k:k in [1..10^7]| #Set(Intseq(k,4)) eq 1]; // Marius A. Burtea, Oct 11 2019

a:= n-> (1+irem(n+2, 3))*(4^iquo(n+2,3)-1)/3:
seq(a(n), n = 0..45);

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 4], {n, 0, 40}, {d, 3}]]](* Vincenzo Librandi, Feb 06 2014 *)
LinearRecurrence[{0,0,5,0,0,-4},{0,1,2,3,5,10},40] (* Harvey P. Dale, Jul 11 2023 *)

G.f.: x*(1+2*x+3*x^2) / ( (x-1)*(4*x^3-1)*(1+x+x^2) ) with a(n) = 5*a(n-3) - 4*a(n-6). - R. J. Mathar, Mar 15 2015

Sum_{n>=1} 1/a(n) = (11/2) * A248721 = 2.31603727318383077512... - Amiram Eldar, Jan 21 2022

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 17, 63, 257, 1023, 4097, 16383, 65537, 262143, 1048577, 4194303, 16777217, 67108863, 268435457, 1073741823, 4294967297, 17179869183, 68719476737, 274877906943, 1099511627777, 4398046511103, 17592186044417, 70368744177663, 281474976710657

Offset: 0

Bruno Berselli, Jan 09 2013

This is the Lucas sequence V(3,-4).

Inverse binomial transform of this sequence is A087451.

Bruno Berselli, Table of n, a(n) for n = 0..200
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. for the same recurrence with initial values (i,i+1): A015521 (Lucas sequence U(3,-4); i=0), A122117 (i=1), A189738 (i=3).
Cf. for similar closed form: A014551 (2^n+(-1)^n), A102345 (3^n+(-1)^n), A087404 (5^n+(-1)^n).
Cf. A052539, A024036.

[n le 1 select n+2 else 3*Self(n)+4*Self(n-1): n in [0..25]];

RecurrenceTable[{a[n] == 3 a[n - 1] + 4 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]

a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+4*a[n-2]$ makelist(a[n], n, 0, 25);

Vec((2-3*x)/((1+x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Jun 26 2015

G.f.: (2-3*x)/((1+x)*(1-4*x)).
a(n) = 4^n+(-1)^n.
a(n) = A086341(A047524(n)) for n>0, a(0)=2.
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 25*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = (2/4^n) * Sum_{k = 0..n} binomial(4*n+1, 4*k). - Peter Bala, Feb 06 2019

A103454 a(n) = 0^n + 4^n - 1.

Original entry on oeis.org

1, 3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663

Offset: 0

Paul Barry, Feb 06 2005

A transform of 4^n under the matrix A103452.

The square of the cotangent of the arcsin of 1/(2^n). - Al Hakanson (hawkuu(AT)excite.com), Feb 23 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A024036, A103452.

[0^n+4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Table[Boole[n==0] +4^n -1, {n,0,40}] (* G. C. Greubel, Jun 21 2021 *)

[1]+[4^n -1 for n in [1..40]] # G. C. Greubel, Jun 21 2021

G.f.: (1 - 2*x + 4*x^2)/((1-x)*(1-4*x));
a(n) = Sum_{k=0..n} A103452(n, k)*4^k;
a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))4^k.
a(n) = A024036(n), n > 0. - R. J. Mathar, Aug 30 2008
E.g.f.: 1 - exp(x) + exp(4*x). - G. C. Greubel, Jun 21 2021
a(n) = 5*a(n-1) - 4*a(n-2). - Wesley Ivan Hurt, Mar 17 2023

A114569 a(n) = 9*4^n - 1.

Original entry on oeis.org

8, 35, 143, 575, 2303, 9215, 36863, 147455, 589823, 2359295, 9437183, 37748735, 150994943, 603979775, 2415919103, 9663676415, 38654705663, 154618822655, 618475290623, 2473901162495, 9895604649983, 39582418599935, 158329674399743, 633318697598975, 2533274790395903

Offset: 0

Al Hakanson (hawkuu(AT)excite.com), Feb 16 2006

Squares of the cotangents of the arcsins of 1/(3*2^n).

			a(2) = 143.

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

Cf. A024036, A097743, A156760, A199208.

[9*4^n-1: n in [0..30]]; // Vincenzo Librandi, Oct 29 2011

Table[(9*4^n) - 1, {n, 0, 23}] (* Stefan Steinerberger, Feb 16 2006 *)

From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 4*a(n-1) + 3, n>0; a(0)=8.
a(n) = 5*a(n-1) - 4*a(n-2), n>1; a(0)=8, a(1)=35.
G.f.: (8-5*x)/(1-5*x+4*x^2). (End)
From Elmo R. Oliveira, May 08 2025: (Start)
E.g.f.: exp(x)*(9*exp(3*x) - 1).
a(n) = A199208(n) - 2. (End)

More terms from Stefan Steinerberger, Feb 16 2006

A179857 Smallest number greater than n having in binary representation exactly twice the number of ones as n has in binary representation.

Original entry on oeis.org

3, 3, 15, 5, 15, 15, 63, 9, 15, 15, 63, 15, 63, 63, 255, 17, 23, 23, 63, 23, 63, 63, 255, 27, 63, 63, 255, 63, 255, 255, 1023, 33, 39, 39, 63, 39, 63, 63, 255, 43, 63, 63, 255, 63, 255, 255, 1023, 51, 63, 63, 255, 63, 255, 255, 1023, 63, 255, 255, 1023, 255, 1023, 1023

Offset: 1

Reinhard Zumkeller, Jul 31 2010

a(n) = Min{m: m > n and A000120(m) = 2*A000120(n)};
a(n) is odd;
n < a(n) < A000290(A062383(n));
a(A000079(n)) = A000051(n);
A024036 and A000225 give record values and where they occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to binary expansion of n

Cf. A000120, A086799.
Cf. A000290, A062383.
Cf. A000051, A000079.
Cf. A000225, A024036.

br2[n_]:=Module[{k=If[EvenQ[n],n+1,n+2],t=2*DigitCount[n,2,1]},While[ DigitCount[ k,2,1]!=t,k=k+2];k]; Array[br2,70] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = my(k=n+1, h=hammingweight(n)); while (hammingweight(k) != 2*h, k++); k; \\ Michel Marcus, Nov 13 2023

Definition clarified by Harvey P. Dale, Sep 20 2016

A282124 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 430", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 3, 15, 11, 63, 43, 255, 171, 1023, 683, 4095, 2731, 16383, 10923, 65535, 43691, 262143, 174763, 1048575, 699051, 4194303, 2796203, 16777215, 11184811, 67108863, 44739243, 268435455, 178956971, 1073741823, 715827883, 4294967295, 2863311531, 17179869183

Offset: 0

Robert Price, Feb 06 2017

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

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 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
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A282121, A282122, A282123.

Cf. A001045, A010684, A024036, A081631, A098646.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 430; 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}]

Conjectures from Colin Barker, Feb 07 2017: (Start)
a(n) = (-1 + 2*(-1)^n - (-1)^n*2^(1+n) + 2^(2+n)) / 3.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)
Conjectures from Paul Curtz, Jun 10 2019: (Start)
a(n) = A001045(n+1)*(period 2: repeat[1, 3]).
a(n+4) = a(n) + 10*A081631(n).
a(2*n+1) = 2^(2*n+2) -1.
a(n+2) = a(n) + A098646(n+1).
(End)

A330941 a(n) is the greatest value whose binary representation can be obtained by interleaving (or shuffling) two copies of the binary representation of n.

Original entry on oeis.org

0, 3, 12, 15, 48, 53, 60, 63, 192, 201, 212, 219, 240, 245, 252, 255, 768, 785, 804, 819, 848, 853, 876, 887, 960, 969, 980, 987, 1008, 1013, 1020, 1023, 3072, 3105, 3140, 3171, 3216, 3237, 3276, 3303, 3392, 3401, 3412, 3435, 3504, 3509, 3548, 3567, 3840, 3857

Offset: 0

Rémy Sigrist, Jan 04 2020

The binary representation of all positive terms are square binary words (see A191755).

			The first terms, alongside the binary representations of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ----------
   0     0       0           0
   1     3       1          11
   2    12      10        1100
   3    15      11        1111
   4    48     100      110000
   5    53     101      110101
   6    60     110      111100
   7    63     111      111111
   8   192    1000    11000000
   9   201    1001    11001001
  10   212    1010    11010100
  11   219    1011    11011011
  12   240    1100    11110000

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Logarithmic scatterplot of the first difference of the first 2^13 terms
Rémy Sigrist, PARI program for A330941
Index entries for sequences related to binary expansion of n

See A330940 for the minimum variant.

Cf. A002001, A024036, A191755, A193020.

PARI
```
See Links section.
```

a(2^k) = 3*4^k = A002001(k+1) for any k >= 0.
a(2^k-1) = 4^k-1 = A024036(k) for any k >= 0.
a(n) >= A330940(n).

cp's OEIS Frontend

A156760 5*4^n-1.

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A188529 Numbers which contain only the digit 3 in their base-4 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1 or 2, otherwise the exception must be the digit 2.

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A248217 a(n) = 8^n - 2^n.

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A048329 Numbers that are repdigits in base 4.

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A201455 a(n) = 3*a(n-1) + 4*a(n-2) for n>1, a(0)=2, a(1)=3.

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A103454 a(n) = 0^n + 4^n - 1.

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A114569 a(n) = 9*4^n - 1.

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A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.