cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A087120 Smallest numbers having in binary representation exactly n maximal groups of consecutive zeros.

Original entry on oeis.org

1, 0, 10, 42, 170, 682, 2730, 10922, 43690, 174762, 699050, 2796202, 11184810, 44739242, 178956970, 715827882, 2863311530, 11453246122, 45812984490, 183251937962, 733007751850, 2932031007402, 11728124029610, 46912496118442
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 14 2003

Keywords

Comments

A087116(a(n))=n and A087116(k)
For n>1, a(n) = A020988(n-1).

Links

Crossrefs

Cf. A087118, A023416, A007088.

Programs

  • Mathematica
    Join[{1,0},NestList[4#+2&,10,25]] (* Harvey P. Dale, Apr 20 2019 *)

Formula

a(0)=1, a(1)=0, a(2)=10, a(n)=4*a(n-1)+2.

A184908 Let S_n be the set of the integers having alternating bit sum equal to -n. There are a(n) primes among the smallest 3n+5 odd numbers of S_n.

Original entry on oeis.org

1, 7, 5, 0, 7, 5, 0, 7, 10, 0, 6, 3, 0, 5, 9, 0, 7, 7, 0, 7, 9, 0, 5, 6, 0, 6, 7, 0, 10, 7, 0, 4, 5, 0, 11, 7, 0, 10, 9, 0, 9, 4, 0, 4, 8, 0, 2, 6, 0, 9, 5, 0, 10, 9, 0, 8, 6, 0, 4, 3, 0, 4, 11, 0, 9, 3, 0, 5, 8, 0, 6, 3, 0, 11, 7, 0, 6, 8, 0, 5, 6
Offset: 0

Views

Author

Washington Bomfim, Jan 27 2011

Keywords

Examples

			The smallest 3n+5 = 8 odd numbers of the set S_1 of the integers having alternating bit sum -1 are 11, 35, 41, 47, 59, 107, 131, and 137, so a(1)=7.

Links

Crossrefs

Cf. A065359, A065085, A184907, A020988.

Programs

  • Maple
    A065359 := proc(n) local dgs ; dgs := convert(n,base,2) ; add( -op(i,dgs)*(-1)^i,i=1..nops(dgs)) ; end proc:
S := proc(n) local ads,k; ads := {} ; for k from 1 by 2 do if A065359(k) = -n then ads := ads union {k} ; end if; if nops(ads) = 3*n+5 then return ads; end if; end do: end proc:
A184908 := proc(n) local ads,a,p; a := 0 ; for p in S(n) do if isprime(p) then a := a+1 ; end if; end do: a ; end proc:
for n from 0 do print(A184908(n)); end do: # slow! R. J. Mathar, Feb 11 2011
  • PARI
    II()={i = (2/3)*(4^n-1) + 1 + 2^(2*n+1); if(isprime(i),an++)};
III()={w = 2^(2*n+3); for(j=1, n+1, i += w; w /= 4; i -= w; if(isprime(i), an++ ))};
IV()={i+=6; if(isprime(i), an++ ); w=4; for(j=1, n, i -= w; w *= 4; i += w; if(isprime(i), an++))};
V()={i += 2^(2*n+4) - 2^(2*n+2); if(isprime(i),an++ );w = i + 2^(2*n+5) - 2^(2*n+4); i = w - 2^(2*n+3) - 2^(2*n+1); if(isprime(i),an++ );w = 2^(2*n+1);for(j=1, n,i += w; w /= 4; i -= w;if(isprime(i),an++ ))};
print1("1, 7, ");for(n=2,80, an=0; II(); III(); IV(); V(); print1(an,", ")) \\ Washington Bomfim, Feb 06 2011

A216649 Triangle T(n,k) in which n-th row lists in increasing order all positive integers with a representation as totally balanced 2n digit binary string such that all consecutive totally balanced substrings are in nondecreasing order; n>=1, 1<=k<=A000081(n+1).

Original entry on oeis.org

2, 10, 12, 42, 44, 52, 56, 170, 172, 180, 184, 204, 212, 216, 232, 240, 682, 684, 692, 696, 716, 724, 728, 744, 752, 820, 824, 852, 856, 872, 880, 920, 936, 944, 976, 992, 2730, 2732, 2740, 2744, 2764, 2772, 2776, 2792, 2800, 2868, 2872, 2900, 2904, 2920, 2928
Offset: 1

Views

Author

Alois P. Heinz, Sep 12 2012

Keywords

Comments

There is a simple bijection between the elements of row n and the rooted trees with n+1 nodes. The tree has a root node. Each matching pair (1,0) in the binary string representation encodes an additional node, the totally balanced substrings encode lists of subtrees.

Examples

			172 is element of row 4, the binary string representation (with totally balanced substrings enclosed in parentheses) is (10)(10)(1(10)0).  The encoded rooted tree is:
.    o
.   /|\
.  o o o
.      |
.      o
Triangle T(n,k) begins:
2;
10,     12;
42,     44,   52,   56;
170,   172,  180,  184,  204,  212,  216,  232,  240;
682,   684,  692,  696,  716,  724,  728,  744,  752,  820,  824, ...
2730, 2732, 2740, 2744, 2764, 2772, 2776, 2792, 2800, 2868, 2872, ...
Triangle T(n,k) in binary:
10;
1010,       1100;
101010,     101100,     110100,     111000;
10101010,   10101100,   10110100,   10111000,   11001100,   11010100, ...
1010101010, 1010101100, 1010110100, 1010111000, 1011001100, 1011010100, ...

Links

Crossrefs

First column gives: A020988.
Last elements of rows give: A020522.
Row lengths are: A000081(n+1).
Subsequence of A014486, A031443.
Cf. A061773, A216349, A216350, A216648.

Programs

  • Maple
    F:= proc(n) option remember; `if`(n=1, [10], sort(map(h->
      parse(cat(1, sort(h)[], 0)), g(n-1, n-1)))) end:
g:= proc(n, i) option remember; `if`(i=1, [[10$n]], [seq(seq(seq(
      [seq (F(i)[w[t]-t+1], t=1..j),v[]], w=combinat[choose](
      [$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])
    end:
b:= proc(n) local h, i, r; h, r:= n/10, 0; for i from 0
      while h>1 do r:= r+2^i*irem(h, 10, 'h') od; r
    end:
T:= proc(n) option remember; map(b, F(n+1))[] end:
seq(T(n), n=1..6);

Formula

T(n,k) = A216648(n+1,k)/2 - 2^(2*n).

A263133 Numbers m such that binomial(4*m + 3, m) is odd.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 10, 11, 15, 21, 23, 31, 42, 43, 47, 63, 85, 87, 95, 127, 170, 171, 175, 191, 255, 341, 343, 351, 383, 511, 682, 683, 687, 703, 767, 1023, 1365, 1367, 1375, 1407, 1535, 2047, 2730, 2731, 2735, 2751, 2815, 3071, 4095, 5461, 5463, 5471, 5503
Offset: 1

Views

Author

Peter Bala, Oct 11 2015

Keywords

Comments

The even terms in the sequence are A020988. If m is a term in the sequence then 2*m + 1 is also a term in the sequence. Repeatedly applying the transformation m -> 2*m + 1 to the terms of A020988 produces all the terms of this sequence. See the example below.
2*a(n) gives the values of m such that binomial(4*m + 6, m) is odd.

Examples

			1) This sequence can be read from Table 1 below in a sequence of 'knight moves' (2 down and 1 to the left) starting from the first two rows. For example, starting at 42 in the first row we jump 42 -> 43 -> 47 -> 63, then return to the second row at 85 and jump 85 -> 87 -> 95 -> 127, followed by 170 -> 171 -> 175 -> 191 -> 255, and so on.
...........................................................
. Table 1. 2^n*ceiling((2^(2*k + 1) - 1)/3) - 1, n,k >= 0 .
...........................................................
  n\k|   0    1    2    3    4    5
  ---+---------------------------------
   0 |   0    2   10   42  170  682 ...
   1 |   1    5   21   85  341  ...
   2 |   3   11   43  171  683  ...
   3 |   7   23   87  343  ...
   4 |  15   47  175  687  ...
   5 |  31   95  351  ...
   6 |  63  191  703  ...
   7 | 127  383  ...
   8 | 255  767  ...
   9 | 511  ...
   ...
The first row of the table is A020988. The columns of the table are obtained by repeatedly applying the transformation m -> 2*m + 1 to the entries in the first row.
2) Alternatively, this sequence can be read from Table 2 below by starting with a number on the top row and moving in a series of 'knight moves' (1 down and 2 to the left) through the table as far as you can, before returning to the next number in the top row and repeating the process. For example, starting at 10 in the first row we move 10 -> 11 -> 15, then return to the top row at 21 and move 21 -> 23 -> 31, before returning to the top row at 42 and so on.
........................................................
. Table 2. (4^n)*ceiling(2^k/3) - 1 for n >= 0, k >= 1 .
........................................................
n\k|    1    2    3    4     5     6     7     8    9   10
---+---------------------------------------------------------
  0|    0    1    2    5    10    21    42    85  170  682...
  1|    3    7   11   23    43    87   171   343  683  ...
  2|   15   31   47   95   175   351   687  1375  ...
  3|   63  127  191  383   703  1407  2751  5503  ...
  4|  255  511  767 1535  2815  5631 11007 22015  ...
  5| 1023 2047 3071 6143 11263 22527 44031 88063  ...
  6| 4095 ...
  ...
The first row of the table is A000975. The columns of the table are obtained by repeatedly applying the transformation m -> 4*m + 3 to the entries in the first row.

Crossrefs

Cf. A000975.
Other odd binomials: A263132 (4*m-1,m), A002450 (4*m+1,m), A020988 (4*m+2,m), A080674 (4*m+4,m), A118113 (3*m-2,m), A003714 (3*m,m).

Programs

  • Magma
    [n: n in [0..6000] | Binomial(4*n+3, n) mod 2 eq 1]; // Vincenzo Librandi, Oct 12 2015
  • Maple
    for n from 1 to 4096 do if mod(binomial(4*n+3, n), 2) = 1 then print(n) end if end do;
  • Mathematica
    Select[Range[0,5600],OddQ[Binomial[4#+3,#]]&] (* Harvey P. Dale, Apr 15 2019 *)
  • PARI
    for(n=0, 1e4, if (binomial(4*n+3, n) % 2 == 1, print1(n", "))) \\ Altug Alkan, Oct 11 2015
  • PARI
    a(n) = my(r,s=sqrtint(4*n-3,&r)); (1<Kevin Ryde, Jul 06 2025
  • Python
    A263133_list = [m for m in range(10**6) if not ~(4*m+3) & m] # Chai Wah Wu, Feb 07 2016

Formula

a(n) = A263132(n) - 1.
m is a term if and only if m AND NOT (4*m+3) = 0 where AND and NOT are bitwise operators. - Chai Wah Wu, Feb 07 2016
a(n) = (2^A000267(n) + 2^A384688(n))/3 - 1, for n >= 1. - Kevin Ryde, Jul 06 2025

Extensions

More terms from Vincenzo Librandi, Oct 12 2015

A266753 Decimal representation of the n-th iteration of the "Rule 163" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 18, 74, 298, 1194, 4778, 19114, 76458, 305834, 1223338, 4893354, 19573418, 78293674, 313174698, 1252698794, 5010795178, 20043180714, 80172722858, 320690891434, 1282763565738, 5131054262954, 20524217051818, 82096868207274, 328387472829098
Offset: 0

Views

Author

Robert Price, Jan 17 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A263919, A266752.

Programs

  • Mathematica
    rule=163; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 20 2016 and Apr 20 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>2.
G.f.: (1-x+2*x^2) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = (7*4^n - 4)/6 for n>1. - Colin Barker, Nov 25 2016 and Apr 20 2019
a(n) = 4*a(n-1) + 2, n>1, conjectured. - Yosu Yurramendi, Jan 22 2017
a(n) = 2*A020988(n) - A020988(n-1) = A020988(n) + 2^(2n-1) for n > 0, conjectured. - Yosu Yurramendi, Jan 24 2017 [n range correction - Karl V. Keller, Jr., May 07 2022]
a(n) = A072197(n-1) + A002450(n), n > 0, conjectured. - Yosu Yurramendi, Mar 03 2017

Extensions

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - N. J. A. Sloane, Jun 13 2022

A342700 For any number n with binary expansion (b(1), b(2), ..., b(k)), the binary expansion of a(n) is (1-floor((b(k)+b(1)+b(2))/2), 1-floor((b(1)+b(2)+b(3))/2), ..., 1-floor((b(k-1)+b(k)+b(1))/2)).

Original entry on oeis.org

0, 0, 2, 0, 7, 0, 0, 0, 15, 6, 10, 0, 3, 0, 0, 0, 31, 14, 30, 12, 23, 4, 16, 0, 7, 6, 2, 0, 3, 0, 0, 0, 63, 30, 62, 28, 63, 28, 56, 24, 47, 14, 42, 8, 35, 0, 32, 0, 15, 14, 14, 12, 7, 4, 0, 0, 7, 6, 2, 0, 3, 0, 0, 0, 127, 62, 126, 60, 127, 60, 120, 56, 127, 62
Offset: 0

Views

Author

Rémy Sigrist, Mar 18 2021

Keywords

Comments

This sequence is a variant of A342698; here the value of the k-th bit of a(n) is the less frequent value in the bit triple centered around the k-th bit of n.

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     2      10         10
   3     0      11          0
   4     7     100        111
   5     0     101          0
   6     0     110          0
   7     0     111          0
   8    15    1000       1111
   9     6    1001        110
  10    10    1010       1010
  11     0    1011          0
  12     3    1100         11
  13     0    1101          0
  14     0    1110          0
  15     0    1111          0

Links

Crossrefs

Cf. A003817, A020988 (fixed points), A342698.

Programs

  • PARI
    a(n) = my (w=#binary(n)); sum(k=0, w-1, ((bittest(n, (k-1)%w)+bittest(n, k%w)+bittest(n, (k+1)%w))<=1) * 2^k)

Formula

a(n) + A342698(n) = A003817(n).
a(n) = n iff n belongs to A020988.

A125725 Numbers whose base-7 representation is 222....2.

Original entry on oeis.org

0, 2, 16, 114, 800, 5602, 39216, 274514, 1921600, 13451202, 94158416, 659108914, 4613762400, 32296336802, 226074357616, 1582520503314, 11077643523200, 77543504662402, 542804532636816, 3799631728457714, 26597422099204000
Offset: 1

Views

Author

Zerinvary Lajos, Feb 02 2007

Keywords

Examples

			base 7.......decimal
0..................0
2..................2
22................16
222..............114
2222.............800
22222...........5602
222222.........39216
2222222.......274514
22222222.....1921600
222222222...13451202
etc...........etc.

Links

Crossrefs

Cf. A007310, A023000, A047522, A165759.
Cf. also A002276, A005610, A020988, A024023, A125831, A125835, A125857 for related or similarly constructed sequences.

Programs

  • GAP
    List([1..25], n-> (7^(n-1) -1)/3); # G. C. Greubel, May 23 2019
  • Magma
    [0] cat [n:n in [1..15000000]| Set(Intseq(n,7)) subset [2]]; // Marius A. Burtea, May 06 2019
  • Magma
    [(7^(n-1)-1)/3: n in [1..25]]; // Marius A. Burtea, May 06 2019
  • Maple
    seq(2*(7^n-1)/6, n=0..25);
  • Mathematica
    FromDigits[#,7]&/@Table[PadLeft[{2},n,2],{n,0,25}]  (* Harvey P. Dale, Apr 13 2011 *)
(7^(Range[25]-1) - 1)/3 (* G. C. Greubel, May 23 2019 *)
  • PARI
    vector(25, n, (7^(n-1)-1)/3) \\ Davis Smith, Apr 04 2019
  • Sage
    [(7^(n-1) -1)/3 for n in (1..25)] # G. C. Greubel, May 23 2019

Formula

a(n) = (7^(n-1) - 1)/3 = 2*A023000(n-1).
a(n) = 7*a(n-1) + 2, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 2*x^2 / ( (1-x)*(1-7*x) ). - R. J. Mathar, Sep 30 2013
From Davis Smith, Apr 04 2019: (Start)
A007310(a(n) + 1) = 7^(n - 1).
A047522(a(n + 1)) = -1*A165759(n). (End)
E.g.f.: (exp(7*x) - 7*exp(x) + 6)/21. - Stefano Spezia, Jan 12 2025

Extensions

Offset corrected by N. J. A. Sloane, Oct 02 2010

A193579 a(n) = 2*4^n + 7.

Original entry on oeis.org

9, 15, 39, 135, 519, 2055, 8199, 32775, 131079, 524295, 2097159, 8388615, 33554439, 134217735, 536870919, 2147483655, 8589934599, 34359738375, 137438953479, 549755813895, 2199023255559, 8796093022215, 35184372088839, 140737488355335, 562949953421319, 2251799813685255
Offset: 0

Views

Author

Brad Clardy, Sep 20 2011

Keywords

Comments

Bisection of A168415 (odd part).

Links

Crossrefs

Cf. A168415, A020988.

Programs

  • Magma
    [2*4^n + 7: n in [0..30]]; // Vincenzo Librandi, Sep 30 2011
  • Mathematica
    2*4^Range[0,30]+7 (* or *) LinearRecurrence[{5,-4},{9,15},30] (* Harvey P. Dale, Jun 13 2020 *)
  • PARI
    a(n) = 2*4^n+7 \\ Felix Fröhlich, Nov 07 2018
  • PARI
    Vec(3*(3 - 10*x)/((1 - x)*(1 - 4*x)) + O(x^20)) \\ Felix Fröhlich, Nov 07 2018

Formula

a(n) = 2^(2n + 1) + 7 = 3*(A020988(n) + 3).
From Bruno Berselli, Sep 20 2011: (Start)
G.f.: 3*(3 - 10*x)/((1 - x)*(1 - 4*x)).
a(n) = A085688(A016969(n)). (End)
E.g.f.: 7*exp(x) + 2*exp(4*x). - Franck Maminirina Ramaharo, Nov 07 2018

A295521 a(n) = (1/n) * Sum_{d|n} mu(n/d)*(4^d - 3^d - 2^d + 1).

Original entry on oeis.org

0, 2, 10, 39, 150, 545, 2010, 7320, 26880, 98775, 365010, 1353185, 5038950, 18830145, 70623958, 265737270, 1002976350, 3796197160, 14406059010, 54801140307, 208932573650, 798218035245, 3055417070010, 11716354754030, 45002103387120, 173117601112575
Offset: 1

Views

Author

Seiichi Manyama, Nov 23 2017

Keywords

Links

Crossrefs

Cf. A001037, A008683, A020988, A027376, A027377.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#] * (4^# - 3^# - 2^# + 1) &] / n; Array[a, 26] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    {a(n) = sumdiv(n, d, moebius(n/d)*(4^d-3^d-2^d+1))/n}

Formula

a(n) = A027377(n) - A027376(n) - A001037(n) for n > 1.

A321358 a(n) = (2*4^n + 7)/3.

Original entry on oeis.org

3, 5, 13, 45, 173, 685, 2733, 10925, 43693, 174765, 699053, 2796205, 11184813, 44739245, 178956973, 715827885, 2863311533, 11453246125, 45812984493, 183251937965, 733007751853, 2932031007405, 11728124029613, 46912496118445, 187649984473773, 750599937895085, 3002399751580333
Offset: 0

Views

Author

Paul Curtz, Nov 07 2018

Keywords

Comments

Difference table:
3, 5, 13, 45, 173, 685, 2733, ... (this sequence)
2, 8, 32, 128, 512, 2048, 8192, ... A004171
6, 24, 96, 384, 1536, 6144, 24576, ... A002023

Links

Crossrefs

Cf. A010701, A010727, A020988, A083594, A002023, A004171, A155980, A171382, A193579.

Programs

  • Mathematica
    a[n_]:= (2*4^n + 7)/3; Array[a, 20, 0] (* or *)
CoefficientList[Series[1/3 (7 E^x + 2 E^(4 x)), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 10 2018 *)
  • PARI
    a(n) = (2*4^n + 7)/3; \\ Michel Marcus, Nov 08 2018
  • PARI
    Vec((3 - 10*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 10 2018

Formula

O.g.f.: (3 - 10*x) / ((1 - x)*(1 - 4*x)). - Colin Barker, Nov 10 2018
E.g.f.: (1/3)*(7*exp(x) + 2*exp(4*x)). - Stefano Spezia, Nov 10 2018
a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 3, a(1) = 5.
a(n) = 4*a(n-1) - 7, a(0) = 3.
a(n) = (2/3)*(4^n-1)/3 + 3.
a(n) = A171382(2*n) = A155980(2*n+2).
a(n) = A193579(n)/3.
a(n) = A007583(n) + 2 = A001045(2*n+1) + 2.

Extensions

More terms from Michel Marcus, Nov 08 2018
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