A004171 -id:A004171 - cp's OEIS frontend

A303421 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 252, 128, 16, 32, 512, 1985, 1988, 512, 32, 64, 2048, 15647, 30897, 15684, 2048, 64, 128, 8192, 123337, 480953, 480960, 123732, 8192, 128, 256, 32768, 972168, 7486281, 14783632, 7486369, 976132, 32768, 256, 512, 131072

Offset: 1

R. H. Hardin, Apr 23 2018

Table starts
...1......2........4...........8.............16...............32
...2......8.......32.........128............512.............2048
...4.....32......252........1985..........15647...........123337
...8....128.....1988.......30897.........480953..........7486281
..16....512....15684......480960.......14783632........454377792
..32...2048...123732.....7486369......454381369......27575294129
..64...8192...976132...116529645....13965759339....1673515027797
.128..32768..7700788..1813851698...429248347970..101563813268522
.256.131072.60752164.28233652317.13193275586412.6163796529251277

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..1. .0..0..0..1
..0..1..0..0. .1..1..1..0. .0..1..0..1. .1..0..1..1. .0..0..0..1
..1..1..1..0. .0..0..1..0. .0..1..0..1. .0..0..0..0. .1..0..0..0
..0..0..1..0. .0..1..0..1. .0..0..0..1. .1..1..1..0. .0..1..0..1
..1..0..0..0. .1..1..0..0. .0..1..1..0. .0..0..0..0. .0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..311

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).
Row 1 is A000079(n-1).
Row 2 is A004171(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 7*a(n-1) +6*a(n-2) +8*a(n-3)
k=4: a(n) = 14*a(n-1) +21*a(n-2) +54*a(n-3) -23*a(n-4) -18*a(n-5) -20*a(n-6)
k=5: [order 16]
k=6: [order 31]
k=7: [order 58]
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 4*a(n-1)
n=3: a(n) = 7*a(n-1) +4*a(n-2) +21*a(n-3) +18*a(n-4)
n=4: [order 9]
n=5: [order 19]
n=6: [order 44]

A303456 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 232, 128, 16, 32, 512, 1690, 1696, 512, 32, 64, 2048, 12340, 22756, 12408, 2048, 64, 128, 8192, 90112, 306448, 306767, 90800, 8192, 128, 256, 32768, 658204, 4129588, 7626768, 4136339, 664512, 32768, 256, 512, 131072

Offset: 1

R. H. Hardin, Apr 24 2018

Table starts
...1......2........4...........8............16..............32
...2......8.......32.........128...........512............2048
...4.....32......232........1690.........12340...........90112
...8....128.....1696.......22756........306448.........4129588
..16....512....12408......306767.......7626768.......189848373
..32...2048....90800.....4136339.....189837638......8727953509
..64...8192...664512....55781418....4726484016....401405461699
.128..32768..4863312...752277525..117683035940..18461936404456
.256.131072.35593024.10145443043.2930192820802.849140799884830

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..0..1. .0..0..1..0. .0..0..0..0. .0..0..0..1
..1..1..1..0. .1..1..0..1. .0..1..0..1. .0..1..0..0. .0..0..1..0
..0..0..1..1. .0..1..0..1. .1..0..0..0. .1..1..1..0. .0..0..1..1
..0..1..1..0. .0..1..0..0. .1..1..1..1. .0..1..1..0. .0..0..0..1
..0..0..0..1. .0..0..1..1. .1..1..0..1. .0..0..1..1. .0..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..180

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).
Row 1 is A000079(n-1).
Row 2 is A004171(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 8*a(n-1) -4*a(n-2) -2*a(n-3) -36*a(n-4) -16*a(n-5)
k=4: [order 15]
k=5: [order 47]
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 4*a(n-1)
n=3: a(n) = 8*a(n-1) -40*a(n-3) +20*a(n-4) +8*a(n-5) -3*a(n-6) +32*a(n-7) for n>8
n=4: [order 15] for n>16
n=5: [order 68] for n>69

A303469 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 29, 32, 8, 16, 105, 170, 128, 16, 32, 384, 948, 1033, 512, 32, 64, 1405, 5237, 9110, 6369, 2048, 64, 128, 5135, 29009, 79377, 89371, 39098, 8192, 128, 256, 18766, 160590, 692636, 1243692, 872026, 240109, 32768, 256, 512, 68589, 888993

Offset: 1

R. H. Hardin, Apr 24 2018

Table starts
...1......2.......4.........8..........16............32..............64
...2......8......29.......105.........384..........1405............5135
...4.....32.....170.......948........5237.........29009..........160590
...8....128....1033......9110.......79377........692636.........6051850
..16....512....6369.....89371.....1243692......17247543.......239939422
..32...2048...39098....872026....19374638.....427097893......9459086839
..64...8192..240109...8511918...302023677...10589284528....373571747330
.128..32768.1476141..83188773..4716032889..263136262937..14793218857797
.256.131072.9071642.812770434.73624207904.6538180319944.585806882371397

			Some solutions for n=5 k=4
..0..1..0..0. .0..0..1..0. .0..0..1..0. .0..0..0..1. .0..0..1..0
..1..1..1..1. .1..0..1..1. .1..1..1..0. .1..0..1..0. .1..0..0..0
..0..0..0..0. .0..1..0..0. .1..1..1..1. .0..0..1..1. .0..1..1..0
..0..1..1..0. .0..1..0..1. .0..0..1..0. .0..0..0..0. .0..1..1..0
..1..0..0..0. .0..0..1..0. .0..0..1..0. .1..1..1..1. .1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..180

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).
Row 1 is A000079(n-1).
Row 2 is A302266.

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 7*a(n-1) -5*a(n-2) +20*a(n-3) -144*a(n-4) +72*a(n-5) for n>6
k=4: [order 20] for n>21
k=5: [order 93] for n>94
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 3*a(n-1) +a(n-2) +4*a(n-3) +4*a(n-4)
n=3: [order 14] for n>15
n=4: [order 49] for n>50

A102900 a(n) = 3a(n-1) + 4a(n-2), a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 7, 25, 103, 409, 1639, 6553, 26215, 104857, 419431, 1677721, 6710887, 26843545, 107374183, 429496729, 1717986919, 6871947673, 27487790695, 109951162777, 439804651111, 1759218604441, 7036874417767, 28147497671065

Offset: 0

Paul Barry, Jan 17 2005

Binomial transform of A102901.

Hankel transform is = 1,6,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A001045, A004171, A046717, A086901, A102901, A247666 (which appears to be the run length transform of this sequence).

a102900 n = a102900_list !! n
a102900_list = 1 : 1 : zipWith (+)
               (map (* 4) a102900_list) (map (* 3) $ tail a102900_list)
-- Reinhard Zumkeller, Feb 13 2015

[n le 2 select 1 else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 28 2015

a[n_]:=(MatrixPower[{{2,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{3, 4}, {1, 1}, 30] (* Vincenzo Librandi, Dec 28 2015 *)

a(n)=([0,1; 4,3]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Mar 28 2016

A102900=BinaryRecurrenceSequence(3,4,1,1)
[A102900(n) for n in range(51)] # G. C. Greubel, Dec 09 2022

G.f.: (1-2*x)/(1-3*x-4*x^2).
a(n) = (2*4^n + 3*(-1)^n)/5.
a(n) = ceiling(4^n/5) + floor(4^n/5) = (ceiling(4^n/5))^2 - (floor(4^n/5))^2.
a(n) + a(n+1) = 2^(2*n+1) = A004171(n).
a(n) = Sum_{k=0..n} binomial(2*n-k, 2*k)*2^k. - Paul Barry, Jan 20 2005
a(n) = upper left term in the 2 X 2 matrix [1,3; 2,2]^n. - Gary W. Adamson, Mar 14 2008
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(8*4^k-3*(-1)^k)/(x*(8*4^k-3*(-1)^k) + (2*4^k+3*(-1)^k)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
a(n) = 2^(2*n-1) - a(n-1), a(1)=1. - Ben Paul Thurston, Dec 27 2015; corrected by Klaus Purath, Aug 02 2020
From Klaus Purath, Aug 02 2020: (Start)
a(n) = 4*a(n-1) + 3*(-1)^n.
a(n) = 6*4^(n-2) + a(n-2), n>=2. (End)

A103333 Number of closed walks on the graph of the (7,4) Hamming code.

Original entry on oeis.org

1, 3, 24, 192, 1536, 12288, 98304, 786432, 6291456, 50331648, 402653184, 3221225472, 25769803776, 206158430208, 1649267441664, 13194139533312, 105553116266496, 844424930131968, 6755399441055744, 54043195528445952, 432345564227567616

Offset: 0

Paul Barry, Jan 31 2005

Counts closed walks of length 2n at the degree 3 node of the graph of the (7,4) Hamming code. With interpolated zeros, counts paths of length n at this node.
a(n+1) = A157176(A016945(n)). - Reinhard Zumkeller, Feb 24 2009
For n>0: a(n) = A083713(n) - A083713(n-1). - Reinhard Zumkeller, Feb 22 2010

David J.C. Mackay, Information Theory, Inference and Learning Algorithms, CUP, 2003, p. 19

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A082412, A103334.

Cf. A000302, A004171. - Vincenzo Librandi, Jan 22 2009

seq((3*8^n+5*`if`(n=0,1,0))/8,n=0..20); # Nathaniel Johnston, Jun 26 2011

Join[{1},NestList[8#&,3,20]] (* Harvey P. Dale, Mar 02 2012 *)

G.f.: (1-5*x)/(1-8*x);
a(n) = (3*8^n + 5*0^n)/8.
a(n) = 8*a(n-1) for n > 0. - Harvey P. Dale, Mar 02 2012

A365808 Numbers k such that A163511(k) is a square.

Original entry on oeis.org

0, 2, 5, 8, 11, 17, 20, 23, 32, 35, 41, 44, 47, 65, 68, 71, 80, 83, 89, 92, 95, 128, 131, 137, 140, 143, 161, 164, 167, 176, 179, 185, 188, 191, 257, 260, 263, 272, 275, 281, 284, 287, 320, 323, 329, 332, 335, 353, 356, 359, 368, 371, 377, 380, 383, 512, 515, 521, 524, 527, 545, 548, 551, 560, 563, 569, 572, 575

Offset: 1

Antti Karttunen, Oct 01 2023

nonn

The sequence is defined inductively as:
 (a) it contains 0 and 2,
   and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 4*a(n) are also included as terms.
Because the inductive definition guarantees that all terms after 0 are of the form 3k+2 (A016789), and because for any n >= 0, n^2 == 0 or 1 (mod 3), (i.e., squares are in A032766), it follows that there are no squares in this sequence after the initial 0.

Index entries for sequences related to binary expansion of n

Cf. A000290, A010052, A032766, A163511, A365807 (characteristic function).
Positions of even terms in A365805.
Sequence A243071(n^2), n >= 1, sorted into ascending order.
Subsequences: A004171, A055010, A365809 (odd terms).
Subsequence of A016789 (after the initial 0).
Cf. also A277335, A365801, A365802, A365806.

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA365808v2(n) = issquare(A163511(n));

isA365808(n) = if(n<=2, !(n%2), if(n%2, isA365808((n-1)/2), if(n%4, 0, isA365808(n/4))));

from itertools import count, islice
def A365808_gen(): # generator of terms
    return map(lambda n:(3*(n+1)>>2)-1,filter(lambda n:n==1 or (n&3==3 and not '00' in bin(n)),count(1)))
A365808_list = list(islice(A365808_gen(),20)) # Chai Wah Wu, Feb 12 2025

A166920 a(n) = 2^n - (1 + (-1)^n)/2.

Original entry on oeis.org

0, 2, 3, 8, 15, 32, 63, 128, 255, 512, 1023, 2048, 4095, 8192, 16383, 32768, 65535, 131072, 262143, 524288, 1048575, 2097152, 4194303, 8388608, 16777215, 33554432, 67108863, 134217728, 268435455, 536870912, 1073741823, 2147483648, 4294967295

Offset: 0

Paul Curtz, Oct 23 2009

Partial sums of A014551. The inverse binomial transform yields a sequence 0,2,-1,5,-7,17,...: zero followed by a sign alternating A014551.
The table of a(n) plus higher order differences in successive rows shows A131577 on the main diagonal.
a(n) = 2^n when n is odd and 2^n-1 when n is even. - Wesley Ivan Hurt, Nov 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A004171, A014551, A024036, A131577, A168361.

a166920 n = a166920_list !! n
a166920_list = scanl (+) 0 a014551_list
-- Reinhard Zumkeller, Jan 02 2013

[2^n -(1+(-1)^n)/2: n in [0..30]]; // Vincenzo Librandi, May 16 2011

A166920:=n->2^n-(1+(-1)^n)/2; seq(A166920(n), n=0..50); # Wesley Ivan Hurt, Nov 15 2013

LinearRecurrence[{2,1,-2},{0,2,3},40] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=2^n-(1+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(2-x)/((1-x)*(1-2*x)*(1+x)).
a(n) = 2^n - (1+(-1)^n)/2.
a(2*n) = A024036(n); a(2*n+1) = A004171(n).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - 2*a(n) = A168361(n).
a(n) = A000225(n+1) - A051049(n) = A014551(n) - A168361(n).
E.g.f.: exp(2*x) - cosh(x). - G. C. Greubel, May 28 2016
a(n) = Sum_{k=1..n+1} Sum_{i=0..n+1} C(n-k,i). - Wesley Ivan Hurt, Sep 22 2017
a(n) = 2*A001045(n) + A000975(n-1) for n>0. - Yuchun Ji, Aug 30 2018

Edited and extended by R. J. Mathar, Mar 02 2010

A302010 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2 or 3 horizontally or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 240, 128, 16, 32, 512, 1808, 1808, 512, 32, 64, 2048, 13616, 25808, 13616, 2048, 64, 128, 8192, 102544, 369040, 368144, 102544, 8192, 128, 256, 32768, 772272, 5276816, 9989376, 5251712, 772272, 32768, 256, 512, 131072

Offset: 1

R. H. Hardin, Mar 30 2018

Table starts
...1......2........4...........8............16...............32
...2......8.......32.........128...........512.............2048
...4.....32......240........1808.........13616...........102544
...8....128.....1808.......25808........369040..........5276816
..16....512....13616......368144.......9989376........270990144
..32...2048...102544.....5251712.....270422672......13918667808
..64...8192...772272....74917424....7320574992.....714887543376
.128..32768..5816080..1068722240..198174358400...36717919842624
.256.131072.43801648.15245681888.5364752820144.1885898831169344

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..1. .0..0..0..0
..1..0..0..1. .1..0..1..0. .0..1..0..1. .1..1..0..0. .0..0..1..0
..0..1..0..0. .0..0..1..1. .1..1..1..0. .0..0..1..0. .0..0..0..1
..1..1..1..1. .1..1..0..0. .0..0..0..0. .0..1..0..0. .1..1..0..1
..0..1..0..1. .0..0..1..0. .0..1..0..0. .1..1..1..0. .0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..479

Column 1 and row 1 are A000079(n-1).
Column 2 and row 2 are A004171(n-1).
Column 3 and row 3 are A301779.

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 7*a(n-1) +4*a(n-2)
k=4: a(n) = 13*a(n-1) +18*a(n-2) +a(n-3) -4*a(n-4)
k=5: a(n) = 24*a(n-1) +82*a(n-2) +34*a(n-3) -90*a(n-4) -40*a(n-5) +37*a(n-6)
k=6: [order 10] for n>12
k=7: [order 17] for n>19
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 4*a(n-1)
n=3: a(n) = 7*a(n-1) +4*a(n-2)
n=4: a(n) = 13*a(n-1) +20*a(n-2) -16*a(n-3) -64*a(n-4) for n>6
n=5: [order 12] for n>15
n=6: [order 32] for n>36
n=7: [order 78] for n>83

A054755 Odd powers of primes of the form q = x^2 + 1 (A002496).

Original entry on oeis.org

2, 5, 8, 17, 32, 37, 101, 125, 128, 197, 257, 401, 512, 577, 677, 1297, 1601, 2048, 2917, 3125, 3137, 4357, 4913, 5477, 7057, 8101, 8192, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401

Offset: 1

Labos Elemer, Apr 25 2000

nonn

A002496 is a subset; the odd power exponent is 1.
From Bernard Schott, Mar 16 2019: (Start)
The terms of this sequence are exactly the integers with only one prime factor and whose Euler's totient is square, so this sequence is a subsequence of A039770. The primitive terms of this sequence are the primes of the form q = x^2 + 1, which are exactly in A002496.
Additionally, the terms of this sequence also have a square cototient, so this sequence is a subsequence of A063752 and A054754.
If q prime = x^2 + 1, phi(q) = x^2, phi(q^(2k+1)) = (x*q^k)^2, and cototient(q) = 1^2, cototient(q^(2k+1)) = (q^k)^2. (End)

			a(20) = 3125 = 5^5, q = 5 = 4^2+1 and Phi(3125) = 2500 = 50^2, cototient(3125) = 3125 - Phi(3125) = 625 = 25^2.

David A. Corneth, Table of n, a(n) for n = 1..18864 (terms <= 10^11)
Bernard Schott, Subfamilies and subsequences

Cf. A000010, A051953, A039770, A063752, A054754, A334745 (with 2 distinct prime factors), A306908 (with 3 distinct prime factors).

Subsequences: A002496 (primitive primes: m^2+1), A004171 (2^(2k+1)), A013710 (5^(2k+1)), A013722 (17^(2k+1)), A262786 (37^(2k+1)).

Select[Range[10^5], And[PrimeNu@ # == 1, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* Michael De Vlieger, Mar 31 2019 *)

isok(m) = (omega(m)==1) && issquare(eulerphi(m)); \\ Michel Marcus, Mar 16 2019

upto(n) = {my(res = List([2]), q); forstep(i = 2, sqrtint(n), 2, if(isprime(i^2 + 1), listput(res, i^2 + 1) ) ); q = #res; forstep(i = 3, logint(n, 2), 2, for(j = 1, q, c = res[j]^i; if(c <= n, listput(res, c) , next(2) ) ) ); listsort(res); res } \\ David A. Corneth, Mar 17 2019

A000010(a(n)) = (q^(2k))*(q-1) and A051953(a(n)) = q^(2k), where q = 1 + x^2 and is prime.

A132020 Decimal expansion of Product_{k>=0} (1 - 1/(2*4^k)).

Original entry on oeis.org

4, 1, 9, 4, 2, 2, 4, 4, 1, 7, 9, 5, 1, 0, 7, 5, 9, 7, 7, 0, 9, 9, 5, 6, 1, 0, 7, 7, 0, 2, 9, 7, 4, 2, 5, 2, 2, 3, 3, 9, 5, 3, 2, 3, 4, 3, 9, 2, 6, 6, 6, 7, 4, 9, 0, 8, 0, 4, 4, 9, 9, 1, 6, 6, 3, 1, 7, 7, 2, 0, 5, 0, 8, 7, 2, 7, 0, 9, 1, 9, 3, 9, 1, 0, 0, 2, 3, 2, 4, 5, 4, 7, 4, 2, 3, 8, 1, 9, 5, 5, 0, 2, 8, 5, 8

Offset: 0

Hieronymus Fischer, Aug 14 2007

This is the limiting probability that a large random symmetric binary matrix is nonsingular (cf. A086812, A048651). In other words, equals Lim_{n->oo} A086812(n)/A006125(n+1).- H. Tracy Hall, Sep 07 2024

			0.41942244179510759770995610770297425223395323439266674908044991663177...

John E. Cremona and R. W. K. Odoni, Some density results for negative Pell equations; an application of graph theory, Journal of the London Mathematical Society s2-39:1 (1989), pp. 16-28.

Cf. A004171, A048651, A098844, A067080, A132019, A132026, A132028, A100221, A000217, A081845.

evalf(1+sum((-1)^n*2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # Robert FERREOL, Feb 23 2020

RealDigits[ Product[1 - 1/(2*4^i), {i, 0, 175}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
RealDigits[QPochhammer[1/2, 1/4], 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)

prodinf(k=0,1-1.>>(2*k+1)) \\ Charles R Greathouse IV, Nov 16 2012

Equals lim inf_{n->oo} Product_{k=0..floor(log_4(n))} floor(n/4^k)*4^k/n.
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^((1/2)*(1+floor(log_4(n)))*floor(log_4(n))).
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^A000217(floor(log_4(n))).
Equals (1/2)*exp(-Sum_{n>0} (4^(-n)*(Sum_{k|n} 1/(k*2^k)))).
Equals lim inf_{n->oo} A132028(n)/A132028(n+1).
Equals Product_{k>0} (1-1/(2^k+1)). - Robert G. Wilson v, May 25 2011
From Robert FERREOL, Feb 23 2020: (Start)
Equals Product_{k>0} (1 + 1/2^k)^(-1) = 2/A081845.
Equals 1 + Sum_{n>=1} (-1)^n*2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). (End)
From Peter Bala, Jan 15 2021: (Start)
Constant C = Sum_{n >= 0} 2^n/Product_{k = 1..n} (1 - 4^k).
Faster converging series:
2*C = (1/2)*Sum_{n >= 0} 2^(-n)/Product_{k = 1..n} (1 - 4^k);
(2^4)*C = 7*Sum_{n >= 0} 2^(-3*n)/Product_{k = 1..n} (1 - 4^k);
(2^9)*C = 7*31*Sum_{n >= 0} 2^(-5*n)/Product_{k = 1..n} (1 - 4^k), and so on.
Slower converging series:
C = -Sum_{n >= 0} 2^(3*n)/Product_{k = 1..n} (1 - 4^k);
7*C = Sum_{n >= 0} 2^(5*n)/Product_{k = 1..n} (1 - 4^k);
7*31*C = -Sum_{n >= 0} 2^(7*n)/Product_{k = 1..n} (1 - 4^k), and so on. (End)
Equals Product_{n>=0} (1 - 1/A004171(n)). - Amiram Eldar, May 09 2023

Name corrected by Charles R Greathouse IV, Nov 16 2012

cp's OEIS Frontend

A303421 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

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A303456 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

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A303469 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

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A102900 a(n) = 3*a(n-1) + 4*a(n-2), a(0)=a(1)=1.

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A103333 Number of closed walks on the graph of the (7,4) Hamming code.

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A365808 Numbers k such that A163511(k) is a square.

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A166920 a(n) = 2^n - (1 + (-1)^n)/2.

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A102900 a(n) = 3a(n-1) + 4a(n-2), a(0)=a(1)=1.