A001611 -id:A001611 - cp's OEIS frontend

A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

0, 1, 2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139

Offset: 0

N. J. A. Sloane, R. K. Guy

nonn

Laatsch (1986) proved that for n >= 2, a(n) gives the smallest number of distinct prime factors in even numbers having an abundancy index > n.
The abundancy index of a number k is sigma(k)/k. - T. D. Noe, May 08 2006
The first differences of this sequence, A005347, begin the same as the Fibonacci sequence A000045. - T. D. Noe, May 08 2006
Equal to A256968 except for n = 2 and n = 3.  See comment in A256968. - Chai Wah Wu, Apr 17 2015

			The products Product_{i=1..k} prime(i)/(prime(i)-1) for k >= 0 start with 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 3.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 0..44
Richard K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Popular Computing (Calabasas, CA), Problem 182 (Suggested by Victor Meally), Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

A001611 is similar but strictly different.

Cf. A000720, A001611, A005580, A060753, A023199, A038110, A072986, A091440, A256968.

(* For speed and accuracy, the second Mathematica program uses 30-digit real numbers and interval arithmetic. *)
prod=1; k=0; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k, {n,0,25}] (* T. D. Noe, May 08 2006 *)
prod=Interval[1]; k=0; Table[While[Max[prod]<=n, k++; p=Prime[k]; prod=N[prod*p/(p-1),30]]; If[Min[prod]>n, k, "too few digits"], {n,0,38}]

a(n)=my(s=1,k); forprime(p=2,, s*=p/(p-1); k++; if(s>n, return(k))) \\ Charles R Greathouse IV, Aug 20 2015

from sympy import nextprime
def a_list(upto: int) -> list[int]:
    L: list[int] = [0]
    count = 1; bn = 1; bd = 1; p = 2
    for k in range(1, upto + 1):
        bn *= p
        bd *= p - 1
        while bn > count * bd:
            L.append(k)
            count += 1
        p = nextprime(p)
    return L
print(a_list(1000))  # Chai Wah Wu, Apr 17 2015, adapted by Peter Luschny, Jan 25 2025

a(n) = smallest k such that A002110(k)/A005867(k) > n. - Artur Jasinski, Nov 06 2008

a(n) = PrimePi(A091440(n)) = A000720(A091440(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

Edited by T. D. Noe, May 08 2006
a(26) added by T. D. Noe, Sep 18 2008
Typo corrected by Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
a(27)-a(36) from Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
Comment corrected by T. D. Noe, Apr 04 2010
a(37)-a(39) from T. D. Noe, Nov 16 2010
Edited and terms a(0)-a(1) prepended by Max Alekseyev, Jan 25 2025

A208698 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 1 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 14, 81, 98, 100, 16, 22, 196, 271, 358, 256, 26, 35, 484, 844, 1309, 1152, 676, 42, 56, 1225, 2706, 5524, 5371, 3910, 1764, 68, 90, 3136, 8977, 24086, 30160, 23637, 12994, 4624, 110, 145, 8100, 30168, 109599, 177488, 177872

Offset: 1

R. H. Hardin Mar 01 2012

Table starts
..2....4.....6......9......14.......22.........35..........56...........90
..4...16....36.....81.....196......484.......1225........3136.........8100
..6...36....98....271.....844.....2706.......8977.......30168.......102384
.10..100...358...1309....5524....24086.....109599......506870......2376964
.16..256..1152...5371...30160...177488....1103081.....6990922.....45002090
.26..676..3910..23637..177872..1415508...12014735...104356568....923279444
.42.1764.12994.101069.1016258.10934750..126827983..1510509752..18362140414
.68.4624.43596.438103.5893862.85697362.1356513169.22125222702.369223577680

			Some solutions for n=4 k=3
..0..1..0....0..0..0....1..0..1....0..0..0....1..1..0....1..1..1....0..0..0
..0..0..0....0..0..0....0..0..0....0..1..1....0..0..0....1..1..1....0..0..0
..1..0..1....1..0..1....0..0..0....0..1..1....0..0..0....0..1..0....1..1..0
..1..0..1....1..0..1....0..1..1....0..1..0....0..1..1....0..1..0....1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..418

Column 1 is A006355(n+2)
Column 2 is A206981
Column 3 is A207462
Column 4 is A207914
Row 1 is A001611(n+2)
Row 2 is A207436
Row 3 is A207939

A117591 a(n) = 2^n + Fibonacci(n).

Original entry on oeis.org

1, 3, 5, 10, 19, 37, 72, 141, 277, 546, 1079, 2137, 4240, 8425, 16761, 33378, 66523, 132669, 264728, 528469, 1055341, 2108098, 4212015, 8417265, 16823584, 33629457, 67230257, 134414146, 268753267, 537385141, 1074573864, 2148829917

Offset: 0

Franklin T. Adams-Watters, Apr 04 2006

a(3n) is even if n>0. - Robert G. Wilson v, Sep 06 2002

3 divides a(8n+1) and a(8n-1). - Enrique Pérez Herrero, Dec 29 2010

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000045, A000079, A001611, A074824, A099036, A212262.

a117591 n = a117591_list !! n
a117591_list = zipWith (+) a000079_list a000045_list
-- Reinhard Zumkeller, Aug 15 2013

[2^n+Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Nov 02 2014

Table[f=Fibonacci[n];2^n+f,{n,1,40,1}] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
CoefficientList[Series[(1-3x^2)/((1-x-x^2)(1-2x)), {x,0,35}], x] (* Vincenzo Librandi, Nov 02 2014 *)

a(n)=2^n + fibonacci(n) \\ Charles R Greathouse IV, Oct 07 2016

[2^n +fibonacci(n) for n in (0..40)] # G. C. Greubel, Jul 05 2021

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

a(n) = A000079(n+1) - A099036(n) = A099036(n) + 2 * A000045(n). - Reinhard Zumkeller, Aug 15 2013

A166470 a(n) = 2^F(n+1)*3^F(n), where F(n) is the n-th Fibonacci number, A000045(n).

Original entry on oeis.org

2, 6, 12, 72, 864, 62208, 53747712, 3343537668096, 179707499645975396352, 600858794305667322270155425185792, 107978831564966913814384922944738457859243070439030784

Offset: 0

Matthew Vandermast, Nov 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..16

Subsequence of A025610 and hence of A003586 and A025487.

Cf. A000045, A000301, A002110, A010098, A115033, A166469, A174666, A230900.

[2^Fibonacci(n+1)*3^Fibonacci(n): n in [0..14]]; // G. C. Greubel, Jul 29 2024

3^First[#] 2^Last[#]&/@Partition[Fibonacci[Range[0,12]],2,1] (* Harvey P. Dale, Aug 20 2012 *)

a(n)=2^fibonacci(n+1)*3^fibonacci(n) \\ Charles R Greathouse IV, Sep 19 2022

[2^fibonacci(n+1)*3^fibonacci(n) for n in range(15)] # G. C. Greubel, Jul 29 2024

a(n) = A000301(n+1)*A010098(n).
For n > 1, a(n) = a(n-1)*a(n-2).
For m > 1, n > 1, A166469(A002110(m)*(a(n)^k)/12) = k*Fibonacci(m+n).
A166469(a(n)) = Fibonacci(n+2) + 1 = A001611(n+2).
a(n) = 2 * A174666(n+1). - Alois P. Heinz, Sep 16 2022
a(n) = 2^(Fibonacci(n+1) + c*Fibonacci(n)), with c=log_2(3). Cf. A000301 (c=1) & A010098 (c=2). - Andrea Pinos, Sep 29 2022
a(n) = A115033(2*n+1). - David Radcliffe, May 31 2025

Typo corrected by Matthew Vandermast, Nov 07 2009

A248156 Inverse Riordan triangle of A106513: Riordan ((1 - 2*x^2 )/(1 + x), x/(1+x)).

Original entry on oeis.org

1, -2, 1, 1, -3, 1, 0, 4, -4, 1, -1, -4, 8, -5, 1, 2, 3, -12, 13, -6, 1, -3, -1, 15, -25, 19, -7, 1, 4, -2, -16, 40, -44, 26, -8, 1, -5, 6, 14, -56, 84, -70, 34, -9, 1, 6, -11, -8, 70, -140, 154, -104, 43, -10, 1, -7, 17, -3, -78, 210, -294, 258, -147, 53, -11, 1, 8, -24, 20, 75, -288, 504, -552, 405, -200, 64, -12, 1

Offset: 0

Wolfdieter Lang, Oct 05 2014

Row sums have o.g.f. (1 - 2*x)/(1 + x): [1, -1, repeat(-1, 1)].

			The triangle T(n,k) begins:
  n\k  0   1   2   3    4    5    6    7    8    9
  0:   1
  1:  -2   1
  2:   1  -3   1
  3:   0   4  -4   1
  4:  -1  -4   8  -5    1
  5:   2   3 -12  13   -6    1
  6:  -3  -1  15 -25   19   -7    1
  7:   4  -2 -16  40  -44   26   -8    1
  8:  -5   6  14 -56   84  -70   34   -9    1
  9:   6 -11  -8  70 -140  154 -104   43  -10    1
  ...
For more rows see the link.
Recurrence from A-sequence: T(5,2) = T(4,1) - T(4,2) = -4 - 8 = -12.
Recurrence from the Z-sequence: T(5,0) = -(2*(-1) + 3*(-4) + 7*8 + 17*(-5) + 41*1) = 2.
Standard recurrence for T(n,0): T(3,0) = -2*T(2,0) - T(1,0) = -2*1 - (-2) = 0.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wolfdieter Lang, First 13 rows of the triangle

Cf. A006232, A083318, A106513, A248157, A248158, A248159, A248160, A248161.
Columns: A248157 (k=0), A248158 (k=1), A248159 (k=2), A248160 (k=3).
Diagonals: A000012 (k=n), A022958(n+3) (k=n-1), -A034856(n-1) (k=n-2), A000297(n-4) (k=n-3), A014309(n-3) (k=n-4).
Sums: (-1)^n*A001611(n) (diagonal), (-1)^n*A083318(n) (alternating sign row).

function T(n,k) // T = A248156
  if k eq n then return 1;
  elif k eq 0 then return (-1)^n*(3-n);
  else return T(n-1,k-1) - T(n-1,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2025

T[n_, k_] := SeriesCoefficient[x^k*(1 - 2*x^2)/(1 + x)^(k + 2), {x, 0, n}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 09 2014 *)
T[n_, k_]:= T[n, k]= If[k==n,1, If[k==0,(-1)^n*(3-n), T[n-1,k-1]-T[n-1,k]]];
Table[T[n,k], {n,0,25}, {k,0,n}]//Flatten (* G. C. Greubel, May 27 2025 *)

def T(n,k): # T = A248156
    if (k==n): return 1
    elif (k==0): return (-1)^n*(3-n)
    else: return T(n-1,k-1) - T(n-1,k)
print(flatten([[T(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, May 27 2025

O.g.f. row polynomials R(n,x) = Sum_{k=0..n} T(n,k)*x^k = [(-z)^n] (1 - 2*z^2)/( (1 + z)*(1 + (1-x)*z)).
O.g.f. column m: x^m*(1 - 2*x^2)/(1 + x)^(m+2), m >= 0.
The A-sequence is [1, -1], implying the recurrence T(n,k) = T(n-1, k-1) - T(n-1, k), n >= k > = 1.
The Z-sequence is -[2, 3, 7, 17, 41, 99, 239, 577, 1393, ...] = A248161, implying the recurrence T(n, 0) = Sum_{k=0..n-1} T(n-1,k)*Z(k). See the W. Lang link under A006232 for Riordan A- and Z-sequences.
The standard recurrence for the sequence for column k=0 is T(0,0) = 1 and T(n,0) = -2*T(n-1,0) - T(n-2,0), n >= 3, with T(1,0) = -2 and T(2,0) = 1.
From G. C. Greubel, May 27 2025: (Start)
Sum_{k=0..n} T(n, k) = (-1)^(n+1) + 2*([n=0] - [n=1]).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = the repeated pattern of [1, -2, 0, 3, -4, 2]. (End)

A301662 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1 or 4 horizontally or vertically adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 4, 4, 5, 8, 6, 6, 6, 8, 13, 9, 9, 9, 9, 13, 21, 14, 15, 13, 15, 14, 21, 34, 22, 26, 24, 24, 26, 22, 34, 55, 35, 46, 45, 46, 45, 46, 35, 55, 89, 56, 83, 89, 99, 99, 89, 83, 56, 89, 144, 90, 151, 182, 223, 254, 223, 182, 151, 90, 144, 233, 145, 276, 373, 528, 696

Offset: 1

R. H. Hardin, Mar 25 2018

Table starts
..1..2...3...5....8...13....21.....34......55......89......144.......233
..2..3...4...6....9...14....22.....35......56......90......145.......234
..3..4...6...9...15...26....46.....83.....151.....276......506.......929
..5..6...9..13...24...45....89....182.....373.....773.....1604......3333
..8..9..15..24...46...99...223....528....1268....3085.....7557.....18564
.13.14..26..45...99..254...696...2013....5953...17878....54126....164525
.21.22..46..89..223..696..2335...8380...31001..116558...442314...1686894
.34.35..83.182..528.2013..8380..37679..175638..833737..4001569..19319239
.55.56.151.373.1268.5953.31001.175638.1036686.6245327.38085186.233683529

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

R. H. Hardin, Table of n, a(n) for n = 1..544

Column 1 is A000045(n+1).

Column 2 is A001611(n+1).

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) -a(n-3)
k=3: a(n) = 2*a(n-1) -a(n-4)
k=4: a(n) = 2*a(n-1) +a(n-2) -a(n-3) -2*a(n-4) +a(n-5)
k=5: [order 11]
k=6: [order 18]
k=7: [order 31]

A020706 Pisot sequences L(4,6), E(4,6).

Original entry on oeis.org

4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142

Offset: 0

David W. Wilson

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

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

Subsequence of A001611, A048577. See A008776 for definitions of Pisot sequences.

I:=[4, 6, 9]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Apr 20 2012

CoefficientList[Series[(4-2*x-3*x^2)/(1-x)/(1-x-x^2),{x,0,40}],x](* Vincenzo Librandi, Apr 20 2012 *)

a(n) = Fib(n+4)+1 = A000045(n+4)+1.
a(n) = 2a(n-1) - a(n-3).
G.f.: (4-2*x-3*x^2)/(1-x)/(1-x-x^2). - Colin Barker, Feb 21 2012

A208007 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 1 1 and 1 1 0 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 14, 81, 92, 81, 14, 22, 196, 221, 221, 196, 22, 35, 484, 618, 536, 618, 484, 35, 56, 1225, 1690, 1711, 1711, 1690, 1225, 56, 90, 3136, 4861, 4993, 7016, 4993, 4861, 3136, 90, 145, 8100, 13900, 16742, 24512, 24512, 16742, 13900

Offset: 1

R. H. Hardin Feb 22 2012

Table starts
..2....4.....6.....9.....14......22.......35........56........90........145
..4...16....36....81....196.....484.....1225......3136......8100......21025
..6...36....92...221....618....1690.....4861.....13900.....40452.....117717
..9...81...221...536...1711....4993....16742.....53411....182247.....608142
.14..196...618..1711...7016...24512...106503....411848...1787438....7238503
.22..484..1690..4993..24512...90232...486443...2001968..10846870...47911153
.35.1225..4861.16742.106503..486443..3569728..18874239.139219755..803098636
.56.3136.13900.53411.411848.2001968.18874239.101519408.989521860.5706548155

			Some solutions for n=4 k=3
..0..1..0....0..0..0....0..1..0....1..0..1....1..0..1....0..0..0....0..1..0
..0..0..0....0..1..1....1..1..0....0..0..0....0..0..0....0..1..1....0..1..0
..0..1..1....0..0..0....0..1..0....1..1..1....0..1..1....0..0..0....0..1..1
..0..0..0....0..1..0....1..1..1....0..0..0....0..0..0....1..0..1....1..1..0

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

Column 1 is A001611(n+2)
Column 2 is A207436
Column 3 is A207559

A001588 a(n) = a(n-1) + a(n-2) - 1.

Original entry on oeis.org

1, 3, 3, 5, 7, 11, 17, 27, 43, 69, 111, 179, 289, 467, 755, 1221, 1975, 3195, 5169, 8363, 13531, 21893, 35423, 57315, 92737, 150051, 242787, 392837, 635623, 1028459, 1664081, 2692539, 4356619, 7049157, 11405775, 18454931, 29860705, 48315635

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.
J. A. H. Hunter and F. D. Parker, Problem B-100, Fib. Quart., 5 (1967), p. 288.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A001611, A001906.

A001588:=-(-1-z+3*z**2)/(z-1)/(z**2+z-1); # conjectured by Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(n-2) + fibonacci(n+1) + 1, n = 0..35); # Zerinvary Lajos, Feb 01 2008

Fibonacci[Range[0,100]]*2+1 (* Vladimir Joseph Stephan Orlovsky, Mar 19 2010 *)
nxt[{a_,b_}]:={b,a+b-1}; NestList[nxt,{1,3},40][[;;,1]] (* Harvey P. Dale, Jun 13 2025 *)

a(n)=2*fibonacci(n)+1 \\ Charles R Greathouse IV, Apr 06 2016

From Henry Bottomley, Feb 20 2001: (Start)
a(n) = 2*Fibonacci(n) + 1 = A000045(n) + A001611(n).
G.f.: (1+x-3x^2)/(1-2*x+x^3). (End)
If n>=4, a(n) = floor(Phi*a(n-1)); Phi = (1 + sqrt(5))/2. - Philippe Deléham, Aug 08 2003
a(n) = F(n-2) + F(n+1) + 1, n >= 0 (where F(n) is the n-th Fibonacci number). - Zerinvary Lajos, Feb 01 2008

A048577 Pisot sequence L(3,4).

Original entry on oeis.org

3, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142

Offset: 0

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Subsequence of A001611. See A008776 for definitions of Pisot sequences.

[Fibonacci(n+3)+1: n in [0..40]]; // Vincenzo Librandi, Apr 27 2012

CoefficientList[Series[(3-2*x-2*x^2)/((1-x)*(1-x-x^2)),{x,0,40}],x] (* Vincenzo Librandi, Apr 27 2012 *)

a(n)=fibonacci(n+3)+1 \\ Charles R Greathouse IV, Apr 16 2012

a(n) = Fib(n+3)+1. a(n) = 2a(n-1) - a(n-3).
a(n) = A020706(n-1), n>0. [From R. J. Mathar, Oct 15 2008]
G.f.: (3-2*x-2*x^2)/((1-x)*(1-x-x^2)). [Colin Barker, Apr 16 2012]

cp's OEIS Frontend

A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A208698 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 1 0 and 1 0 1 vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A117591 a(n) = 2^n + Fibonacci(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Sage

Formula

A166470 a(n) = 2^F(n+1)*3^F(n), where F(n) is the n-th Fibonacci number, A000045(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A248156 Inverse Riordan triangle of A106513: Riordan ((1 - 2*x^2 )/(1 + x), x/(1+x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A301662 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1 or 4 horizontally or vertically adjacent elements, with upper left element zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A020706 Pisot sequences L(4,6), E(4,6).

Views

Author

Keywords

References

Links

Crossrefs