A000918 -id:A000918 - cp's OEIS frontend

A105279 a(0)=0; a(n) = 10*a(n-1) + 10.

0, 10, 110, 1110, 11110, 111110, 1111110, 11111110, 111111110, 1111111110, 11111111110, 111111111110, 1111111111110, 11111111111110, 111111111111110, 1111111111111110, 11111111111111110, 111111111111111110, 1111111111111111110, 11111111111111111110, 111111111111111111110

Offset: 0

Alexandre Wajnberg, Apr 25 2005

a(n) is the smallest even number with digits in {0,1} having digit sum n; in other words, the base 10 reading of the binary string of A000918(n). Cf. A069532. - Jason Kimberley, Nov 02 2011

Also, except for a(0), the binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 19 2017

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

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Robert Price, Diagrams of first 20 stages of the Cellular Automata.
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.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000918, A002275, A029858, A052379, A052386, A069532, A080674, A124166, A161770.
Row n=10 of A228275.
Partial sums of A178500.

a105279 n = a105279_list !! n
a105279_list = iterate ((* 10) . (+ 1)) 0
-- Reinhard Zumkeller, Feb 05 2012

[-10/9+(10/9)*10^n: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

NestList[10*(# + 1) &, 0, 25] (* Paolo Xausa, Jul 17 2024 *)

a(n) = (10/9)*(10^n - 1), with n>=0.
a(n) = Sum_{k=1..n} 10^k.
Repunits times 10: a(n) = 10 * A002275(n). - Reinhard Zumkeller, Feb 05 2012
From Stefano Spezia, Sep 15 2023: (Start)
O.g.f.: 10*x/((1 - x)*(1 - 10*x)).
E.g.f.: 10*exp(x)*(exp(9*x) - 1)/9. (End)
From Elmo R. Oliveira, Jun 18 2025: (Start)
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = A124166(n)/10.
a(n) = A161770(n)/100 for n >= 1. (End)

A139257 Twice Mersenne primes A000668(n).

Original entry on oeis.org

6, 14, 62, 254, 16382, 262142, 1048574, 4294967294, 4611686018427387902, 1237940039285380274899124222, 324518553658426726783156020576254, 340282366920938463463374607431768211454

Offset: 1

Omar E. Pol, Apr 23 2008

nonn

Radicals of even perfect numbers. - Charles R Greathouse IV, Feb 01 2013

Amiram Eldar, Table of n, a(n) for n = 1..18
Florian Luca and Carl Pomerance, On the radical of a perfect number, New York Journal of Math., 16 (2010), 23-30; alternative link.

Cf. A000043, A000668, A000918, A060590, A095121, A100484, A139256.

2*(2^MersennePrimeExponent[Range[15]]-1) (* Harvey P. Dale, Jan 05 2020 *)

apply(p->2*(2^p-1),select(p->ispseudoprime(2^p-1),primes(40))) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = 2*A000668(n).

a(n) = A000918(1 + A000043(n)) = A095121(A000043(n)). - Omar E. Pol, Jun 07 2012

Corrected and extended by Joerg Arndt, Jun 07 2012.

A164874 Triangle read by rows: T(1,1)=2; T(n,k)=2*T(n-1,k)+1, 1<=k

Original entry on oeis.org

2, 5, 6, 11, 13, 14, 23, 27, 29, 30, 47, 55, 59, 61, 62, 95, 111, 119, 123, 125, 126, 191, 223, 239, 247, 251, 253, 254, 383, 447, 479, 495, 503, 507, 509, 510, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 2046

Offset: 1

Reinhard Zumkeller, Aug 29 2009

All terms contain exactly 1 zero in binary representation.

			Initial rows:
   1:                             2
   2:                        5        6
   3:                  11        13        14
   4:             23        27       29        30
   5:        47        55        59        61        62
   6:    95       111       119      123       125       126
also in binary representation:
                                 10
                            101       110
                      1011      1101      1110
                 10111     11011     11101     11110
           101111    110111    111011    111101    111110
      1011111   1101111   1110111   1111011   1111101   1111110 .

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A030130, A023416, A059673, A055010, A086224, A036563, A000918.

a164874 n k = a164874_tabl !! (n-1) !! (k-1)
a164874_row n = a164874_tabl !! (n-1)
a164874_tabl = map reverse $ iterate f [2] where
   f xs@(x:_) = (2 * x + 2) : map ((+ 1) . (* 2)) xs
-- Reinhard Zumkeller, Mar 31 2015

A164874row[n_] := 2^(n + 1) - 1 - BitShiftRight[2^n, Range[n]];
Array[A164874row, 10] (* Paolo Xausa, Jun 13 2025 *)

from math import isqrt
def A164874(n): return (1<<(a:=(isqrt(n<<3)+1>>1)+1))-(1<<(a*(a-1)>>1)-n)-1 # Chai Wah Wu, May 21 2025

T(n,k) = 2^(n+1) - 2^(n-k) - 1, 1 <= k <= n.
T(n,k) = A030130(n*(n-1)/2 + k + 1);
A023416(T(n,k)) = 1, 1<=k<=n;
A059673(n) = sum of n-th row;
T(n,1) = A055010(n);
T(n,2) = A086224(n-2) for n > 1;
T(n,n-1) = A036563(n+1) for n > 1;
T(n,n) = A000918(n+1).

A165326 a(0)=a(1)=1, a(n) = -a(n-1) for n > 1.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1

Offset: 0

Philippe Deléham, Sep 14 2009

Inverse binomial transform of A040000(n) = 1,2,2,2,2,...; binomial transform of (-1)^(n+1)*A000918(n) = 1,0,-2,6,-14,30,-62,... - Philippe Deléham, Sep 16 2009

This is also the Z-sequence of the Riordan triangle A105809. See the W. Lang link under A006232 for Riordan A- and Z-sequences. - Wolfdieter Lang, Oct 04 2014

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

Cf. A033999.

PadRight[{1},120,{-1,1}] (* Harvey P. Dale, Dec 04 2012 *)
Join[{1},LinearRecurrence[{-1},{1},83]] (* Ray Chandler, Aug 12 2015 *)

G.f.: (1+2*x)/(1+x).
E.g.f.: 2-exp(-x).
a(n) = -a(n-1). - Wesley Ivan Hurt, Apr 23 2021

A281773 Number of distinct topologies on an n-set that have exactly 4 open sets.

Original entry on oeis.org

0, 0, 1, 9, 43, 165, 571, 1869, 5923, 18405, 56491, 172029, 521203, 1573845, 4742011, 14266989, 42882883, 128812485, 386765131, 1160950749, 3484162963, 10455110325, 31370573851, 94122207309, 282387593443, 847204723365, 2541698056171, 7625261940669

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

			a(3) = 9 because we have: {{}, {c}, {a,b}, {a,b,c}} with 3 labelings and {{}, {c}, {b,c}, {a,b,c}} with 6 labelings.

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

Partial sums are given in A298564.

CoefficientList[Series[x^2*(1 + 3 x)/((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 27}], x] (* Michael De Vlieger, Jan 21 2018 *)

a(n) = stirling(n,2,2) + 3!*stirling(n,3,2) \\ Colin Barker, Jan 30 2017

concat(vector(2), Vec(x^2*(1 + 3*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = A000392(n+1) + 3*A000392(n).
E.g.f.: (exp(x)-1)^3 + (exp(x)-1)^2/2!.
From Colin Barker, Jan 30 2017: (Start)
G.f.: x^2*(1 + 3*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3.
a(n) = 2 - 5*2^(n-1) + 3^n for n>0. (End)

A281774 Number of distinct topologies on an n-set with exactly 6 open sets.

Original entry on oeis.org

0, 0, 0, 6, 72, 630, 4680, 31206, 193032, 1131990, 6386760, 35025606, 188061192, 993760950, 5187840840, 26831095206, 137770476552, 703455087510, 3576115150920, 18117222864006, 91536570671112, 461496288791670, 2322770028381000, 11675109032796006

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

LinearRecurrence[{15,-85,225,-274,120},{0,0,0,6,72,630},30] (* Harvey P. Dale, Oct 22 2018 *)

a(n) = 3!*stirling(n, 3, 2) + 3*4!*stirling(n, 4, 2)/2 + 5!*stirling(n, 5, 2) \\ Colin Barker, Jan 30 2017

concat(vector(3), Vec(6*x^3*(1 - 3*x + 10*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 3! Stirling2(n, 3) + 3/2*4! Stirling2(n, 4) + 5! Stirling2(n, 5).
From Colin Barker, Jan 30 2017: (Start)
a(n) = 2 - 2^(2+n) - 7*2^(2*n-1) + 5*3^n + 5^n for n>5.
a(n) = 15*a(n-1) - 85*a(n-2) + 225*a(n-3) - 274*a(n-4) + 120*a(n-5) for n>5.
G.f.: 6*x^3*(1 - 3*x + 10*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)).
(End)

A066884 Square array read by upward antidiagonals where the n-th row contains the positive integers with n binary 1's.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 15, 11, 6, 8, 31, 23, 13, 9, 16, 63, 47, 27, 14, 10, 32, 127, 95, 55, 29, 19, 12, 64, 255, 191, 111, 59, 30, 21, 17, 128, 511, 383, 223, 119, 61, 39, 22, 18, 256, 1023, 767, 447, 239, 123, 62, 43, 25, 20, 512, 2047, 1535, 895, 479, 247, 125, 79, 45, 26, 24, 1024

Offset: 1

Jared Benjamin Ricks (jaredricks(AT)yahoo.com), Jan 21 2002

This is a permutation of the positive integers; the inverse permutation is A067587.

			Column: 1   2   3   4   5   6
-----------------------------
Row 1:| 1   2   4   8  16  32
Row 2:| 3   5   6   9  10  12
Row 3:| 7  11  13  14  19  21
Row 4:|15  23  27  29  30  39
Row 5:|31  47  55  59  61  62
Row 6:|63  95 111 119 123 125

Ivan Neretin, Table of n, a(n) for n = 1..8001 (126 antidiagonals)
Vladimir Dobric, M. Skyers, and L. J. Stanley, Polynomial Time Computable Triangular Arrays For Almost Sure Convergence, arXiv preprint arXiv:1603.04896 [math.PR], 2016. [Shows that this sequence is in P-TIME]
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A067587.
Selected rows: A000079 (1), A018900 (2), A014311 (3), A014312 (4), A014313 (5), A023688 (6), A023689 (7), A023690 (8), A023691 (9), A038461 (10), A038462 (11), A038463 (12). For decimal analogs, see A011557 and A038444-A038452.
Selected columns: A000225 (1), A055010 (2).
Selected diagonals: A036563 (main), A000918 (1st upper), A153894 (2nd upper). [Franklin T. Adams-Watters, Apr 22 2009]
Cf. A067576 (the same array read by downward antidiagonals).
Antidiagonal sums give A361074.

a = {}; Do[ a = Append[a, Last[ Take[ Take[ Select[ Range[2^12], Count[ IntegerDigits[ #, 2], 1] == j - i + 1 & ], j], i]]], {j, 1, 11}, {i, 1, j}]; a

Corrected and extended by Henry Bottomley, Jan 27 2002

A281775 Number of distinct topologies on an n-set that have exactly 7 open sets.

Original entry on oeis.org

0, 0, 0, 0, 54, 780, 7830, 67620, 535374, 3992940, 28483110, 196316340, 1317106494, 8650141500, 55853351190, 355770438660, 2241509994414, 13998294536460, 86795899256070, 535048203626580, 3282628800655134, 20061393719417820, 122212221633141750

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A281774, A028244, A281775, A281776, A281777, A281778, A281779, A281780.

a(n) = 9*4!*stirling(n, 4, 2)/4 + 2*5!*stirling(n, 5, 2) + 6!*stirling(n, 6, 2) \\ Colin Barker, Jan 30 2017

concat(vector(4), Vec(6*x^4*(9 - 59*x + 150*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 9/4*4! Stirling2(n, 4) + 2*5! Stirling2(n, 5) + 6! Stirling2(n, 6).
From Colin Barker, Jan 30 2017: (Start)
G.f.: 6*x^4*(9 - 59*x + 150*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)).
a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>6.
a(n) = -5 + 17*2^(n-1) - 3^(2+n) + 29*4^(n-1) - 4*5^n + 6^n for n>0. (End)

A281776 Number of distinct topologies on an n-set that have exactly 8 open sets.

Original entry on oeis.org

0, 0, 0, 1, 54, 955, 11760, 122941, 1175034, 10595215, 91506420, 763624081, 6194818014, 49084747075, 381338401080, 2914184784421, 21965095364994, 163656285828535, 1207613518375740, 8838842878371961, 64253768864671974, 464416229729871595, 3340518964319750400

Offset: 0

Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

concat(vector(3), Vec(x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = Stirling2(n, 3) + 2*4! Stirling2(n, 4) + 15/4*5! Stirling2(n, 5) + 5/2*6! Stirling2(n, 6) + 7! Stirling2(n, 7).
From Colin Barker, Jan 30 2017: (Start)
a(n) = 13/4 - 19*2^(n-1) + 44*3^(n-1) - 2^(n-1)*3^(2+n) - 57*4^(n-1) + (39*5^n)/4 + 7^n for n>0.
G.f.: x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
(End)

A281777 Number of distinct topologies on an n-set that have exactly 9 open sets.

Original entry on oeis.org

0, 0, 0, 0, 20, 800, 14260, 189280, 2181060, 23241120, 235737620, 2308206560, 21979728100, 204477713440, 1864504348980, 16707856095840, 147469451067140, 1284607771225760, 11063319237792340, 94343562846289120, 797685042851814180, 6694943490279586080

Offset: 0

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{0,0,0,0,20,800,14260,189280,2181060},30] (* Harvey P. Dale, Aug 19 2020 *)

concat(vector(4), Vec(20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

a(n) = 5/6*4! Stirling2(n, 4) + 5*5! Stirling2(n, 5) + 11/2*6! Stirling2(n, 6) + 3*7! Stirling2(n, 7) + 8! Stirling2(n, 8).

G.f.: 20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)). - Colin Barker, Jan 30 2017

cp's OEIS Frontend

A105279 a(0)=0; a(n) = 10*a(n-1) + 10.

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A139257 Twice Mersenne primes A000668(n).

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A164874 Triangle read by rows: T(1,1)=2; T(n,k)=2*T(n-1,k)+1, 1<=k

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A165326 a(0)=a(1)=1, a(n) = -a(n-1) for n > 1.

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A281773 Number of distinct topologies on an n-set that have exactly 4 open sets.

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A281774 Number of distinct topologies on an n-set with exactly 6 open sets.

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A066884 Square array read by upward antidiagonals where the n-th row contains the positive integers with n binary 1's.

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