A005061 -id:A005061 - cp's OEIS frontend

A212943 T(n,k)=Number of nXk 0..k-1 arrays with no column j greater than or equal to than column j-1 in all rows.

1, 1, 1, 1, 7, 1, 1, 181, 37, 1, 1, 10311, 9019, 175, 1, 1, 1016501, 6470341, 331489, 781, 1, 1, 152747323, 10058484751, 2509306671, 10669771, 3367, 1, 1, 32383630189, 28744943858947, 52311221188001, 801439905901, 320396041, 14197, 1, 1

Offset: 1

R. H. Hardin May 31 2012

Table starts
.1...1........1............1..................1........................1
.1...7......181........10311............1016501................152747323
.1..37.....9019......6470341........10058484751...........28744943858947
.1.175...331489...2509306671.....52311221188001......2438624218076957695
.1.781.10669771.801439905901.212180664326328751.153322267564381742818531

			Some solutions for n=4 k=4
..1..3..2..2....0..0..0..3....1..3..0..1....1..3..2..1....1..3..3..3
..1..1..2..2....0..1..1..2....0..0..1..0....1..1..0..3....2..3..0..3
..0..1..3..1....0..2..2..1....0..1..1..3....2..3..3..0....0..2..0..2
..3..1..0..3....2..1..0..2....3..0..3..1....1..0..3..0....2..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..191

Column 2 is A005061

cf. A212930

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 7*a(n-1) -12*a(n-2)
k=3: a(n) = 55*a(n-1) -936*a(n-2) +4860*a(n-3)
k=4: a(n) = 631*a(n-1) -144700*a(n-2) +15035200*a(n-3) -702208000*a(n-4) +11468800000*a(n-5)
The coefficient of a(n-1) is A209668(k) (through at least k=1..7)

A359576 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 15, 37, 17, 1, 31, 175, 197, 41, 1, 63, 781, 1985, 1041, 99, 1, 127, 3367, 18621, 22193, 5503, 239, 1, 255, 14197, 167337, 433809, 247759, 29089, 577, 1, 511, 58975, 1461797, 8057905, 10056959, 2764991, 153769, 1393, 1, 1023, 242461, 12519345, 144769425, 384479935, 232824241, 30856705, 812849, 3363, 1

Offset: 1

Andrew Howroyd, Jan 06 2023

The grid has m rows and n columns.
"Path" refers to a sequence of L(eft), R(ight), U(p), D(own) steps (edge connectivity like in fixed polyominoes), self-avoiding, starting anywhere in the first row and ending anywhere in the last row. The path does not need to step on all 1's of the array. The path has obviously at least m-1 steps. - R. J. Mathar, Jun 21 2023
Note that the total would be smaller if Up steps were disallowed (as in the original comment above); the smallest grid size for which this phenomenon occurs is 4 X 5. The total number of 4 X 5 and 5 X 5 grids would be 433801 instead of 433809 and 10056087 instead of 10056959, respectively, without Up steps. - Caleb Stanford, Feb 01 2024
Each row and each column satisfies a linear recurrence with constant coefficients. - Pontus von Brömssen, Feb 05 2025

			Array begins:
====================================================================
m\n| 1   2      3        4          5            6             7
---+----------------------------------------------------------------
1  | 1   3      7       15         31           63           127 ...
2  | 1   7     37      175        781         3367         14197 ...
3  | 1  17    197     1985      18621       167337       1461797 ...
4  | 1  41   1041    22193     433809      8057905     144769425 ...
5  | 1  99   5503   247759   10056959    384479935   14142942975 ...
6  | 1 239  29089  2764991  232824241  18287614751 1374273318721 ...
7  | 1 577 153769 30856705 5388274121 868972410929 ...
  ...
All the 37 2 X 3 binary arrays:
001 001 001 001
001 011 101 111 plus 4 copies left-right flipped
.
010 010 010 010
010 011 110 111
.
011 011 011 011 011 011
001 010 011 101 110 111 plus 6 copies left-right flipped
.
101 101 101 101 101 101
001 011 100 101 110 111
.
111 111 111 111 111 111 111
001 010 011 100 101 110 111 - _R. J. Mathar_, Jun 21 2023

Samuel Dittmer, Hiram Golze, Grant Molnar, and Caleb Stanford, Puzzle and Proof: A Decade of Problems from the Utah Math Olympiad, CRC Press, 2025, p. 51.

Caleb Stanford, Rust program to compute the sequence.
Utah Math Olympiad 2022, Problem 6.

Main diagonal is A365988.
Rows 1..19 are A000225, A005061, A069361, A069362, A069363-A069377.
Columns 1..20 are A000012, A001333(n+1), A069378, A069379, A069380-A069395.
Cf. A359574, A359575.

One additional diagonal of terms added by Caleb Stanford, Feb 05 2024

A069362 Number of 4 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 41, 1041, 22193, 433809, 8057905, 144769425, 2541013617, 43843180113, 746691527217, 12588144461329, 210502738714097, 3497001564166609, 57781030561348017, 950437243856526737, 15574913193760097649, 254416775893204873553, 4144677558181255455025

Offset: 1

R. H. Hardin, Mar 22 2002

Colin Barker, Table of n, a(n) for n = 1..800
Index entries for linear recurrences with constant coefficients, signature (35,-378,1264,-1272,-128).

Row 4 of A359576.

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+6*x-16*x^2-8*x^3)/((1-16*x)*(1-19*x+ 74*x^2 -80*x^3-8*x^4)))); // G. C. Greubel, Apr 22 2018

Rest[CoefficientList[Series[x*(1+6*x-16*x^2-8*x^3)/((1-16*x)*(1-19*x+ 74*x^2 -80*x^3-8*x^4)), {x,0,50}],x]] (* G. C. Greubel, Apr 22 2018 *)
LinearRecurrence[{35,-378,1264,-1272,-128},{1,41,1041,22193,433809},20] (* Harvey P. Dale, Jan 01 2019 *)

Vec(x*(1 + 6*x - 16*x^2 - 8*x^3) / ((1 - 16*x)*(1 - 19*x + 74*x^2 - 80*x^3 - 8*x^4)) + O(x^30)) \\ Colin Barker, Oct 12 2017

G.f.: x*(1 +6*x -16*x^2 -8*x^3)/((1 -16*x)*(1 -19*x +74*x^2 -80*x^3 - 8*x^4)).

A090888 Matrix defined by a(n,k) = 3^nFibonacci(k) - 2^nFibonacci(k-2), read by antidiagonals.

Original entry on oeis.org

1, 2, 0, 4, 1, 1, 8, 5, 3, 1, 16, 19, 9, 4, 2, 32, 65, 27, 14, 7, 3, 64, 211, 81, 46, 23, 11, 5, 128, 665, 243, 146, 73, 37, 18, 8, 256, 2059, 729, 454, 227, 119, 60, 29, 13, 512, 6305, 2187, 1394, 697, 373, 192, 97, 47, 21, 1024, 19171, 6561, 4246, 2123, 1151, 600, 311

Offset: 0

Ross La Haye, Feb 12 2004; revised Sep 24 2004, Sep 10 2005

a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1).
a(n,1) = a(n-1,1) + a(n-1,3) for n > 0; a(n,1) = A001047(n) = 2^(2n) - A083324(n); a(n,2) = A000244(n) = 2^(2n) - A005061(n); a(n,3) = 2a(n-1,4) for n > 0; a(n,3) = A027649(n); a(n,4) = A083313(n+1); a(n,5) = A084171(n+1).
Sum[a(n-k,k), {k,0,n}] = A098703(n+1), antidiagonal sums.
Let R, S and T be binary relations on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x, xSy if x is a subset of y and xTy if x is a proper subset of y. Then a(n,3) = |R|, a(n,2) = |S| and a(n,1) = |T|. Note that a binary relation W on P(A) can be defined also such that for every element x, y of P(A) xWy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. A090802(n,1) = |W|. Also, a(n,0) = |P(A)|.

			   1    0    1    1    2    3    5    8    13    21    34
   2    1    3    4    7   11   18   29    47    76   123
   4    5    9   14   23   37   60   97   157   254   411
   8   19   27   46   73  119  192  311   503   814  1317
  16   65   81  146  227  373  600  973  1573  2546  4119
  32  211  243  454  697 1151 1848 2999  4847  7846 12693
  64  665  729 1394 2123 3517 5640 9157 14797 23954 38751
a(5,3) = 454 because Fibonacci(3) = 2, Fibonacci(1) = 1 and (2 * 3^5) - (1 * 2^5) = 454.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Ross La Haye, Binary relations on the power set of an n-element set, JIS 12 (2009) 09.2.6, table 4.
Eric Weisstein, Fibonacci Number
Eric Weisstein, Lucas Number

Table[3^(n - k) Fibonacci@ k - 2^(n - k) Fibonacci[k - 2], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 28 2015 *)

a(n, k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2).
a(n, 0) = 2^n, a(n, 1) = 3^n - 2^n, a(n, k) = a(n, k-1) + a(n, k-2) for k > 1.
a(0, k) = Fibonacci(k-1), a(1, k) = Lucas(k), a(n, k) = 5a(n-1, k) - 6a(n-2, k) for n > 1.
O.g.f. (by rows) = (-2^n + (2^(n+1) - 3^n)x)/(-1+x+x^2). - Ross La Haye, Mar 30 2006
a(n,1) - a(n,0) = A003063(n+1). - Ross La Haye, Jun 22 2007
Binomial transform (by columns) of A118654. - Ross La Haye, Jun 22 2007

More terms from Ray Chandler, Oct 27 2004

A167762 a(n) = 2a(n-1)+3a(n-2)-6*a(n-3) starting a(0)=a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, 2, 7, 14, 37, 74, 175, 350, 781, 1562, 3367, 6734, 14197, 28394, 58975, 117950, 242461, 484922, 989527, 1979054, 4017157, 8034314, 16245775, 32491550, 65514541, 131029082, 263652487, 527304974, 1059392917, 2118785834, 4251920575, 8503841150

Offset: 0

Paul Curtz, Nov 11 2009

Inverse binomial transform yields two zeros followed by A077917 (a signed variant of A127864).
a(n) mod 10 is zero followed by a sequence with period length 8: 0, 1, 2, 7, 4, 7, 4, 5 (repeat).
a(n) is the number of length n+1 binary words with some prefix w such that w contains three more 1's than 0's and no prefix of w contains three more 0's than 1's. - Geoffrey Critzer, Dec 13 2013
From Gus Wiseman, Oct 06 2023: (Start)
Also the number of subsets of {1..n} with two distinct elements summing to n + 1. For example, the a(2) = 1 through a(5) = 14 subsets are:
  {1,2}  {1,3}    {1,4}      {1,5}
         {1,2,3}  {2,3}      {2,4}
                  {1,2,3}    {1,2,4}
                  {1,2,4}    {1,2,5}
                  {1,3,4}    {1,3,5}
                  {2,3,4}    {1,4,5}
                  {1,2,3,4}  {2,3,4}
                             {2,4,5}
                             {1,2,3,4}
                             {1,2,3,5}
                             {1,2,4,5}
                             {1,3,4,5}
                             {2,3,4,5}
                             {1,2,3,4,5}
The complement is counted by A038754.
Allowing twins gives A167936, complement A108411.
For n instead of n + 1 we have A365544, complement A068911.
The version for all subsets (not just pairs) is A366130.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Cf. A024495, A167710.
First differences are A167936, complement A108411.
Cf. A004526, A004737, A008967, A038754, A046663, A068911, A088809, A093971, A365376, A365544, A366130.

LinearRecurrence[{2,3,-6},{0,0,1},40] (* Harvey P. Dale, Sep 17 2013 *)
CoefficientList[Series[x^2/((2 x - 1) (3 x^2 - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 17 2013 *)
Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],n+1]&]],{n,0,10}] (* Gus Wiseman, Oct 06 2023 *)

a(n) mod 9 = A153130(n), n>3 (essentially the same as A154529, A146501 and A029898).
a(n+1)-2*a(n) = 0 if n even, = A000244((1+n)/2) if n odd.
a(2*n) = A005061(n). a(2*n+1) = 2*A005061(n).
G.f.: x^2/((2*x-1)*(3*x^2-1)). a(n) = 2^n - A038754(n). - R. J. Mathar, Nov 12 2009
G.f.: x^2/(1-2*x-3*x^2+6*x^3). - Philippe Deléham, Nov 11 2009

Edited and extended by R. J. Mathar, Nov 12 2009

A085350 Binomial transform of poly-Bernoulli numbers A027649.

Original entry on oeis.org

1, 5, 23, 101, 431, 1805, 7463, 30581, 124511, 504605, 2038103, 8211461, 33022991, 132623405, 532087943, 2133134741, 8546887871, 34230598205, 137051532983, 548593552421, 2195536471151, 8785632669005, 35152991029223

Offset: 0

Paul Barry, Jun 24 2003

Binomial transform is A085351.

a(n) mod 10 = period 4:repeat 1,5,3,1 = A132400. - Paul Curtz, Nov 13 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -12).

a(n-1) = A080643(n)/2 = A081674(n+1) - A081674(n).

Cf. A000244 (3^n).

[2*4^n-3^n: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

LinearRecurrence[{4,9,-36},{1,5,23},30] (* Harvey P. Dale, Nov 30 2011 *)
LinearRecurrence[{7, -12},{1, 5},23] (* Ray Chandler, Aug 03 2015 *)

G.f.: (1-2x)/((1-3x)(1-4x)).
E.g.f.: 2exp(4x) - exp(3x).
a(n) = 2*4^n-3^n.
From Paul Curtz, Nov 13 2009: (Start)
a(n) = 4*a(n-1) + 9*a(n-2) - 36*a(n-3);
a(n) = 4*a(n-1) + 3^(n-1), both like A005061 (note for A005061 dual formula a(n) = 3*a(n-1) + 4^(n-1) = 3*a(n-1) + A000302(n-1)).
a(n) = 3*a(n-1) + 2^(2n+1) = 3*a(n-1) + A004171(n).
a(n) = A005061(n) + A000302(n).
b(n) = mix(A005061, A085350) = 0,1,1,5,7,23,... = differences of (A167762 = 0,0,1,2,7,14,37,...); b(n) differences = A167784. (End)

A277965 Numbers whose largest decimal digit is 3.

Original entry on oeis.org

3, 13, 23, 30, 31, 32, 33, 103, 113, 123, 130, 131, 132, 133, 203, 213, 223, 230, 231, 232, 233, 300, 301, 302, 303, 310, 311, 312, 313, 320, 321, 322, 323, 330, 331, 332, 333, 1003, 1013, 1023, 1030, 1031, 1032, 1033, 1103, 1113, 1123, 1130, 1131, 1132

Offset: 1

Colin Barker, Nov 06 2016

Number of terms less than 10^n is 4^n - 3^n. - Chai Wah Wu, Nov 06 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)

Cf. A007088, A277964, A277966.
Cf. A005061 (4^n - 3^n).
Cf. A106099 (subsequence of primes).

Filtered([1..450],n->Maximum(ListOfDigits(n))=3); # Muniru A Asiru, Mar 01 2019

A277965Q = Max[IntegerDigits[#]] == 3 &; Select[Range[1200], A277965Q] (* JungHwan Min, Nov 06 2016 *)

L=List(); for(n=1, 10000, if(vecmax(digits(n))==3, listput(L, n))); Vec(L)

A051588 Number of 3 X n binary matrices such that any 2 rows have a common 1.

Original entry on oeis.org

0, 1, 15, 175, 1827, 17791, 164955, 1475335, 12844707, 109581871, 920591595, 7643833495, 62904774387, 514168732351, 4180996130235, 33864296127655, 273465115692867, 2203291473841231, 17721094011796875, 142344054436901815

Offset: 0

Vladeta Jovovic, Goran Kilibarda

G. C. Greubel, Table of n, a(n) for n = 0..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (in Russian), Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.
Index entries for linear recurrences with constant coefficients, signature (23,-194,712,-960).

Cf. A005061.

List([0..20], n-> 8^n -3*6^n +3*5^n -4^n); # G. C. Greubel, Nov 12 2019

[8^n -3*6^n +3*5^n -4^n: n in [0..20]]; // G. C. Greubel, Nov 12 2019

A051588:=n->8^n-3*6^n+3*5^n-4^n: seq(A051588(n), n=0..30); # Wesley Ivan Hurt, May 03 2017

Table[8^n-3*6^n+3*5^n-4^n, {n,0,20}] (* or *) LinearRecurrence[{23,-194, 712,-960}, {0,1,15,175}, 20] (* Harvey P. Dale, Mar 07 2012 *)

vector(21, n, m=n-1; 8^m -3*6^m +3*5^m -4^m) \\ G. C. Greubel, Oct 06 2017

[8^n -3*6^n +3*5^n -4^n for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = 8^n - 3*6^n + 3*5^n - 4^n.
a(0)=0, a(1)=1, a(2)=15, a(3)=175, a(n) = 23*a(n-1) -194*a(n-2) + 712*a(n-3) -960*a(n-4). - Harvey P. Dale, Mar 07 2012
G.f.: x*(24*x^2-8*x+1)/((4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). - Colin Barker, Nov 05 2012
E.g.f.: exp(8*x) -3*exp(6*x) +3*exp(5*x) -exp(4*x). - G. C. Greubel, Nov 12 2019

A255463 a(n) = 34^n - 23^n.

Original entry on oeis.org

1, 6, 30, 138, 606, 2586, 10830, 44778, 183486, 747066, 3027630, 12228618, 49268766, 198137946, 795740430, 3192527658, 12798808446, 51281327226, 205383589230, 822309197898, 3291561314526, 13173218826906, 52713796014030, 210917946175338, 843860071059006, 3376005143308986, 13505715150454830

Offset: 0

N. J. A. Sloane and Doron Zeilberger, Feb 23 2015

a(n-1) is also the number of n-digit numbers whose largest decimal digit is 3. - Stefano Spezia, Nov 15 2023

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 No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
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 (7,-12).

Cf. A255462.

First differences of 4^n - 3^n = A005061(n). See A257285, A257286, A257287, A257288, A257289 for first differences of 5^n - 4^n, ..., 9^n - 8^n. - M. F. Hasler, May 04 2015

[3*4^n-2*3^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Table[3 4^n - 2 3^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)

a(n)=3*4^n-2*3^n \\ M. F. Hasler, May 04 2015

G.f.: (1-x)/((1-3*x)*(1-4*x)).
a(n+1) = 7*a(n) - 12*a(n-1) with a(0)=1, a(1)=6.
a(n) = A255462(2^n-1).
E.g.f.: exp(3*x)*(3*exp(x) - 2). - Stefano Spezia, Nov 15 2023

Simpler definition from N. J. A. Sloane, Mar 10 2015

A087490 Primes p such that 4^p - 3^p is composite.

Original entry on oeis.org

5, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 293, 307

Offset: 1

Cino Hilliard, Oct 26 2003

nonn

Primes not in A059801. - Robert Israel, Nov 03 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005061, A059801.

Primes p such that k^p - (k-1)^p is composite: A087489 (k=3), this sequence (k=4), A087685 (k=5), A087749 (k=6), A087759 (k=7), A087763 (k=8), A087894 (k=9), A087895 (k=10).

filter:= p -> isprime(p) and not isprime(4^p-3^p):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Nov 03 2024

Select[Prime[Range[70]],CompositeQ[4^#-3^#]&] (* Harvey P. Dale, Mar 14 2025 *)

apmb(a,b,n) = { forprime(x=2,n, y=a^x-b^x; if(!ispseudoprime(y), print1(x","); ) ) }

Offset corrected by Mohammed Yaseen, Jul 18 2022

cp's OEIS Frontend

A212943 T(n,k)=Number of nXk 0..k-1 arrays with no column j greater than or equal to than column j-1 in all rows.

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A359576 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with a path of adjacent 1's from top row to bottom row.

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A069362 Number of 4 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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A090888 Matrix defined by a(n,k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2), read by antidiagonals.

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A167762 a(n) = 2*a(n-1)+3*a(n-2)-6*a(n-3) starting a(0)=a(1)=0, a(2)=1.

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Mathematica

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A085350 Binomial transform of poly-Bernoulli numbers A027649.

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Magma

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A277965 Numbers whose largest decimal digit is 3.

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A051588 Number of 3 X n binary matrices such that any 2 rows have a common 1.

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A090888 Matrix defined by a(n,k) = 3^nFibonacci(k) - 2^nFibonacci(k-2), read by antidiagonals.

A167762 a(n) = 2a(n-1)+3a(n-2)-6*a(n-3) starting a(0)=a(1)=0, a(2)=1.

A255463 a(n) = 34^n - 23^n.