A000034 -id:A000034 - cp's OEIS frontend

A274913 Square array read by antidiagonals upwards in which each new term is the least positive integer distinct from its neighbors.

1, 2, 3, 1, 4, 1, 2, 3, 2, 3, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Omar E. Pol, Jul 11 2016

This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the square array we have that:
Antidiagonal sums give the positive terms of A008851.
Odd-indexed rows give A010684.
Even-indexed rows give A010694.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed antidiagonals give the initial terms of A010685.
Even-indexed antidiagonals give the initial terms of A010693.
Main diagonal gives A010685.
This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the triangle we have that:
Row sums give the positive terms of A008851.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed diagonals give A010684.
Even-indexed diagonals give A010694.
Odd-indexed rows give the initial terms of A010685.
Even-indexed rows give the initial terms of A010693.
Odd-indexed antidiagonals give the initial terms of A010684.
Even-indexed antidiagonals give the initial terms of A010694.

			The corner of the square array begins:
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, ...
1, 3, 1, 3, ...
2, 4, 2, ...
1, 3, ...
2, ...
...
The sequence written as a triangle begins:
1;
2, 3;
1, 4, 1;
2, 3, 2, 3;
1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3, 2, 3;
...

Cf. A000034, A008851, A010684, A010685, A010693, A010694, A010702, A274912, A274921.

Table[1 + Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

a(n) = A274912(n) + 1.

A283393 a(n) = gcd(n^2-1, n^2+9).

Original entry on oeis.org

1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10

Offset: 0

Bruno Berselli, Mar 07 2017

Periodic with period 10.
Similar sequences with formula gcd(n^2-1, n^2+k):
k= 1:  1,  2, 1, 2, 1,  2, 1,  2, 1,  2, 1,  2, 1, ...  (A000034)
k= 3:  1,  4, 1, 4, 1,  4, 1,  4, 1,  4, 1,  4, 1, ...  (A010685)
k= 5:  1,  6, 3, 2, 3,  6, 1,  6, 3,  2, 3,  6, 1, ...  (A129203, start 6)
k= 7:  1,  8, 1, 8, 1,  8, 1,  8, 1,  8, 1,  8, 1, ...  (A010689)
k= 9:  1, 10, 1, 2, 5,  2, 5,  2, 1, 10, 1, 10, 1, ...  (this sequence)
k=11:  1, 12, 3, 4, 3, 12, 1, 12, 3,  4, 3, 12, 1, ...  (A129197, start 12)
Connection between the values of a(n) and the last digit of n:
. if n ends with 0, 2 or 8, then a(n) = 1;
. if n ends with 1 or 9, then a(n) = 10;
. if n ends with 3, 5 or 7, then a(n) = 2;
. if n ends with 4 or 6, then a(n) = 5.
Also, continued fraction expansion of (57 + sqrt(4579))/114.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A000034, A010685, A010689, A129197, A129203.

&cat [[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]^^10];

Table[PolynomialGCD[n^2 - 1, n^2 + 9], {n, 0, 100}]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 10, 1, 2, 5, 2, 5, 2, 1, 10}, 100]

makelist(gcd(n^2-1, n^2+9), n, 0, 100);

Vec((1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10) + O(x^100)) \\ Colin Barker, Mar 08 2017

Python
```
[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]*10
```

[gcd(n^2-1, n^2+9) for n in range(100)]

G.f.: (1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10).

A052612 Expansion of e.g.f. x*(2+x)/(1-x^2).

Original entry on oeis.org

0, 2, 2, 12, 24, 240, 720, 10080, 40320, 725760, 3628800, 79833600, 479001600, 12454041600, 87178291200, 2615348736000, 20922789888000, 711374856192000, 6402373705728000, 243290200817664000, 2432902008176640000, 102181884343418880000, 1124000727777607680000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Stirling transform of (-1)^n*a(n-1) = [0,2,-2,12,-24,...] is A052856(n-1) =[0,2,4,14,76,...]. - Michael Somos, Mar 04 2004

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 557.

Cf. A000034, A000142, A191662.

spec := [S,{S=Prod(Z,Union(Sequence(Z),Sequence(Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[x (2+x)/(1-x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 10 2018 *)
Join[{0}, Table[1/2 (3 - (-1)^n) n!, {n, 20}]] (* David Trimas, Jul 28 2023 *)

a(n)=if(n<0,0,n!*polcoeff((x^2+2*x)/(1-x^2)+x*O(x^n),n))

PARI
```
a(n)=if(n<1,0,n!*(n%2+1))
```

a(n)= n! / gcd(n, n * (n + 1) / 2) \\ Andrew S. Plewe, Jan 09 2006

Recurrence: {a(0)=0, a(1)=2, a(2)=2, (-2-n^2-3*n)*a(n)+a(n+2)=0}.
Sum(1/2*(2+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2))*n!.
E.g.f.: x*(x+2)/(1-x^2).
a(2n+1) = 2*(2n+1)!, a(2n) = (2n)!, if n>0.
a(n) = n! if n is even, 2*n! otherwise. a(n) = n!*A000034(n).
a(n) = n! / gcd(n, T(n)) where T(n) is the n-th triangular number. - Andrew S. Plewe, Jan 09 2006
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = sinh(1)/2 + cosh(1) - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1)/2 - cosh(1) + 1. (End)
a(0)=0, a(n) = (1/2)*(3 - (-1)^n)*n! if n>0. - David Trimas, Jul 28 2023
a(n) = 2 * A191662(n) for n>=1. - Alois P. Heinz, Sep 05 2023

a(20)-a(22) from Alois P. Heinz, Sep 05 2023

A105475 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds.

Original entry on oeis.org

1, 2, 1, 1, 4, 1, 2, 6, 6, 1, 1, 8, 15, 8, 1, 2, 11, 26, 28, 10, 1, 1, 12, 42, 64, 45, 12, 1, 2, 16, 60, 122, 130, 66, 14, 1, 1, 16, 82, 208, 295, 232, 91, 16, 1, 2, 21, 108, 324, 582, 621, 378, 120, 18, 1, 1, 20, 135, 480, 1035, 1404, 1176, 576, 153, 20, 1, 2, 26, 170, 675

Offset: 1

Emeric Deutsch, Apr 09 2005

Riordan array ((1+2x)/(1-x^2),x(1+2x)/(1-x^2)). Factorizes as ((1+2x)/(1-x^2),x)*(1,x(1+2x)/(1-x^2)). Row sums A105476 form an eigensequence for ((1+2x)/(1-x^2),x). - Paul Barry, Feb 10 2011
Triangle T(n,k), 1<=k<=n, given by (0, 2, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012
Also the convolution triangle of A000034. - Peter Luschny, Oct 08 2022

			T(4,2) = 6 because we have (1,3), (3,1), (2,2), (2,2'), (2',2) and (2',2').
Triangle begins:
1;
2, 1;
1, 4,  1;
2, 6,  6, 1;
1, 8, 15, 8, 1;
Triangle (0, 2, -3/2, -1/2, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 2, 1
0, 1, 4, 1
0, 2, 6, 6, 1
0, 1, 8, 15, 8, 1
0, 2, 11, 26, 28, 10, 1
0, 1, 12, 42, 64, 45, 12, 1

Alois P. Heinz, Rows n = 1..141, flattened

Row sums yield A105476.

Cf. Diagonals: A000007, A000034, A000012, A005843, A000384, A100504.

G:=t*z*(1+2*z)/(1-t*z-z^2-2*t*z^2): Gser:=simplify(series(G,z=0,14)): for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      expand(add((2-irem(i, 2))*b(n-i)*x, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, k), k=1..n))(b(n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Oct 16 2013
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> [1, 2][irem(n-1, 2) + 1]); # Peter Luschny, Oct 08 2022

max = 14; g = t*z*(1 + 2*z)/(1 - t*z - z^2 - 2*t*z^2); gser = Series[g, {z, 0, max}]; coes = CoefficientList[gser, {z, t}]; Table[ Table[ coes[[n, k]], {k, 2, n}], {n, 2, max}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Maple *)

A123344 Expansion of (1+3x)/(1+2x).

Original entry on oeis.org

1, 1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, -2048, 4096, -8192, 16384, -32768, 65536, -131072, 262144, -524288, 1048576, -2097152, 4194304, -8388608, 16777216, -33554432, 67108864, -134217728, 268435456, -536870912, 1073741824, -2147483648

Offset: 0

Philippe Deléham, Oct 11 2006

Inverse binomial transform of A000034.

Hankel transform is [1,-3,0,0,0,0,0,0,0,0,...].

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A011782 (unsigned version).

[1] cat [(-2)^(n-1): n in [1..35]]; // Vincenzo Librandi, Feb 14 2014

a:=n->mul(-2, k=0..n): seq(a(n), n=-2..30); # Zerinvary Lajos, Jan 22 2008

Table[(-2)^(n - Sign[n]), {n, 0, 30}] (* Wesley Ivan Hurt, Feb 01 2014 *)
Join[{1},LinearRecurrence[{-2},{1},32]] (* Ray Chandler, Aug 12 2015 *)
Join[{1},NestList[-2#&,1,40]] (* Harvey P. Dale, Aug 24 2019 *)

x='x+O('x^50); Vec((1+3*x)/(1+2*x)) \\ G. C. Greubel, Oct 12 2017

a(0)=1, a(n) = (-2)^(n-1) for n>0.
G.f.: (1+3*x)/(1+2*x).
G.f.: 1/U(0)  where U(k)=  1 - x*(k+4) + x*(k+3)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
E.g.f.: (3 - exp(-2*x))/2. - G. C. Greubel, Oct 12 2017
a(n) = numerator((1/2 - n)!/sqrt(Pi)). - Peter Luschny, Jun 21 2020

A141023 a(n) = 2^n - (3-(-1)^n)/2.

Original entry on oeis.org

0, 0, 3, 6, 15, 30, 63, 126, 255, 510, 1023, 2046, 4095, 8190, 16383, 32766, 65535, 131070, 262143, 524286, 1048575, 2097150, 4194303, 8388606, 16777215, 33554430, 67108863, 134217726, 268435455, 536870910, 1073741823, 2147483646, 4294967295, 8589934590, 17179869183

Offset: 0

Paul Curtz, Jul 29 2008

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

Cf. A062510 (first differences).

[2^n -(3-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

Range[0,20]! CoefficientList[Series[D[(Cosh[x]-1)(Exp[x]-1), x], {x,0,20}], x]  (* Geoffrey Critzer, Dec 03 2011 *)
LinearRecurrence[{2, 1, -2}, {0, 0, 3}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)
Table[2^n - (3 - (-1)^n)/2, {n, 0, 34}] (* Alonso del Arte, Feb 14 2012 *)

x='x+O('x^50); concat([0,0], Vec(3*x^2/((x-1)*(2*x-1)*(1+x)))) \\ G. C. Greubel, Oct 10 2017

a(n) = A000079(n) - A000034(n).
a(n) = 3*A000975(n-1).
G.f.: 3*x^2/( (x-1)*(2*x-1)*(1+x) ). - R. J. Mathar, Jul 07 2011

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.

Original entry on oeis.org

0, 3, 1, 15, 3, 35, 6, 63, 10, 99, 15, 143, 21, 195, 28, 255, 36, 323, 45, 399, 55, 483, 66, 575, 78, 675, 91, 783, 105, 899, 120, 1023, 136, 1155, 153, 1295, 171, 1443, 190, 1599, 210, 1763, 231, 1935, 253, 2115, 276, 2303, 300, 2499, 325

Offset: 0

Paul Curtz, Oct 21 2011

See, in A181318(n), A060819(n)*A060819(n+p): A060819(n)^2, A064038(n), a(n), A160050(n), A061037(n), A178242(n). The second differences a(n+2)-2*a(n+1)+a(n) = -5, 16, -26, 44, -61, 86, -110, 142, -173, 212, -250, 296, -341, 394, -446, 506, taken modulo 9 are periodic with the palindromic period 4, 7, 1, 8, 2, 5, 7, 7, 7, 5, 2, 8, 1, 7, 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. quadrisections A014105, A001539, A000384, A141759.

Cf. A060819, A000217, A000466, A142705, A000034, A005563, A010689, A028552, A050534.

List([0..60], n -> n*(n+2)*(9-7*(-1)^n)/16); # G. C. Greubel, Feb 21 2019

[n*(n+2)*(9-7*(-1)^n)/16: n in [0..60]]; // Vincenzo Librandi, Nov 25 2011

A198148:=n->n*(n+2)*(9-7*(-1)^n)/16; seq(A198148(k), k=0..100); # Wesley Ivan Hurt, Oct 16 2013

LinearRecurrence[{0,3,0,-3,0,1},{0,3,1,15,3,35},60] (* Vincenzo Librandi, Nov 25 2011 *)

a(n)=n*(n+2)*(9-7*(-1)^n)/16 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n+2)*(9-7*(-1)^n)/16 for n in (0..60)] # G. C. Greubel, Feb 21 2019

a(n) = A060819(n)*A060819(n+2).
a(2n) = n*(n+1)/2 = A000217(n).
a(2n+1) = (2*n+1)*(2*n+3) = A000466(n+1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6), n>5.
a(n+1) - a(n) = (7*(-1)^n *(2*n^2+6*n+3) +18*n +27)/16.
a(n) = A142705(n) / A000034(n+1).
a(n) = A005563(n) / A010689(n+1). - Franklin T. Adams-Watters, Oct 21 2011
G.f. x*(3 +x +6*x^2 -x^4)/(1-x^2)^3. - R. J. Mathar, Oct 25 2011
a(n)*a(n+1) = a(A028552(n)) = A050534(n+2). - Bruno Berselli, Oct 26 2011
a(n) = numerator( binomial((n+2)/2,2) ). - Wesley Ivan Hurt, Oct 16 2013
E.g.f.: x*((24+x)*cosh(x) + (3+8*x)*sinh(x))/8. - G. C. Greubel, Sep 20 2018
Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Aug 12 2022

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24

Offset: 1

T. D. Noe, Dec 17 2011

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

T. D. Noe, Rows n = 1..60, flattened

Cf. A062117, A201908, A201909, A201910, A201911.

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019

nn = 10; p = 2; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

A213268 Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 2, 2, 8, 2, 1, 1, 4, 1, 16, 16, 1, 2, 1, 8, 8, 32, 16, 1, 1, 4, 4, 16, 8, 64, 64, 1, 2, 2, 8, 4, 32, 32, 128, 16, 1, 1, 4, 2, 16, 16, 64, 8, 256, 256, 1, 2, 1, 8, 8, 32, 4, 128, 128, 512, 256, 1, 1, 4, 4, 16, 2, 64, 64, 256, 128, 1024, 1024

Offset: 0

Paul Curtz, Jun 08 2012

Starting from any sequence a(k) in the first row, define the array T(n,k) of the inverse semi-binomial transform  by T(0,k) = a(k), T(n,k) = T(n-1,k+1) -T(n-1,k)/2, n>=1.
Here, where the first row is the nonnegative integers, the array is
0        1      2      3    4      5      6      7     8   =A001477(n)
1       3/2     2    5/2    3    7/2      4    9/2     5   =A026741(n+2)/A000034(n)
1       5/4   3/2    7/4    2    9/4    5/2   11/4     3   =A060819(n+4)/A176895(n)
3/4     7/8     1    9/8  5/4   11/8    3/2   13/8   7/4   =A106609(n+6)/A205383(n+6)
1/2    9/16   5/8  11/16  3/4  13/16    7/8  15/16     1   =A106617(n+8)/TBD
5/16  11/32   3/8  13/32 7/16  15/32    1/2  17/32  9/16
3/16  13/64  7/32  15/64  1/4  17/64   9/32  19/64  5/16
7/64 15/128   1/8 17/128 9/64 19/128   5/32 21/128 11/64
1/16 17/256 9/128 19/256 5/64 21/256 11/128 23/256  3/32.
The first column contains 0, followed by fractions A000265/A084623, that is Oresme numbers n/2^n multiplied by 2 (see A209308).

			The array of denominators starts:
  1   1   1   1   1   1   1   1   1   1   1 ...
  1   2   1   2   1   2   1   2   1   2   1 ...
  1   4   2   4   1   4   2   4   1   4   2 ...
  4   8   1   8   4   8   2   8   4   8   1 ...
  2  16   8  16   4  16   8  16   1  16   8 ...
16  32   8  32  16  32   2  32  16  32   8 ...
16  64  32  64   4  64  32  64  16  64  32 ...
64 128   8 128  64 128  32 128  64 128  16 ...
16 256 128 256  64 256 128 256  32 256 128 ...
256 512 128 512 256 512  64 512 256 512 128 ...
All entries are powers of 2.

A213268frac := proc(n,k)
        if n = 0 then
                return k ;
        else
                return procname(n-1,k+1)-procname(n-1,k)/2 ;
        end if;
end proc:
A213268 := proc(n,k)
        denom(A213268frac(n,k)) ;
end proc: # R. J. Mathar, Jun 30 2012

T[0, k_] := k; T[n_, k_] := T[n, k] = T[n-1, k+1] - T[n-1, k]/2; Table[T[n-k, k] // Denominator, {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2014 *)

A216917 Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 12, 3, 2, 1, 1, 60, 3, 2, 1, 1, 1, 60, 15, 4, 3, 2, 1, 1, 420, 15, 20, 3, 6, 1, 1, 1, 840, 105, 20, 15, 12, 1, 2, 1, 1, 2520, 105, 140, 15, 12, 1, 6, 1, 1, 1, 2520, 315, 280, 105, 12, 5, 12, 3, 2, 1, 1, 27720, 315, 280, 105, 84

Offset: 1

Peter Luschny, Oct 02 2012

T(N,n) is the least common multiple of all integers up to N that are relatively prime to n.

Replacing LCM in the definition with "product" gives the Gauss factorial A216919.

			   n | N=0 1 2 3  4  5  6   7   8    9   10
-----+-------------------------------------
   1 |   1 1 2 6 12 60 60 420 840 2520 2520
   2 |   1 1 1 3  3 15 15 105 105  315  315
   3 |   1 1 2 2  4 20 20 140 280  280  280
   4 |   1 1 1 3  3 15 15 105 105  315  315
   5 |   1 1 2 6 12 12 12  84 168  504  504
   6 |   1 1 1 1  1  5  5  35  35   35   35
   7 |   1 1 2 6 12 60 60  60 120  360  360
   8 |   1 1 1 3  3 15 15 105 105  315  315
   9 |   1 1 2 2  4 20 20 140 280  280  280
  10 |   1 1 1 3  3  3  3  21  21   63   63
  11 |   1 1 2 6 12 60 60 420 840 2520 2520
  12 |   1 1 1 1  1  5  5  35  35   35   35
  13 |   1 1 2 6 12 60 60 420 840 2520 2520

t[, 0] = 1; t[n, k_] := LCM @@ Select[Range[k], CoprimeQ[#, n]&]; Table[t[n - k + 1, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

def A216917(N, n):
    return lcm([j for j in (1..N) if gcd(j, n) == 1])
for n in (1..13): [A216917(N,n) for N in (0..10)]

For n > 0:
A(n,1) = A003418(n);
A(n,2^k) = A217858(n) for k > 0;
A(n,3^k) = A128501(n-1) for k > 0;
A(2,n) = A000034(n);
A(3,n) = A129203(n-1);
A(4,n) = A129197(n-1);
A(n,n) = A038610(n);
A(floor(n/2),n) = A124443(n);
A(n,1)/A(n,n) = A064446(n);
A(n,1)/A(n,2) = A053644(n).

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