A076032 -id:A076032 - cp's OEIS frontend

A103517 Expansion of (1+2*x-x^2)/(1-x)^2.

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 0

Paul Barry, Feb 09 2005

Row sums of A103516.
Also the number of maximal and maximum cliques in the (n+1) X (n+1) rook graph. - Eric W. Weisstein, Sep 14 2017
Also the number of maximal and maximum independent vertex sets in the (n+1) X (n+1) rook complement graph. - Eric W. Weisstein, Sep 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A103516.
Essentially the same as A004277, A005843, A051755, and A076032. - R. J. Mathar, Jul 31 2010
Cf. A272651 (for which this sequence is a conjectured continuation for large n).

[1] cat [2*n+2 : n in [1..60]]; // Wesley Ivan Hurt, Dec 07 2016

a := n -> 2*(n + 1) - 0^n: seq(a(n), n = 0..62); # Peter Luschny, May 12 2023

CoefficientList[Series[(-z^2 + 2*z + 1)/(z - 1)^2, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
CoefficientList[Series[2/(x - 1)^2 - 1, {x, 0, 62}], x] (* Robert G. Wilson v, Jan 29 2015 *)
Join[{1}, 2 Range[2, 20]] (* Eric W. Weisstein, Sep 14 2017 *)
Join[{1}, LinearRecurrence[{2, -1}, {4, 6}, 20]] (* Eric W. Weisstein, Sep 14 2017 *)

a(n) = 2*n + 2 - 0^n.
a(n) = Sum_{k=0..n} 0^(k(n-k))*(n+1).
Equals binomial transform of [1, 3, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008
a(n) = 2*a(n-1) - a(n-2) for n > 2. - Eric W. Weisstein, Sep 14 2017
G.f.: (1 + 2*x - x^2)/(-1 + x)^2. - Eric W. Weisstein, Sep 14 2017

A076034 Group the natural numbers so that the n-th group contains the smallest set of n relatively prime numbers: (1), (2, 3), (4, 5, 7), (6, 11, 13, 17), (8, 9, 19, 23, 25), (10, 21, 29, 31, 37, 41), ...

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 13, 17, 8, 9, 19, 23, 25, 10, 21, 29, 31, 37, 41, 12, 35, 43, 47, 53, 59, 61, 14, 15, 67, 71, 73, 79, 83, 89, 16, 27, 49, 55, 97, 101, 103, 107, 109, 18, 65, 77, 113, 127, 131, 137, 139, 149, 151, 20, 33, 91, 157, 163, 167, 173, 179, 181, 191, 193

Offset: 1

Amarnath Murthy, Oct 01 2002

			The triangle begins:
   1;
   2, 3;
   4,  5,  7;
   6, 11, 13, 17;
   8,  9, 19, 23, 25;
  10, 21, 29, 31, 37, 41;
  ...

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

Cf. A075383, A076032, A076033, A095167.

S:=[$1..1000]: Res:= NULL:
for n from 1 to 20 do
  A:= [S[1]]; R:= 1; count:= 1;
  for k from 2 while count < n do
    if andmap(t -> igcd(t,S[k])=1, A) then count:= count+1; A:= [op(A),S[k]]; R:= R,k; fi
  od;
  S:= subsop(op(map(t -> t=NULL, [R])),S);
  Res:= Res, op(A);
od:
Res; # Robert Israel, Dec 04 2022

Perl
```
# See Links section.
```

More terms from David Wasserman, Jan 29 2005

Crossrefs added by Paul Tek, Oct 24 2015

A076033 Final members of groups in A076034.

Original entry on oeis.org

1, 3, 7, 17, 25, 41, 61, 89, 109, 151, 193, 239, 281, 349, 409, 479, 557, 619, 709, 811, 907, 1013, 1109, 1229, 1327, 1481, 1601, 1741, 1889, 2063, 2221, 2381, 2551, 2729, 2917, 3137, 3331, 3539, 3733, 3943, 4177, 4441, 4663, 4937, 5171, 5437, 5683, 5923

Offset: 1

Amarnath Murthy, Oct 01 2002

nonn

Cf. A076032, A076034.

More terms from David Wasserman, Jan 29 2005

cp's OEIS Frontend

A103517 Expansion of (1+2*x-x^2)/(1-x)^2.

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A076034 Group the natural numbers so that the n-th group contains the smallest set of n relatively prime numbers: (1), (2, 3), (4, 5, 7), (6, 11, 13, 17), (8, 9, 19, 23, 25), (10, 21, 29, 31, 37, 41), ...

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A076033 Final members of groups in A076034.

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