A016933 a(n) = 6*n + 2.

2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004

Exponents n>1 for which 1 - x + x^n is reducible. - Ron Knott, Oct 13 2016

For the Collatz problem, these are the descenders' values that require division by 2. - Fred Daniel Kline, Jan 19 2017

For n > 3, also the number of (not necessarily maximal) cliques in the n-helm graph. - Eric W. Weisstein, Nov 29 2017

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See p. 9.
• Tanya Khovanova, Recursive Sequences.
• Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
• Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
• Eric Weisstein's World of Mathematics, Clique.
• Eric Weisstein's World of Mathematics, Helm Graph.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A008588, A016921, A016945, A016957, A016969, A017569, A016777, A016813.

Programs

Haskell

a016933 = (+ 2) . (* 6)  -- Reinhard Zumkeller, Jul 05 2013

Maple

a[1]:=2:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..47); # Zerinvary Lajos, Mar 16 2008

Mathematica

Range[2, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[6 n + 2, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
6 Range[0, 20] + 2 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{2, -1}, {8, 14}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[2 (1 + 2 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

PARI

a(n)=6*n+2 \\ Charles R Greathouse IV, Jul 10 2016

Sage

[i+2 for i in range(280) if gcd(i,6) == 6] # Zerinvary Lajos, May 20 2009

Formula

A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008

A157176(a(n)) = A013730(n). - Reinhard Zumkeller, Feb 24 2009

A089911(2*a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013

a(n) = 2*(6*n-1) - a(n-1) (with a(0)=2). - Vincenzo Librandi, Nov 20 2010

G.f.: 2*(1+2*x)/(1-x)^2. - Colin Barker, Jan 08 2012

a(n) = (3 * A016813(n) + 1) / 2.- Fred Daniel Kline, Jan 20 2017

a(n) = A016789(A005843(n)). - Felix Fröhlich, Jan 20 2017

Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 + log(2)/6. - Amiram Eldar, Dec 10 2021

a(n) = 2 * A016777(n). - Alois P. Heinz, Dec 27 2023

From Elmo R. Oliveira, Mar 08 2024: (Start)

a(n) = 2*a(n-1) - a(n-2) for n >= 2.

E.g.f.: 2*exp(x)*(1 + 3*x). (End)