A047264 Numbers that are congruent to 0 or 5 mod 6.

0, 5, 6, 11, 12, 17, 18, 23, 24, 29, 30, 35, 36, 41, 42, 47, 48, 53, 54, 59, 60, 65, 66, 71, 72, 77, 78, 83, 84, 89, 90, 95, 96, 101, 102, 107, 108, 113, 114, 119, 120, 125, 126, 131, 132, 137, 138, 143, 144, 149, 150, 155, 156, 161, 162, 167, 168, 173, 174

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Values of n for which Sum_{k=1..n} k*Fibonacci(k) is even (n > 0). Example: 5 is in the sequence because Sum_{k=1..5} k*Fibonacci(k) = 1*1 + 2*1 + 3*2 + 4*3 + 5*5 = 46. - Emeric Deutsch, Mar 28 2005

For a(n) is the n-th Tower of Hanoi move, the smallest disc (#1) is on peg A. If n == (1,2) mod 6, the disc is on peg C; and if n == (3,4) mod 6, the disc is on peg B. Disc #1 rotates C,B,A,C,B,A,C,B,A,... All discs start at "0" on peg A. Disc #1 is on peg A again for moves (5,6), (11,12), (17,18), ... - Gary W. Adamson, Jun 23 2012

Examples

			From _Vincenzo Librandi_, Aug 05 2010: (Start)
a(2) = 6*2 - 0 - 7 = 5;
a(3) = 6*3 - 5 - 7 = 6;
a(4) = 6*4 - 6 - 7 = 11. (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Herta T. Freitag, Problem B-776: An Even Sum, Fibonacci Quarterly, Vol. 32, No. 5 (1994), p. 468; An Even Sum, Solution to Problem B-77 by Paul S. Bruckman, ibid., Vol. 34, No. 1 (1996), p. 85.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Complement of A047227.

Programs

Maple

c:=proc(n) if n mod 6 = 0 or n mod 6 = 5 then n else fi end: seq(c(n),n=0..149); # Emeric Deutsch, Mar 28 2005

Mathematica

Select[Range[0, 149], MemberQ[{0, 5}, Mod[#, 6]] &] (* or *)
Fold[Append[#1, 6 #2 - Last@ #1 - 7] &, {0}, Range[2, 50]] (* or *)
Rest@ CoefficientList[Series[x^2*(5 + x)/((1 + x) (x - 1)^2), {x, 0, 50}], x] (* Michael De Vlieger, Jan 12 2018 *)

PARI

forstep(n=0,200,[5,1],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

PARI

a(n) = 3*n - 2 + (-1)^n \\ David Lovler, Aug 04 2022

Formula

a(n) = 3*n + (-1)^n - 2.

a(n) = 6*n - a(n-1) - 7 (with a(1)=0). - Vincenzo Librandi, Aug 05 2010

G.f.: x^2*(5+x) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

Let b(1)=0, b(2)=1 and b(k+2) = b(k+1) - b(k) + k^2; then a(n) is the sequence of integers such that b(a(n)) is a square = (a(n) + 1)^2. - Benoit Cloitre, Sep 04 2002

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=A007283(k) for k > 0. - Philippe Deléham, Oct 17 2011

Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(3)/4 - sqrt(3)*Pi/12. - Amiram Eldar, Dec 13 2021

E.g.f.: 1 + (3*x - 2)*exp(x) + exp(-x). - David Lovler, Aug 08 2022