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A047264
Numbers that are congruent to 0 or 5 mod 6.
9
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
1,2
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
LINKS
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.
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
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
Complement of A047227.
Sequence in context: A046608 A228357 A215033 * A343405 A369273 A277095
KEYWORD
nonn,easy
STATUS
approved