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%I #68 Aug 19 2022 22:40:45
%S 0,5,6,11,12,17,18,23,24,29,30,35,36,41,42,47,48,53,54,59,60,65,66,71,
%T 72,77,78,83,84,89,90,95,96,101,102,107,108,113,114,119,120,125,126,
%U 131,132,137,138,143,144,149,150,155,156,161,162,167,168,173,174
%N Numbers that are congruent to 0 or 5 mod 6.
%C 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
%C 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
%H Michael De Vlieger, <a href="/A047264/b047264.txt">Table of n, a(n) for n = 1..10000</a>
%H Herta T. Freitag, <a href="https://www.fq.math.ca/Scanned/32-5/elementary32-5.pdf">Problem B-776: An Even Sum</a>, Fibonacci Quarterly, Vol. 32, No. 5 (1994), p. 468; <a href="https://www.fq.math.ca/Scanned/34-1/elementary34-1.pdf">An Even Sum</a>, Solution to Problem B-77 by Paul S. Bruckman, ibid., Vol. 34, No. 1 (1996), p. 85.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = 3*n + (-1)^n - 2.
%F a(n) = 6*n - a(n-1) - 7 (with a(1)=0). - _Vincenzo Librandi_, Aug 05 2010
%F G.f.: x^2*(5+x) / ( (1+x)*(x-1)^2 ). - _R. J. Mathar_, Oct 08 2011
%F 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
%F 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
%F Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(3)/4 - sqrt(3)*Pi/12. - _Amiram Eldar_, Dec 13 2021
%F E.g.f.: 1 + (3*x - 2)*exp(x) + exp(-x). - _David Lovler_, Aug 08 2022
%e From _Vincenzo Librandi_, Aug 05 2010: (Start)
%e a(2) = 6*2 - 0 - 7 = 5;
%e a(3) = 6*3 - 5 - 7 = 6;
%e a(4) = 6*4 - 6 - 7 = 11. (End)
%p 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
%t Select[Range[0, 149], MemberQ[{0, 5}, Mod[#, 6]] &] (* or *)
%t Fold[Append[#1, 6 #2 - Last@ #1 - 7] &, {0}, Range[2, 50]] (* or *)
%t Rest@ CoefficientList[Series[x^2*(5 + x)/((1 + x) (x - 1)^2), {x, 0, 50}], x] (* _Michael De Vlieger_, Jan 12 2018 *)
%o (PARI) forstep(n=0,200,[5,1],print1(n", ")) \\ _Charles R Greathouse IV_, Oct 17 2011
%o (PARI) a(n) = 3*n - 2 + (-1)^n \\ _David Lovler_, Aug 04 2022
%Y Complement of A047227.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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