A198796 -id:A198796 - cp's OEIS frontend

A254132 a(0)=1 and a(1)=2, then each term is x + y + x*y where x and y are the 2 last terms.

1, 2, 5, 17, 107, 1943, 209951, 408146687, 85691213438975, 34974584955819144511487, 2997014624388697307377363936018956287, 104819342594514896999066634490728502944926883876041385836543

Offset: 0

Michel Marcus, Jan 26 2015

nonn

			a(0) = 1, a(1) = 2, a(2) = 1+2+(1*2) = 5, a(3) = 2+5+(2*5) = 17.

Ana Rechtman, Janvier 2015, 3ème défi, (in French), Images des Mathématiques, CNRS, 2015.
Ana Rechtman, Solution, (in French), Images des Mathématiques, CNRS, 2015.

Cf. A000045 (Fibonacci), A063896 (similar, with initial values 0,1).

Cf. A198796 (2^n*3^(n+1)-1).

a254132[0]=1;a254132[n_]:=2^Fibonacci[n-1]*3^Fibonacci[n]-1;
a254132/@Range[0,11] (* Ivan N. Ianakiev, Jan 27 2015 *)

lista(nn) = {x = 1; y = 2; print1(x, ", ", y, ", "); for (j=1, nn, z = x + y + x*y; print1(z, ", "); x = y; y = z;);}

a(n) = if (!n, 1, 2^fibonacci(n)*3^fibonacci(n+1) - 1);

a(n) = a(n-1) + a(n-2) + a(n-1)*a(n-2).
a(0) = 1 and a(n) = 2^Fibonacci(n)*3^Fibonacci(n+1) - 1 (see 2nd link).
a(n) == 8 mod 9, for n > 2. - Ivan N. Ianakiev, Jan 27 2015

A377624 a(n) is the number of iterations of x -> 6*x + 5 until (# composites reached) = (# primes reached), starting with prime(n).

Original entry on oeis.org

17, 5, 1, 3, 9, 11, 15, 1, 1, 3, 7, 5, 5, 7, 1, 1, 5, 1, 1, 7, 5, 5, 9, 1, 5, 1, 1, 9, 3, 13, 1, 1, 5, 9, 1, 11, 5, 7, 1, 1, 1, 13, 5, 7, 9, 1, 1, 1, 3, 1, 1, 9, 3, 3, 1, 3, 7, 1, 9, 1, 1, 1, 5, 5, 1, 13, 1, 3, 9, 3, 1, 1, 3, 17, 1, 1, 5, 1, 3, 9, 1, 5, 5, 1

Offset: 1

Clark Kimberling, Nov 20 2024

nonn

For a guide to related sequences, see A377609.

			Starting with prime(1) = 2, we have 6*2+5 = 17, then 6*17+5 = 107, etc., resulting in a chain 2, 17, 107, 647, 3887, 23327, 139967, 839807, 5038847, 30233087, 181398527, 1088391167, 6530347007, 39182082047, 235092492287, 1410554953727, 8463329722367, 50779978334207 having 9 primes and 9 composites. Since every initial subchain has fewer composites than primes, a(1) = 18-1 = 17. (For more terms from the mapping x -> 6x-5, see A198796.)

Cf. A377609, A198796.

chain[{start_, u_, v_}] := If[CoprimeQ[u, v] && start*u + v != start,
   NestWhile[Append[#, u*Last[#] + v] &, {start}, !
      Count[#, ?PrimeQ] == Count[#, ?(! PrimeQ[#] &)] &], {}];
chain[{Prime[1], 6, -5}]
Map[Length[chain[{Prime[#], 6, -5}]] &, Range[1, 100]] - 1
(* Peter J. C. Moses, Oct 31 2024 *)

cp's OEIS Frontend

A254132 a(0)=1 and a(1)=2, then each term is x + y + x*y where x and y are the 2 last terms.

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