A199412 -id:A199412 - cp's OEIS frontend

A261399 a(1)=1; thereafter a(n) = (2/5)(96^(n-2)+1).

1, 4, 22, 130, 778, 4666, 27994, 167962, 1007770, 6046618, 36279706, 217678234, 1306069402, 7836416410, 47018498458, 282110990746, 1692665944474, 10155995666842, 60935974001050, 365615844006298, 2193695064037786, 13162170384226714, 78973022305360282, 473838133832161690

Offset: 1

N. J. A. Sloane, Aug 19 2015

Partial sums of A081341. - Klaus Purath, Jul 28 2020

K. Hong, H. Lee, H. J. Lee and S. Oh, Small knot mosaics and partition matrices, J. Phys. A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT], 2013-2014. See Cor. 2.
Index entries for linear recurrences with constant coefficients, signature (7,-6).
Index entries for sequences related to knots

The number 22, the third term here, is the same 22 seen in A261400 and illustrated in a link in that entry.

Cf. A199412.

G.f.: x-2*x^2*(-2+3*x) / ( (6*x-1)*(x-1) ). - R. J. Mathar, Aug 19 2015
a(n) = 2*A199412(n-2), n>1. - R. J. Mathar, Aug 19 2015
From Klaus Purath, Jul 28 2020: (Start)
a(n) = 7*a(n-1) - 6*a(n-2), n > 2.
a(n) = 6*a(n-1) - 2, n > 1.
a(n) = 3*6^(n-2) + a(n-1), n > 1.
(End)

A377621 a(n) is the number of iterations of x -> 6*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

Original entry on oeis.org

11, 7, 7, 3, 1, 1, 3, 5, 5, 5, 1, 1, 1, 3, 3, 5, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 3, 5, 3, 3, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 9, 3, 1, 1, 5, 7, 1, 9, 1, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 5, 1, 7, 3, 9, 7, 3, 1, 1, 1, 1, 1, 1, 7, 3

Offset: 1

Clark Kimberling, Nov 20 2024

nonn

For a guide to related sequences, see A377609.

			Starting with prime(1) = 2, we have 6*2-1 = 11, then 6*11-1 = 65, etc., resulting in a chain 2, 11, 65, 389, 2333, 13997, 83981, 503885, 3023309, 18139853, 108839117, 653034701 having 6 primes and 6 composites. Since every initial subchain has fewer composites than primes, a(1) = 12-1 = 11. (For more terms from the mapping x -> 6x-1, see A199412.)

Cf. A377609, A199412.

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, -1}]
Map[Length[chain[{Prime[#], 6, -1}]] &, Range[1, 100]] - 1
(* Peter J. C. Moses, Oct 31 2024 *)

cp's OEIS Frontend

A261399 a(1)=1; thereafter a(n) = (2/5)(96^(n-2)+1).

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A377621 a(n) is the number of iterations of x -> 6*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

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cp's OEIS Frontend

A261399 a(1)=1; thereafter a(n) = (2/5)*(9*6^(n-2)+1).

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Formula

A377621 a(n) is the number of iterations of x -> 6*x - 1 until (# composites reached) = (# primes reached), starting with prime(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A261399 a(1)=1; thereafter a(n) = (2/5)(96^(n-2)+1).