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A255138 a(n) = (1 + 2^n*(3 + 2*(-1)^n))/3. 3
2, 1, 7, 3, 27, 11, 107, 43, 427, 171, 1707, 683, 6827, 2731, 27307, 10923, 109227, 43691, 436907, 174763, 1747627, 699051, 6990507, 2796203, 27962027, 11184811, 111848107, 44739243, 447392427, 178956971 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Let N_1 be the set of odd natural numbers. Let F : N_1 -> N_1 be the map defined by F(x) = (3*x+1)/2^v(3*x+1) (see A075677), where v(y) denotes the 2-adic valuation of y. Let F^(k)(x) denote the k-fold iteration of F and defined by the recurrence F^(k)(x) = F(F^(k-1)(x)), with initial condition F^(0)(x) = x. Then, for n>0, a(n) is the least m in N_1 such that F^(n)(4*m-3) == 1 (mod 4).

Let k == 1 mod 4, and k(r) be the r-th iteration at which k appears in a Collatz sequence. When n >= 2 and k(r) == [2^(n+1) - a(n)] mod 2^(n+1), then n is the number of halving steps following k(r+1). For instance, since a(5) = 11, there are 5 halving steps following k(r+1) when k(r) == 53 mod 64, because 2^(5+1) = 64 and 64-11 = 53; e.g., k(r) = 117: 117 -> 352 -> 176 -> 88 -> 44 -> 22 -> 11. - Bob Selcoe, Feb 09 2017

LINKS

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

FORMULA

a(2*n) = A136412(n); a(2*n+1) = A007583(n).

G.f.: (2-x-2*x^2)/((x-1)*(2*x-1)*(2*x+1)). - R. J. Mathar, Jul 25 2015

a(n) = a(n-1)+4*a(n-2)-4*a(n-3) for n>2. - Wesley Ivan Hurt, Nov 05 2015

a(n) = 4*a(n-2) - 1. - Bob Selcoe, Feb 09 2017

MAPLE

A255138:=n->(1 + 2^n*(3 + 2*(-1)^n))/3: seq(A255138(n), n=0..50); # Wesley Ivan Hurt, Nov 05 2015

MATHEMATICA

a[n_] := (1 + 2^n*(3 + 2*(-1)^n))/3; Table[a[n], {n, 0, 29}]

PROG

(PARI) vector(30, n, n--; (1 + 2^n*(3 + 2*(-1)^n))/3) \\ Altug Alkan, Nov 05 2015

(MAGMA) [(1 + 2^n*(3 + 2*(-1)^n))/3: n in [0..50]]; // Wesley Ivan Hurt, Nov 05 2015

CROSSREFS

Cf. A007583, A075677, A136412.

Sequence in context: A258235 A021050 A194797 * A115629 A296461 A144696

Adjacent sequences:  A255135 A255136 A255137 * A255139 A255140 A255141

KEYWORD

nonn,easy

AUTHOR

L. Edson Jeffery, May 04 2015

STATUS

approved

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Last modified September 15 12:04 EDT 2019. Contains 327078 sequences. (Running on oeis4.)