A209724 -id:A209724 - cp's OEIS frontend

A153972 a(n) = 2^n + 6.

7, 8, 10, 14, 22, 38, 70, 134, 262, 518, 1030, 2054, 4102, 8198, 16390, 32774, 65542, 131078, 262150, 524294, 1048582, 2097158, 4194310, 8388614, 16777222, 33554438, 67108870, 134217734, 268435462, 536870918, 1073741830, 2147483654

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Old name was "a(n+1) = 2*(a(n) - 2) - 2, a(0) = 7".

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A209724.

[2^n+6: n in [0..40]]; // Vincenzo Librandi, May 13 2014

a=7; lst={a}; Do[a=(a-2)*2-2; AppendTo[lst,a], {n,30}]; lst
NestList[ 2# - 6 &, 7, 31] (* Robert G. Wilson v, Nov 28 2012 *)
CoefficientList[Series[(7 - 13 x)/((1 - x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, May 13 2014 *)
LinearRecurrence[{3,-2}, {7, 8}, 25] (* G. C. Greubel, Sep 01 2016 *)

a(n)=2^n+6 \\ Charles R Greathouse IV, Oct 07 2015

a(n+1) = A209724(2*n-1). - L. Edson Jeffery, Nov 28 2012
From Vincenzo Librandi, May 13 2014: (Start)
G.f.: (7 - 13*x)/((1 - x)*(1 - 2*x)).
a(n) = 2^n + 6.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1. (End)
E.g.f.: exp(2*x) + 6*exp(x). - G. C. Greubel, Sep 01 2016
a(n) = 2*a(n-1) - 6 with a(0) = 7. - Elmo R. Oliveira, Nov 11 2023

A338026 a(1) = 8; for n > 1, a(n) is the largest integer m such that m = ((2xa(n-1)) / (x+1)) - x, with x a positive nontrivial divisor of m.

Original entry on oeis.org

8, 9, 10, 12, 15, 20, 28, 42, 66, 110, 190, 342, 630, 1190, 2278, 4422, 8646, 17030, 33670, 66822, 132900, 264758, 528034, 1053990, 2105077, 4205820, 8405840, 16803405, 33595212, 67173930, 134324628, 268616475, 537185908, 1074305622, 2148516546

Offset: 1

Enric Reverter i Bigas, Oct 07 2020

nonn

There are no primes in the sequence and, excepting for a(1), no powers of 2.
For each n > 1 there are two integers f, g such that f*g = a(n) and f + g = 2*a(n-1) - a(n) - 1. (Empirical observation)
Excluding the condition that each term should be the largest one, the terms which satisfies the remaining conditions performs an irregular infinite net.
If the condition "be the largest term" is replaced for "be the smallest one" with the rest of conditions remaining, A209724 sequence is obtained. (This is true, at least, from a'(1) to a'(100))
Similar sequences are obtained with a'(1) a nonprime integer larger than 6, either a power of two or not. However, if a'(1) is not a power of two, there exist at least, one integer: a'(0) = ((a'(1)+x)*(x+1)) / (2*x) with x a positive nontrivial divisor of a'(1). Example: a'(1) = 26, a'(0) = 21, a'(-1) = 16. (As a'(-1) is a power of 2 there is not an a'(-2) term). If a'(1) = 6, a'(0) = a'(1) = a'(2) = ... = a'(k) = 6.

			a(5) = 15 = ((2*3*12) / 4) - 3 or ((2*5*12) / 6) - 5 = 15; Also 14 = ((2*2*12) / 3) - 2, but 15 is larger.

Enric Reverter i Bigas, Graphic example

Cf. A209724.

w[n_] := Module[{x, p}, Max[p /. List@ToRules@Reduce[p == (2 n*x)/(x + 1) - x == x*y && x > 1 && y > 1, p, Integers]]]; n := 8; k := {n}; m = 1; While[m < 35, {AppendTo[k, w[n]], n = w[n]}; m++]; k

cp's OEIS Frontend

A153972 a(n) = 2^n + 6.

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A338026 a(1) = 8; for n > 1, a(n) is the largest integer m such that m = ((2xa(n-1)) / (x+1)) - x, with x a positive nontrivial divisor of m.

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cp's OEIS Frontend

A153972 a(n) = 2^n + 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A338026 a(1) = 8; for n > 1, a(n) is the largest integer m such that m = ((2*x*a(n-1)) / (x+1)) - x, with x a positive nontrivial divisor of m.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

A338026 a(1) = 8; for n > 1, a(n) is the largest integer m such that m = ((2xa(n-1)) / (x+1)) - x, with x a positive nontrivial divisor of m.