A072261 a(n) = 4*a(n-1) + 1, a(1)=7.

7, 29, 117, 469, 1877, 7509, 30037, 120149, 480597, 1922389, 7689557, 30758229, 123032917, 492131669, 1968526677, 7874106709, 31496426837, 125985707349, 503942829397, 2015771317589, 8063085270357, 32252341081429, 129009364325717, 516037457302869

Offset: 1

Author

N. Rathankar (rathankar(AT)yahoo.com), Jul 08 2002

Keywords

• nonn
• easy

Comments

These are the integers N which on application of the Collatz function yield the number 11. The Collatz function: if N is an odd number then (3N+1)/2^r yields a positive odd integer for some value of r (which in this case is 11).

These numbers reach 11 in Collatz function iteration after 2(n+1) steps and so end in 1 after exactly 2n+18 steps. - Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 08 2004

Numbers whose binary representation is 111 together with n - 1 times 01. For example, a(4) = 469 = 111010101 (2). - Omar E. Pol, Nov 24 2012

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-4).
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A002450, A072197, A072257, A072258, A072259, A072260, A072262, A099730, A178415, A347834.

Programs

GAP

List([1..25], n-> (11*4^n -2)/6); # G. C. Greubel, Jan 14 2020

Magma

[(11*4^n -2)/6: n in [1..25]]; // G. C. Greubel, Jan 14 2020

Maple

seq(coeff(series(x*(7-6*x)/((1-x)*(1-4*x)), x, n+1), x, n), n = 1..25); # G. C. Greubel, Jan 14 2020

Mathematica

a[n_]:= 4a[n-1] +1; a[1]=7; Table[a[n], {n,25}]
NestList[4#+1&,7,30] (* or *) LinearRecurrence[{5,-4},{7,29},30] (* Harvey P. Dale, Sep 04 2023 *)

PARI

Vec(x*(7-6*x)/((1-x)*(1-4*x)) + O(x^25)) \\ Colin Barker, Oct 27 2019

Sage

[(11*4^n -2)/6 for n in (1..25)] # G. C. Greubel, Jan 14 2020

Formula

a(n) = (11*4^n - 2)/6 = 22*A002450(n-1) + 7. - Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 08 2004

From Colin Barker, Oct 27 2019: (Start)

G.f.: x*(7 - 6*x) / ((1 - x)*(1 - 4*x)).

a(n) = 5*a(n-1) - 4*a(n-2) for n>2. (End)

E.g.f.: (-9 - 2*exp(x) + 11*exp(4*x))/6. - G. C. Greubel, Jan 14 2020

a(n) = a(n-1) + 11*2^(2*n-3), for n >= 2, with a(1) = 7. - Wolfdieter Lang, Aug 16 2021

a(n) = A178415(4, n) = A347834(3, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021

Extensions

Edited and extended by Robert G. Wilson v, Jul 17 2002

More terms from Colin Barker, Oct 27 2019