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 A072197 a(n) = 4*a(n-1) + 1 with a(0) = 3. 23
 3, 13, 53, 213, 853, 3413, 13653, 54613, 218453, 873813, 3495253, 13981013, 55924053, 223696213, 894784853, 3579139413, 14316557653, 57266230613, 229064922453, 916259689813, 3665038759253, 14660155037013, 58640620148053, 234562480592213, 938249922368853 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := 2, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n - 1) = (-1)^n*charpoly(A, -2). - Milan Janjic, Jan 26 2010 Numbers whose binary representation is 11 together with n times 01. For example, 213 = 11010101 (2). - Omar E. Pol, Nov 22 2012 The Collatz-function starting with a(n) will terminate at 1 after 2*n + 7 steps. This is because 3*a(n) + 1 = 5*2^(2n + 1), and the Collatz-function starting with 5 terminates at 1 after 5 additional steps.  So for example, a(2) = 53; Collatz sequence starting with 53 follows: 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 (11 steps). - Bob Selcoe, Apr 03 2015 a(n) is also the sum of the numerator and denominator of the binary fractions 0.1, 0.101, 0.10101, 0.1010101... Thus 0.1 = 1/2 with 1 + 2 = 3, 0.101 = 1/2 + 1/8 = 5/8 with 5 + 8 = 13; 0.10101 = 1/2 + 1/8 + 1/32 = 21/32 with 21 + 32 = 53. - J. M. Bergot, Sep 28 2016 a(n), for n >= 2, is also the smallest odd number congruent to 5 modulo 8 for which the modifies reduced Collatz map given in A324036 has n consecutive extra steps compared to the reduced Collatz map given in A075677. Nicolas Vaillant, Philippe Delarue, - Wolfdieter Lang, May 09 2019 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (5,-4). FORMULA a(n) = (10*4^n - 1)/3 = 10*A002450(n) + 3. - Henry Bottomley, Dec 02 2002 a(n) = 5*a(n - 1) - 4*a(n - 2), n > 1. - Vincenzo Librandi, Oct 31 2011 a(n) = 2^(2(n + 1)) - (2^(2n + 1) + 1)/3 = A000302(n + 1) - A007583(n). - Vladimir Pletser, Apr 12 2014 a(n) = (5*2^(2n + 1) - 1)/3. - Bob Selcoe, Apr 03 2015 G.f.: (3-2*x) / ((1-x)*(1-4*x)). - Colin Barker, Sep 28 2016 a(n) = A020988(n) + A020988(n+1) + 1 = 2*(A002450(n) + A002450(n+1)) + 1. - Yosu Yurramendi, Jan 24 2017 EXAMPLE a(1) = 13 because a(0) = 3 and 4 * 3 + 1 = 13. a(2) = 53 because a(1) = 13 and 4 * 13 + 1 = 53. a(3) = 213 because a(2) = 53 and 4 * 53 + 1 = 213. MAPLE A072197:=n->(10*4^n - 1)/3: seq(A072197(n), n=0..30); # Wesley Ivan Hurt, Sep 29 2016 MATHEMATICA Table[(10(4^n) - 1)/3, {n, 0, 19}] (* Alonso del Arte, Nov 22 2012 *) NestList[4#+1&, 3, 30] (* Harvey P. Dale, Mar 09 2019 *) PROG (MAGMA) [(10*4^n-1)/3: n in [0..30]]; // Vincenzo Librandi, Oct 31 2011 (PARI) a(n)=10*4^n\3 \\ Charles R Greathouse IV, Apr 06 2016 (PARI) Vec((3-2*x)/((1-x)*(1-4*x)) + O(x^30)) \\ Colin Barker, Sep 28 2016 CROSSREFS Cf. A000302, A002450, A007583, A075677, A324036. Sequence in context: A082376 A065059 A198584 * A065838 A052990 A151209 Adjacent sequences:  A072194 A072195 A072196 * A072198 A072199 A072200 KEYWORD nonn,easy AUTHOR N. Rathankar (rathankar(AT)yahoo.com), Jul 03 2002 EXTENSIONS More terms from Henry Bottomley, Dec 02 2002 STATUS approved

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Last modified September 18 09:54 EDT 2019. Contains 327168 sequences. (Running on oeis4.)