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a(n) = (5*4^n - 8)/6.
7

%I #37 Sep 08 2022 08:45:09

%S 2,12,52,212,852,3412,13652,54612,218452,873812,3495252,13981012,

%T 55924052,223696212,894784852,3579139412,14316557652,57266230612,

%U 229064922452,916259689812,3665038759252,14660155037012,58640620148052,234562480592212,938249922368852

%N a(n) = (5*4^n - 8)/6.

%C These numbers have a simple binary pattern: 10,1100,110100,11010100,1101010100, ... i.e., the n-th term has a binary expansion 1(10){n-1}0, that is, there are n-1 10's between the most significant 1 and the least significant 0.

%H Vincenzo Librandi, <a href="/A080675/b080675.txt">Table of n, a(n) for n = 1..170</a>

%H Andrei Asinowski, Cyril Banderier, Benjamin Hackl, <a href="https://www.semanticscholar.org/paper/On-extremal-cases-of-pop-stack-sorting-Asinowski/4b17b76c31559d6d8126b8f1126fdb357f3a5df0">On extremal cases of pop-stack sorting</a>, Permutation Patterns (Zürich, Switzerland, 2019).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).

%F a(1)=2, a(2)=12, a(n)=5*a(n-1)-4*a(n-2). - _Harvey P. Dale_, Oct 16 2012

%t (5*4^Range[30]-8)/6 (* or *) LinearRecurrence[{5,-4},{2,12},30] (* _Harvey P. Dale_, Oct 16 2012 *)

%o (Magma) [(5*4^n-8)/6: n in [1..40] ]; // _Vincenzo Librandi_, Apr 28 2011

%o (PARI) a(n)=(5*4^n-8)/6 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y a(n) = A072197(n-1) - 1 = A014486(|A106191(n)|). a(n) = A079946(A020988(n-2)) for n>=2. Cf. also A122229.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Mar 02 2003

%E Further comments added by _Antti Karttunen_, Sep 14 2006