A165665 - OEIS

%I #29 Jul 26 2024 21:16:31

%S 1,8,40,176,736,3008,12160,48896,196096,785408,3143680,12578816,

%T 50323456,201310208,805273600,3221159936,12884770816,51539345408,

%U 206157905920,824632672256,3298532786176,13194135339008,52776549744640

%N a(n) = (3*2^n - 2) * 2^n.

%C Binomial transform of A058481. Second binomial transform of (A082505 without initial term 0). Third binomial transform of A010686.

%C Partial sums are in A060867.

%C a(n) is the sum of the odd numbers taken progressively by moving through them by 2^n-tuples. a(0)=1; a(1) = 3+5=8; a(2) = 7+9+11+13 = 40; a(3) = 15+17+19+21+23+25+27+29 = 176; a(n) = sum_{k=0,1,..,A000225(n)} (A000225(n+1)+2*k).  - _J. M. Bergot_, Dec 06 2014

%C The number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 23 2016

%H Vincenzo Librandi, <a href="/A165665/b165665.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).

%F a(n) = 6*a(n-1)-8*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

%F a(n) = 8*A010036(n-1) for n > 0.

%F G.f.: (2*x+1)/((1-2*x)*(1-4*x)).

%F E.g.f.: 3*e^(4*x) - 2*e^(2*x). - _Robert Israel_, Dec 15 2014

%t Table[(3*2^n-2)2^n,{n,0,30}] (* or  *) LinearRecurrence[{6,-8},{1,8},30] (* _Harvey P. Dale_, Nov 18 2020 *)

%o (Magma) [ (3*2^n-2)*2^n: n in [0..23] ];

%o (PARI) a(n)=(3*2^n-2)*2^n \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A058481, A082505, A010686 (repeat 1, 5), A060867, A010036, A124647.

%K nonn,easy

%O 0,2

%A _Klaus Brockhaus_, Sep 24 2009