|
%I #29 Sep 08 2022 08:45:44
%S 8,24,56,120,248,504,1016,2040,4088,8184,16376,32760,65528,131064,
%T 262136,524280,1048568,2097144,4194296,8388600,16777208,33554424,
%U 67108856,134217720,268435448,536870904,1073741816,2147483640
%N a(n) = 8*(2^n - 1).
%C Fifth diagonal of the array which contains m-acci numbers in the m-th row.
%C The base array is constructed from m-acci numbers starting each with 1, 1, and 2 and filling one row of the table (see the examples.)
%C The main and the upper diagonals of the table are the powers of 2, A000079.
%C The first subdiagonal is essentially A000225, followed by essentially A036563.
%C The next subdiagonal is this sequence here, followed by A159742, A159743, A159744, A159746, A159747, A159748.
%C a(n) written in base 2: 1000, 11000, 111000, 1111000, ..., i.e., n times 1 and 3 times 0 (A161770). - _Jaroslav Krizek_, Jun 18 2009
%C a(n) = A173787(n+3,3) = A175166(2*n)/A175161(n). - _Reinhard Zumkeller_, Feb 28 2010
%C Also numbers for which n^8/(n+8) is an integer. - _Vicente Izquierdo Gomez_, Jan 03 2013
%H G. C. Greubel, <a href="/A159741/b159741.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2).
%F From _R. J. Mathar_, Apr 22 2009: (Start)
%F a(n) = 3*a(n-1) - 2*a(n-2).
%F a(n) = 8*(2^n-1).
%F G.f.: 8*x/((2*x-1)*(x-1)). (End)
%F From _Jaroslav Krizek_, Jun 18 2009: (Start)
%F a(n) = Sum_{i=3..(n+2)} 2^i.
%F a(n) = Sum_{i=1..n} 2^(i+2).
%F a(n) = a(n-1) + 2^(n+2) for n >= 2. (End)
%e From _R. J. Mathar_, Apr 22 2009: (Start)
%e The base table is
%e .1..1....1....1....1....1....1....1....1....1....1....1....1....1
%e .1..1....1....1....1....1....1....1....1....1....1....1....1....1
%e .2..2....2....2....2....2....2....2....2....2....2....2....2....2
%e .0..2....3....4....4....4....4....4....4....4....4....4....4....4
%e .0..2....5....7....8....8....8....8....8....8....8....8....8....8
%e .0..2....8...13...15...16...16...16...16...16...16...16...16...16
%e .0..2...13...24...29...31...32...32...32...32...32...32...32...32
%e .0..2...21...44...56...61...63...64...64...64...64...64...64...64
%e .0..2...34...81..108..120..125..127..128..128..128..128..128..128
%e .0..2...55..149..208..236..248..253..255..256..256..256..256..256
%e .0..2...89..274..401..464..492..504..509..511..512..512..512..512
%e .0..2..144..504..773..912..976.1004.1016.1021.1023.1024.1024.1024
%e .0..2..233..927.1490.1793.1936.2000.2028.2040.2045.2047.2048.2048
%e .0..2..377.1705.2872.3525.3840.3984.4048.4076.4088.4093.4095.4096
%e Columns: A000045, A000073, A000078, A001591, A001592 etc. (End)
%p T := proc(n,m) option remember ; if n < 0 then 0; elif n <= 1 then 1; elif n = 2 then 2; else add(procname(n-i,m),i=1..m) ; fi: end: A159741 := proc(n) T(n+4,n+1) ; end: seq(A159741(n),n=1..40) ; # _R. J. Mathar_, Apr 22 2009
%t Table[8(2^n-1),{n,60}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%t LinearRecurrence[{3,-2},{8,24},30] (* _Harvey P. Dale_, Jan 01 2019 *)
%o (PARI) a(n)=8*(2^n-1) \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [8*(2^n -1): n in [1..50]]; // _G. C. Greubel_, May 22 2018
%K nonn,easy
%O 1,1
%A Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009
%E More terms from _R. J. Mathar_, Apr 22 2009
%E Edited by Al Hakanson (hawkuu(AT)gmail.com), May 11 2009
%E Comments claiming negative entries deleted by _R. J. Mathar_, Aug 24 2009
| 