|
%I #32 Sep 08 2022 08:45:49
%S 1,10,55,253,1081,4465,18145,73153,293761,1177345,4713985,18865153,
%T 75479041,301953025,1207885825,4831690753,19327057921,77308821505,
%U 309236465665,1236948221953,4947797606401,19791199862785,79164818325505,316659311050753,1266637319700481
%N a(n) = (3*2^(n-1)-1)*(3*2^n-1).
%C A subsequence of the triangular numbers A000217.
%H Vincenzo Librandi, <a href="/A169720/b169720.txt">Table of n, a(n) for n = 0..1000</a>
%H Alice V. Kleeva, <a href="/A169720/a169720a.jpg">Grid for this sequence</a>
%H Alice V. Kleeva, <a href="/A169720/a169720b.jpg">Illustration of initial terms</a>
%H Robert Munafo, <a href="http://www.mrob.com/pub/math/seq-a169720.html">Sequence A169720, and two others by Alice V. Kleeva</a>
%H Robert Munafo, <a href="/A169720/a169720.pdf">Sequence A169720, and two others by Alice V. Kleeva</a> [Cached copy, in pdf format, included with permission]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F G.f.: (1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)). - _Paul D. Hanna_, Apr 29 2010
%F a(n) = A000217(A033484(n)). - _Mitch Harris_, Dec 02 2012
%F a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - _Vincenzo Librandi_, Dec 03 2012
%F a(n) = (3*A169726(n)-1)/2. - _L. Edson Jeffery_, Dec 03 2012
%F a(n) = A006095(n+2) +3*A006095(n+1) - A006905(n). - _R. J. Mathar_, Dec 04 2016
%t CoefficientList[Series[(1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -14, 8}, {1, 10, 55}, 30] (* _Vincenzo Librandi_, Dec 03 2012 *)
%o (PARI) a(n)=polcoeff((1+3*x-x^2)/((1-x)*(1-2*x)*(1-4*x)+x*O(x^n)),n) \\ _Paul D. Hanna_, Apr 29 2010
%o (Magma) I:=[1, 10, 55]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Dec 03 2012
%Y Cf. A169721-A169727, A033484, A000217.
%K nonn,easy
%O 0,2
%A Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010
|
