A179903 - OEIS

%I #27 Jan 30 2023 12:49:32

%S 1,3,8,21,46,87,148,233,346,491,672,893,1158,1471,1836,2257,2738,3283,

%T 3896,4581,5342,6183,7108,8121,9226,10427,11728,13133,14646,16271,

%U 18012,19873,21858,23971,26216,28597,31118,33783,36596,39561,42682,45963

%N (1, 3, 5, 7, 9, ...) convolved with (1, 0, 3, 5, 7, 9, ...).

%H Vincenzo Librandi, <a href="/A179903/b179903.txt">Table of n, a(n) for n = 0..1000</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: (1 + 3*x + 8*x^2 + 21*x^3 + ...) = (1 + 3*x + 5*x^2 + 7*x^3 + 9*x^4 + ...) * (1 + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 + ...).

%F From _R. J. Mathar_, Aug 13 2010: (Start)

%F a(n) = 2 + A005900(n), n > 0.

%F G.f.: -(1 + x)*(x^3 - 4*x^2 + 2*x - 1)/(x - 1)^4. (End)

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jul 04 2012

%e a(5) = 46 = (9, 7, 5, 3, 1) dot (1, 0, 3, 5, 7) = 9 + 0 + 15 + 15 + 7.

%t CoefficientList[Series[-(1+x)*(x^3-4*x^2+2*x-1)/(x-1)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)

%t Join[{1},Table[ListConvolve[Range[1,2n+1,2],Join[{1,0},Range[3,2n-1,2]]],{n,50}]// Flatten] (* or *) LinearRecurrence[{4,-6,4,-1},{1,3,8,21,46},50] (* _Harvey P. Dale_, Jan 30 2023 *)

%o (Magma) I:=[1, 3, 8, 21, 46]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jul 04 2012

%Y Cf. A005900.

%K nonn,easy

%O 0,2

%A _Gary W. Adamson_, Jul 31 2010