A318777 - OEIS

%I #19 Aug 04 2020 01:45:42

%S 1,1,1,1,1,3,5,7,9,11,17,27,41,59,81,115,169,251,369,531,761,1099,

%T 1601,2339,3401,4923,7121,10323,15001,21803,31649,45891,66537,96539,

%U 140145,203443,295225,428299,621377,901667,1308553,1899003,2755601,3998355,5801689,8418795

%N Coefficients in expansion of 1/(1 - x - 2*x^5).

%C The coefficients in the expansion of 1/(1 - x - 2*x^5) are given by the sequence generated by the row sums in triangle A318775.

%C Coefficients in expansion of 1/(1 - x - 2*x^5) are given by the sum of numbers along "fourth Layer" skew diagonals pointing top-right in triangle A013609 ((1+2x)^n) and by the sum of numbers along "fourth Layer" skew diagonals pointing top-left in triangle A038207 ((2+x)^n), see links.

%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

%H Zagros Lalo, <a href="/A318777/a318777.pdf">Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n.</a>

%H Zagros Lalo, <a href="/A318777/a318777_1.pdf">Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n.</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,2).

%F a(0)=1, a(n) = a(n-1) + 2 * a(n-5) for n = 0,1...; a(n)=0 for n < 0.

%F G.f.: -1/(2*x^5 + x - 1). - _Chai Wah Wu_, Aug 03 2020

%p seq(coeff(series((1-x-2*x^5)^(-1),x,n+1), x, n), n = 0 .. 50); # _Muniru A Asiru_, Sep 26 2018

%t a[0] = 1; a[n_] := a[n] = If[n < 0, 0, a[n - 1] + 2 * a[n - 5]];Table[a[n], {n, 0, 45}] // Flatten

%t LinearRecurrence[{1, 0, 0, 0, 2}, {1, 1, 1, 1, 1}, 46]

%t CoefficientList[Series[1/(1 - x - 2 x^5), {x, 0, 45}], x]

%o (GAP) a:=[1,1,1,1,1,3];; for n in [7..50] do a[n]:=a[n-1]+2*a[n-5]; od; a; # _Muniru A Asiru_, Sep 26 2018

%Y Cf. A318775, A318776.

%Y Cf. A013609, A038207.

%Y Essentially a duplicate of A143447.

%K nonn,easy

%O 0,6

%A _Zagros Lalo_, Sep 25 2018