login
Expansion of (1 + x^2)*(1 + 3*x + x^2) / ((1 - x)*(1 - x^3)).
3

%I #14 Jan 27 2018 06:32:36

%S 1,4,6,10,14,16,20,24,26,30,34,36,40,44,46,50,54,56,60,64,66,70,74,76,

%T 80,84,86,90,94,96,100,104,106,110,114,116,120,124,126,130,134,136,

%U 140,144,146,150,154,156,160,164,166,170,174,176,180,184,186,190,194,196,200,204,206,210,214,216,220

%N Expansion of (1 + x^2)*(1 + 3*x + x^2) / ((1 - x)*(1 - x^3)).

%C Appears to be the coordination sequence for a tetravalent node in the bex tiling (or net).

%H Colin Barker, <a href="/A298784/b298784.txt">Table of n, a(n) for n = 0..1000</a>

%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/bex">The bex tiling (or net)</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F a(0)=1; thereafter, a(3*k) = 10*k, a(3*k+1) = 10*k+4, a(3*k+2) = 10*k+6.

%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. - _Colin Barker_, Jan 27 2018

%p f4:=proc(n)

%p if n=0 then 1

%p elif (n mod 3) = 0 then 10*n/3

%p elif (n mod 3) = 1 then (10*n+2)/3

%p else (10*n-2)/3; fi; end;

%p [seq(f4(n),n=0..80)];

%o (PARI) Vec((1 + x^2)*(1 + 3*x + x^2) / ((1 - x)^2*(1 + x + x^2)) + O(x^100)) \\ _Colin Barker_, Jan 27 2018

%Y Cf. A298785, A298786.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jan 26 2018