This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227430 #36 Mar 24 2019 07:50:57
%S A227430 0,0,1,3,6,10,15,21,29,45,90,220,561,1365,3095,6555,13110,25126,46971,
%T A227430 87381,164921,320001,640002,1309528,2707629,5592405,11450531,23166783,
%U A227430 46333566,91869970,181348455,357913941,708653429,1410132405,2820264810,5662052980
%N A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).
%C A227430 Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
%C A227430 0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n): after 0, it is A192080.
%C A227430 0, 0, 0, 0, 1, 5, 15, 35, 70, 126,... e(n)
%C A227430 0, 0, 0, 1, 4, 10, 20, 35, 56, 85,... f(n)
%C A227430 0, 0, 1, 3, 6, 10, 15, 21, 29, 45,... a(n)
%C A227430 0, 1, 2, 3, 4, 5, 6, 8, 16, 45,... b(n)
%C A227430 1, 1, 1, 1, 1, 1, 2, 8, 29, 85,... c(n)
%C A227430 0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n).
%C A227430 a(n) + d(n) = A024495(n),
%C A227430 b(n) + e(n) = A131708(n),
%C A227430 c(n) + f(n) = A024493(n).
%C A227430 a(n) - d(n) = 0, 0, 1, 3, 6, 9, 9, 0,... A057083(n-2)
%C A227430 b(n) - e(n) = 0, 1, 2, 3, 3, 0, -9, -27,... A057682(n)
%C A227430 c(n) - f(n) = 1, 1, 1, 0, -3, -9, -18, -27,... A057681(n)
%C A227430 d(n) - a(n) = 0, 0, -1, -3, -6, -9, -9, 0,... -A057083(n-2)
%C A227430 e(n) - b(n) = 0, -1, -2, -3, -3, 0, 9, 27,... -A057682(n)
%C A227430 f(n) - c(n) = -1, -1, -1, 0, 3, 9, 18, 27,... -A057681(n).
%C A227430 The first column is A131531(n).
%C A227430 The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)?
%H A227430 Seiichi Manyama, <a href="/A227430/b227430.txt">Table of n, a(n) for n = 0..3000</a>
%H A227430 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6).
%F A227430 a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
%F A227430 a(n) = A024495(n) - A192080(n-5) for n>4.
%F A227430 G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - _Ralf Stephan_, Jul 13 2013
%F A227430 a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - _Seiichi Manyama_, Mar 23 2019
%e A227430 a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.
%t A227430 Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* _Harvey P. Dale_, Dec 17 2014 *)
%o A227430 (PARI) {a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ _Seiichi Manyama_, Mar 23 2019
%K A227430 nonn,easy
%O A227430 0,4
%A A227430 _Paul Curtz_, Jul 11 2013
%E A227430 Definition uses the g.f. of Ralf Stephan.
%E A227430 More terms from _Harvey P. Dale_, Dec 17 2014