A236333 The (n-2)-th (n>=3) triple of terms gives coefficients of double trinomial P_n(x) = ((n-2)^2*x^2 + n*x + 2)/2 (see comment).

1, 3, 2, 4, 4, 2, 9, 5, 2, 16, 6, 2, 25, 7, 2, 36, 8, 2, 49, 9, 2, 64, 10, 2, 81, 11, 2, 100, 12, 2, 121, 13, 2, 144, 14, 2, 169, 15, 2, 196, 16, 2, 225, 17, 2, 256, 18, 2, 289, 19, 2, 324, 20, 2, 361, 21, 2, 400, 22, 2, 441, 23, 2, 484, 24, 2, 529, 25, 2, 576, 26, 2, 625, 27, 2, 676, 28, 2, 729, 29, 2

Offset: 3

Author

Vladimir Shevelev, Jan 22 2014

Keywords

• nonn
• easy

Comments

Let {G_n(k)}_(k>=0) be sequence of n-gonal numbers. Then G_n(P_n(k)) = G_n(P_n(k)-1) + G_n((n-2)*k+1).

Examples

			Let n=5, k=4. Then G_5(k)=k*(3*k-1)/2 (Cf. A000326) and the double trinomial 2*P_5(x)= 9*x^2+5*x+2, P_5(4)=(9*4^2+5*4+2)/2=83,
Thus, we have G_5(83)=G_5(82)+G_5(13), or 83*124 = 41*245 + 13*19 = 10292.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Crossrefs

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.

Programs

Magma

I:=[1,3,2,4,4,2,9,5,2]; [n le 9 select I[n] else 3*Self(n-3)-3*Self(n-6)+Self(n-9): n in [1..90]]; // Vincenzo Librandi, Feb 02 2014

Mathematica

a[n_]:=Which[Mod[n,3]==0,n^2/9,Mod[n,3]==1,(n+5)/3,True,2]; Map[a,Range[3,103]]
CoefficientList[Series[(-1-3 x-2 x^2-x^3+5 x^4+4 x^5-2 x^7-2 x^8)/((-1+x)^3 (1+x+x^2)^3),{x,0,100}],x]

PARI

Vec(-x^3*(2*x^8+2*x^7-4*x^5-5*x^4+x^3+2*x^2+3*x+1)/((x-1)^3*(x^2+x+1)^3) + O(x^100)) \\ Colin Barker, Jan 23 2014

Formula

If n==0 (mod 3), then a(n) = n^2/9;

if n==1 (mod 3), then a(n) = (n+5)/3;

if n==2 (mod 3), then a(n) = 2.

G.f.: -x^3*(2*x^8+2*x^7-4*x^5-5*x^4+x^3+2*x^2+3*x+1) / ((x-1)^3*(x^2+x+1)^3). - Colin Barker, Jan 23 2014