A090197 a(n) = n^3 + 6*n^2 + 6*n + 1.

1, 14, 45, 100, 185, 306, 469, 680, 945, 1270, 1661, 2124, 2665, 3290, 4005, 4816, 5729, 6750, 7885, 9140, 10521, 12034, 13685, 15480, 17425, 19526, 21789, 24220, 26825, 29610, 32581, 35744, 39105, 42670, 46445, 50436, 54649, 59090, 63765

Offset: 0

Author

Philippe Deléham, Jan 22 2004

Keywords

• nonn
• easy

Comments

N(4,n) where N(4,x) is the 4th Narayana polynomial.

Number of corona of a triangle of order n surrounded by triangles of order n. - Craig Knecht, Oct 30 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Craig Knecht, Area of the coronal frames of T2.
• Craig Knecht, Corona of a triangle of order n.
• Craig Knecht, 100 frame animation for n=4 triangle surrounding itself.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

For N(3,n), see A028387.

Cf. A001263, A016921.

Programs

Magma

I:=[1, 14, 45, 100]; [n le 4 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, Jun 24 2012

Mathematica

LinearRecurrence[{4,-6, 4, -1},{1,14,45,100},40] (* Vincenzo Librandi, Jun 24 2012 *)

PARI

n^3+6*n^2+6*n+1 \\ Charles R Greathouse IV, Jan 17 2012

Formula

a(n) = N(4,n) = Sum_{k>0} A001263(4, k)*n^(k-1) = (n+1)*(n^2+5*n+1).

G.f.: (1 + 10*x - 5*x^2) / (x-1)^4. - R. J. Mathar, Sep 07 2011

a(n) + A016921(n+1) = (n+2)^3. - Bruno Berselli, Jun 24 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 24 2012

E.g.f.: exp(x)*(1 + 13*x + 9*x^2 + x^3). - Stefano Spezia, Nov 22 2024