A058794 Row 3 of A007754.

2, 18, 52, 110, 198, 322, 488, 702, 970, 1298, 1692, 2158, 2702, 3330, 4048, 4862, 5778, 6802, 7940, 9198, 10582, 12098, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 32672, 35838, 39202, 42770, 46548, 50542, 54758, 59202, 63880

Offset: 0

Author

Christian G. Bower, Dec 02 2000

Keywords

• nonn
• easy

Comments

For n >= 2, a(n) is the number of ways a triangle with side length n-1 can be completely surrounded by diamonds. See illustrations in Links. - Craig Knecht, Oct 08 2024

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Craig Knecht, Number of ways a single triangle can be surrounded by diamonds.
• Craig Knecht, The 110 ways the T2 triangle can be surrounded by diamonds.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A007754, A154560.

Programs

GAP

List([0..40], n -> n^3+6*n^2+9*n+2); # G. C. Greubel, Nov 29 2018

Magma

[n^3+6*n^2+9*n+2: n in [0..40]]; // Vincenzo Librandi, Sep 22 2016

Maple

seq(sum(n^2-3, k=1..n), n=2..40); # Zerinvary Lajos, Jan 28 2008
seq ((n^3)-3*n, n=2..40); # Zerinvary Lajos, Jun 17 2008

Mathematica

LinearRecurrence[{4,-6,4,-1}, {2,18,52,110}, 40] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[n^3 + 6 n^2 + 9 n + 2, {n, 0, 40}] (* Bruno Berselli, Jan 10 2015 *)

PARI

vector(40, n, n--; n^3+6*n^2+9*n+2) \\ G. C. Greubel, Nov 29 2018

Sage

[(n^3+6*n^2+9*n+2) for n in range(40)] # G. C. Greubel, Nov 29 2018

Formula

a(n) = n^3 + 6*n^2 + 9*n + 2.

G.f.: 2*(1 + 5*x - 4*x^2 + x^3)/(1-x)^4. - Colin Barker, Jan 10 2012

a(n) = (n + 2)*(n^2 + 4*n + 1) = 2*A154560(n). - Bruno Berselli, Jan 10 2015

E.g.f.: (2 + 16*x + 9*x^2 + x^3)*exp(x). - G. C. Greubel, Nov 29 2018