A242658 a(n) = 3*n^2 - 3*n + 2.

2, 2, 8, 20, 38, 62, 92, 128, 170, 218, 272, 332, 398, 470, 548, 632, 722, 818, 920, 1028, 1142, 1262, 1388, 1520, 1658, 1802, 1952, 2108, 2270, 2438, 2612, 2792, 2978, 3170, 3368, 3572, 3782, 3998, 4220, 4448, 4682, 4922, 5168, 5420, 5678, 5942, 6212, 6488, 6770, 7058, 7352

Offset: 0

Author

N. J. A. Sloane, May 30 2014

Keywords

• nonn
• easy

Comments

An exercise in Smith (1950), my secondary school algebra book.

For n > 0, also the number of (not necessarily maximal) cliques in the (n-1)-triangular grid graph. - Eric W. Weisstein, Nov 29 2017

References

• C. Smith, A Treatise on Algebra, Macmillan, London, 5th ed., 1950, p. 429, Example 2(i).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Clique
• Eric Weisstein's World of Mathematics, Triangular Grid Graph
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

A077588 is the same except for the initial term.

Cf. A002378, A242659.

Programs

Magma

[3*n^2 - 3*n + 2: n in [0..70]]; // Vincenzo Librandi, Sep 05 2016

Mathematica

Table[3 n^2 - 3 n + 2, {n, 0, 100}] (* Vincenzo Librandi, Sep 05 2016 *)
LinearRecurrence[{3, -3, 1}, {2, 8, 20}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[-2 (1 - 2 x + 4 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

PARI

a(n) = 3*n^2-3*n+2 \\ Altug Alkan, Sep 05 2016

Formula

From Chai Wah Wu, May 30 2016: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.

G.f.: 2*(-4*x^2 + 2*x - 1)/(x - 1)^3. (End)

E.g.f.: exp(x)*(2 + 3*x^2). - Stefano Spezia, Dec 27 2021

a(n) = A002378(n) + A002378(n-1) + A002378(n-2). - Peter Bala, Jun 11 2024