A081272 Downward vertical of triangular spiral in A051682.

1, 25, 85, 181, 313, 481, 685, 925, 1201, 1513, 1861, 2245, 2665, 3121, 3613, 4141, 4705, 5305, 5941, 6613, 7321, 8065, 8845, 9661, 10513, 11401, 12325, 13285, 14281, 15313, 16381, 17485, 18625, 19801, 21013, 22261, 23545, 24865, 26221, 27613, 29041, 30505

Offset: 0

Author

Paul Barry, Mar 15 2003

Keywords

• nonn
• easy

Comments

Reflection of A081271 in the horizontal A051682.

Binomial transform of (1, 24, 36, 0, 0, 0, .....).

One of the six verticals of a triangular spiral which starts with 1 (see the link). Other verticals are A060544, A081589, A080855, A157889, A038764. - Yuriy Sibirmovsky, Sep 18 2016.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
• Yuriy Sibirmovsky, Six verticals of the triangular spiral.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A051682.

Cf. A062741, A060544, A081589, A080855, A157889, A038764.

Programs

Mathematica

Table[n^2 + (n + 1)^2, {n, 0, 300, 3}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 25, 85}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
Table[n^2 + (n + 1)^2, {n, 0, 150, 3}] (* Vincenzo Librandi, Aug 07 2013 *)

PARI

x='x+O('x^99); Vec((1+22*x+13*x^2)/(1-x)^3) \\ Altug Alkan, Sep 18 2016

Formula

a(n) = C(n, 0) + 24*C(n, 1) + 36*C(n, 2).

a(n) = 18*n^2 + 6*n + 1.

G.f.: (1 + 22*x + 13*x^2)/(1 - x)^3.

E.g.f.: exp(x)*(1 + 24*x + 18*x^2). - Stefano Spezia, Mar 07 2023