A125201 a(n) = 8*n^2 - 7*n + 1.

1, 2, 19, 52, 101, 166, 247, 344, 457, 586, 731, 892, 1069, 1262, 1471, 1696, 1937, 2194, 2467, 2756, 3061, 3382, 3719, 4072, 4441, 4826, 5227, 5644, 6077, 6526, 6991, 7472, 7969, 8482, 9011, 9556, 10117, 10694, 11287, 11896, 12521, 13162, 13819, 14492, 15181, 15886

Offset: 0

Author

Reinhard Zumkeller, Nov 24 2006

Keywords

• nonn
• easy

Comments

Central terms of the triangle in A125199.

Sequence found by reading the line from 2, in the direction 2, 19, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 05 2011

Maximum number of regions that can be obtained in the plane by drawing n copies of a "strict long-legged M" letter. - N. J. A. Sloane, Aug 01 2025

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Two strict long-legged M's can divide the plane into a(2) = 19 regions.
• N. J. A. Sloane, Three strict long-legged M's can divide the plane into a(3) = 52 regions.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A051870, A125199.

Programs

Magma

[8*n^2-7*n+1:n in [1..44]]; // Vincenzo Librandi, Dec 27 2010

Mathematica

Table[8*n^2 - 7*n + 1, {n, 44}] (* Arkadiusz Wesolowski, Feb 15 2012 *)

PARI

a(n)=8*n^2-7*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 1 + A051870(n). - Omar E. Pol, Sep 05 2011

From Arkadiusz Wesolowski, Dec 25 2011: (Start)

a(1) = 2, a(n) = a(n-1) + 16*n - 15.

a(n) = 2*a(n-1) - a(n-2) + 16 with a(1) = 2 and a(2) = 19.

G.f.: (1 - x + 16*x^2)/(1 - x)^3. (End)

Sum_{n>=1} 1/a(n) = (psi(9/16+sqrt(17)/16) - psi(9/16-sqrt(17)/16))/sqrt(17) = 0.61242052... - R. J. Mathar, Apr 22 2024

From Elmo R. Oliveira, Oct 31 2024: (Start)

E.g.f.: exp(x)*(8*x^2 + x + 1) - 1.

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

Extensions

a(0) = 1 added by N. J. A. Sloane, Aug 01 2025 (this will require several additional changes).