A145018 a(n) = (n^2 - n + 8)/2.

4, 5, 7, 10, 14, 19, 25, 32, 40, 49, 59, 70, 82, 95, 109, 124, 140, 157, 175, 194, 214, 235, 257, 280, 304, 329, 355, 382, 410, 439, 469, 500, 532, 565, 599, 634, 670, 707, 745, 784, 824, 865, 907, 950, 994, 1039, 1085, 1132, 1180, 1229, 1279, 1330, 1382, 1435

Offset: 1

Author

Jayanth (mergujayanth(AT)yahoo.com), Sep 29 2008

Keywords

• nonn
• easy

Comments

The previous name was "a(1) = 4; then add 1 to the first number, then 2, then 3 and so on".

Numbers m such that 8m-31 is a square. - Bruce J. Nicholson, Jul 25 2017

a(n) is the minimal number of vertices for a polyhedron with at least one vertex of degree k and at least one k-gonal face for each k=3..n+2. - Riccardo Maffucci, Aug 03 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• R. W. Maffucci, Self-dual polyhedra of given degree sequence, arXiv:2108.01058 [math.CO], 2021.
• Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217.

Programs

Magma

[(n^2 - n + 8)/2 : n in [1..50]]; // Wesley Ivan Hurt, Mar 25 2020

Maple

A145018:=n->(n^2 - n + 8)/2: seq(A145018(n), n=1..100); # Wesley Ivan Hurt, Jul 25 2017

Mathematica

Nest[Append[#, #[[-1]] + Length@ #] &, {4}, 66] (* or *)
Rest@ CoefficientList[Series[x (4 - 7 x + 4 x^2)/(1 - x)^3, {x, 0, 67}], x] (* Michael De Vlieger, Jan 23 2019 *)

PARI

x='x+O('x^50); Vec(x*(4 -7*x +4*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 18 2017

Sage

[4+binomial(n,2) for n in range(1, 68)] # Zerinvary Lajos, Mar 12 2009

Formula

a(n) = (n^2 - n + 8)/2. - Benoit Cloitre.

From R. J. Mathar, Oct 01 2008: (Start)

G.f.: x*(4 -7*x +4*x^2)/(1-x)^3.

a(n) = a(n-1) + n - 1.

a(n) = 4 + A000217(n-1). (End)

a(n) = 4 + C(n,2), n>=1. - Zerinvary Lajos, Mar 12 2009

Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(31)*Pi/2)/sqrt(31). - Amiram Eldar, Dec 13 2022

Extensions

More terms from Alexander R. Povolotsky, Sep 29 2008

Edited by Benoit Cloitre and R. J. Mathar, Sep 30 2008

New name from Hugo Pfoertner, Aug 03 2021