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a(n) = (n^2 - n + 8)/2.
11

%I #71 Dec 13 2022 02:14:42

%S 4,5,7,10,14,19,25,32,40,49,59,70,82,95,109,124,140,157,175,194,214,

%T 235,257,280,304,329,355,382,410,439,469,500,532,565,599,634,670,707,

%U 745,784,824,865,907,950,994,1039,1085,1132,1180,1229,1279,1330,1382,1435

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

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

%C Numbers m such that 8m-31 is a square. - _Bruce J. Nicholson_, Jul 25 2017

%C 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

%H G. C. Greubel, <a href="/A145018/b145018.txt">Table of n, a(n) for n = 1..1000</a>

%H R. W. Maffucci, <a href="https://arxiv.org/abs/2108.01058">Self-dual polyhedra of given degree sequence</a>, arXiv:2108.01058 [math.CO], 2021.

%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

%H Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = (n^2 - n + 8)/2. - _Benoit Cloitre_.

%F From _R. J. Mathar_, Oct 01 2008: (Start)

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

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

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

%F a(n) = 4 + C(n,2), n>=1. - _Zerinvary Lajos_, Mar 12 2009

%F Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(31)*Pi/2)/sqrt(31). - _Amiram Eldar_, Dec 13 2022

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

%t Nest[Append[#, #[[-1]] + Length@ #] &, {4}, 66] (* or *)

%t Rest@ CoefficientList[Series[x (4 - 7 x + 4 x^2)/(1 - x)^3, {x, 0, 67}], x] (* _Michael De Vlieger_, Jan 23 2019 *)

%o (Sage)[4+binomial(n,2) for n in range(1, 68)] # _Zerinvary Lajos_, Mar 12 2009

%o (PARI) x='x+O('x^50); Vec(x*(4 -7*x +4*x^2)/(1-x)^3) \\ _G. C. Greubel_, Feb 18 2017

%o (Magma) [(n^2 - n + 8)/2 : n in [1..50]]; // _Wesley Ivan Hurt_, Mar 25 2020

%Y Cf. A000217.

%K nonn,easy

%O 1,1

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

%E More terms from _Alexander R. Povolotsky_, Sep 29 2008

%E Edited by _Benoit Cloitre_ and _R. J. Mathar_, Sep 30 2008

%E New name from _Hugo Pfoertner_, Aug 03 2021