cp's OEIS Frontend

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

%I A145018 #71 Dec 13 2022 02:14:42
%S A145018 4,5,7,10,14,19,25,32,40,49,59,70,82,95,109,124,140,157,175,194,214,
%T A145018 235,257,280,304,329,355,382,410,439,469,500,532,565,599,634,670,707,
%U A145018 745,784,824,865,907,950,994,1039,1085,1132,1180,1229,1279,1330,1382,1435
%N A145018 a(n) = (n^2 - n + 8)/2.
%C A145018 The previous name was "a(1) = 4; then add 1 to the first number, then 2, then 3 and so on".
%C A145018 Numbers m such that 8m-31 is a square. - _Bruce J. Nicholson_, Jul 25 2017
%C A145018 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 A145018 G. C. Greubel, <a href="/A145018/b145018.txt">Table of n, a(n) for n = 1..1000</a>
%H A145018 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 A145018 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 A145018 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 A145018 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A145018 a(n) = (n^2 - n + 8)/2. - _Benoit Cloitre_.
%F A145018 From _R. J. Mathar_, Oct 01 2008: (Start)
%F A145018 G.f.: x*(4 -7*x +4*x^2)/(1-x)^3.
%F A145018 a(n) = a(n-1) + n - 1.
%F A145018 a(n) = 4 + A000217(n-1). (End)
%F A145018 a(n) = 4 + C(n,2), n>=1. - _Zerinvary Lajos_, Mar 12 2009
%F A145018 Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(31)*Pi/2)/sqrt(31). - _Amiram Eldar_, Dec 13 2022
%p A145018 A145018:=n->(n^2 - n + 8)/2: seq(A145018(n), n=1..100); # _Wesley Ivan Hurt_, Jul 25 2017
%t A145018 Nest[Append[#, #[[-1]] + Length@ #] &, {4}, 66] (* or *)
%t A145018 Rest@ CoefficientList[Series[x (4 - 7 x + 4 x^2)/(1 - x)^3, {x, 0, 67}], x] (* _Michael De Vlieger_, Jan 23 2019 *)
%o A145018 (Sage)[4+binomial(n,2) for n in range(1, 68)] # _Zerinvary Lajos_, Mar 12 2009
%o A145018 (PARI) x='x+O('x^50); Vec(x*(4 -7*x +4*x^2)/(1-x)^3) \\ _G. C. Greubel_, Feb 18 2017
%o A145018 (Magma) [(n^2 - n + 8)/2 : n in [1..50]]; // _Wesley Ivan Hurt_, Mar 25 2020
%Y A145018 Cf. A000217.
%K A145018 nonn,easy
%O A145018 1,1
%A A145018 Jayanth (mergujayanth(AT)yahoo.com), Sep 29 2008
%E A145018 More terms from _Alexander R. Povolotsky_, Sep 29 2008
%E A145018 Edited by _Benoit Cloitre_ and _R. J. Mathar_, Sep 30 2008
%E A145018 New name from _Hugo Pfoertner_, Aug 03 2021