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%I M3436 #157 Aug 31 2023 16:06:49
%S 1,4,12,28,55,96,154,232,333,460,616,804,1027,1288,1590,1936,2329,
%T 2772,3268,3820,4431,5104,5842,6648,7525,8476,9504,10612,11803,13080,
%U 14446,15904,17457,19108,20860,22716,24679,26752,28938,31240,33661,36204,38872,41668,44595,47656,50854,54192
%N a(n) = (n+1)*(n^2+n+2)/2; g.f.: (1 + 2*x^2) / (1 - x)^4.
%C Enumerates certain paraffins.
%C a(n) is the (n+1)st (n+3)-gonal number. - _Floor van Lamoen_, Oct 20 2001
%C Sum of n terms of an arithmetic progression with the first term 1 and the common difference n: a(1)=1, a(2) = 1+3, a(3) = 1+4+7, a(4) = 1+5+9+13, etc. - _Amarnath Murthy_, Mar 25 2004
%C This is identical to: first triangular number A000217, 2nd square number A000290, 3rd pentagonal number A000326, 4th hexagonal number A000384, 5th heptagonal number A000566, 6th octagonal number A000567, ..., (n+1)-th (n+3)-gonal number = main diagonal of rectangular array T(n,k) of polygonal numbers, by diagonals, referred to in A086271. - _Jonathan Vos Post_, Dec 19 2007
%C Also (n + 1)! times the determinant of the n X n matrix given by m(i,j) = (i+1)/i if i=j and otherwise 1. For example, (6 + 1)!*Det[{{2,1,1,1,1,1}, {1,3/2,1,1,1,1},{1,1,4/3,1,1,1}, {1,1,1,5/4,1,1}, {1,1,1,1,6/5,1}, {1,1,1,1,1,7/6}}] = 154 = a(6). - _John M. Campbell_, May 20 2011
%C a(n-1) = N_2(n), n>=1, is the number of 2-faces of n planes in generic position in three-dimensional space. See comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - _Wolfdieter Lang_, May 27 2011
%C For n>2, a(n) is 2 * (average cycle weight of primitive Hamiltonian cycles on a simply weighted K_n) (see link). - _Jon Perry_, Nov 23 2014
%C a(n) is the partial sums of A104249. - _J. M. Bergot_, Dec 28 2014
%C Sum of the numbers in the 1st column of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - _Wesley Ivan Hurt_, May 17 2021
%C From _Enrique Navarrete_, Mar 27 2023: (Start)
%C a(n) is the number of ordered set partitions of an (n+1)-set into 2 sets such that the first set has 0, 1, or 2 elements, the second set has no restrictions, and we choose an element from the second set. For n=4, the a(4) = 55 set partitions of [5] are the following (where the element selected from the second set is in parentheses):
%C { }, {(1), 2, 3, 4, 5} (5 of these);
%C {1}, {(2), 3, 4, 5} (20 of these);
%C {1, 2}, {(3), 4, 5} (30 of these). (End)
%D V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_2.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H William A. Tedeschi, <a href="/A006000/b006000.txt">Table of n, a(n) for n = 0..10000</a>
%H S. M. Losanitsch, <a href="http://dx.doi.org/10.1002/cber.189703002144">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926.
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
%H P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-23.1.290">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-382. [See p. 301].
%H Jon Perry, <a href="/A006000/a006000_1.pdf">Weighted Hamiltonian Cycles</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = Sum_{j=1..n+1} (binomial(0,0*j) + binomial(n+1,2)). - _Zerinvary Lajos_, Jul 25 2006
%F a(n-1) = n + (n^3 - n^2)/2 = n + n*T(n-1) where T(n-1) is a triangular number, n >= 1. - _William A. Tedeschi_, Aug 22 2010
%F a(n) = A002817(n)*4/n for n > 0. - _Jon Perry_, Nov 21 2014
%F E.g.f.: (1 + x)*(2 + 4*x + x^2)*exp(x)/2. - _Robert Israel_, Nov 24 2014
%F a(n) = A057145(n+3,n+1). - _R. J. Mathar_, Jul 28 2016
%F a(n) = A000124(n) * (n+1). - _Alois P. Heinz_, Aug 31 2023
%p A006000:=(1+2*z**2)/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation
%t a[n_]:=(n^3-n^2)/2+n; Table[a[n],{n,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2010 *)
%t CoefficientList[Series[(1 + 2 x^2) / (1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 21 2014 *)
%o (Python)
%o def a(n):
%o return n + (n**3 - n**2)//2 # _William A. Tedeschi_, Aug 22 2010
%Y Cf. A000124.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
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