|
%I #73 Jan 01 2023 09:45:42
%S -1,2,6,11,17,24,32,41,51,62,74,87,101,116,132,149,167,186,206,227,
%T 249,272,296,321,347,374,402,431,461,492,524,557,591,626,662,699,737,
%U 776,816,857,899,942,986,1031,1077,1124,1172,1221,1271,1322
%N a(n) = (n^2 + 5*n - 2)/2.
%C If Y_i (i=1,2,3,4) are 2-blocks of an n-set X then, for n>=8, a(n-3) is the number of (n-2)-subsets of X intersecting each Y_i (i=1,2,3,4). - _Milan Janjic_, Nov 09 2007
%C Numbers m > -3 such that 8*m + 33 is a square. - _Bruno Berselli_, Aug 20 2015
%C a(n-1) yields the second Betti number of a path graph on n vertices. - _Samuel J. Bevins_, Nov 27 2022
%H G. C. Greubel, <a href="/A046691/b046691.txt">Table of n, a(n) for n = 0..5000</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H P. Di Francesco, O. Golinelli and E. Guitter, <a href="http://arXiv.org/abs/hep-th/9506030">Meander, folding and arch statistics</a>, arXiv:hep-th/9506030, 1995.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (-1 + 5*x - 3*x^2)/(1 - x)^3.
%F a(n) = a(n-1) + n + 2 with a(0) = -1. - _Vincenzo Librandi_, Nov 18 2010
%F a(n) = 3*A000096(n-1) - 2*A000096(n-2), with A000096(-2)=A000096(-1)=-1. - _Bruno Berselli_, Dec 17 2014
%F a(n) = 2*A000217(n) - A000217(n-2), with A000217(-2)=1, A000217(-1)=0. - _Bruno Berselli_, Oct 13 2016
%F E.g.f.: (1/2)*(x^2 + 6*x - 2)*exp(x). - _G. C. Greubel_, Jul 13 2017
%F a(n) + a(n+1) = A028884(n). - _R. J. Mathar_, Mar 13 2021
%F a(n) = A000217(n+2) - 4. - _G. C. Greubel_, Jul 31 2022
%F Sum_{n>=0} 1/a(n) = 7/12 + 2*Pi*tan(sqrt(33)*Pi/2)/sqrt(33). - _Vaclav Kotesovec_, Dec 31 2022
%p seq(binomial(n+3, 2)-4, n=0..49); # _Zerinvary Lajos_, Jan 13 2007
%t Table[(n^2 +5n -2)/2, {n, 0, 50}] (* _Bruno Berselli_, Dec 17 2014 *)
%o (PARI) a(n)=(n^2+5*n-2)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [Binomial(n+3,2) -4: n in [0..50]]; // _G. C. Greubel_, Jul 31 2022
%o (SageMath) [(n^2 +5*n -2)/2 for n in (0..50)] # _G. C. Greubel_, Jul 31 2022
%Y Triangular numbers (A000217) minus 4. Cf. A027379.
%Y Cf. A000096, A000217, A002522, A028884.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_
|
