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%I #59 Sep 27 2023 13:42:11
%S 4,13,26,43,64,89,118,151,188,229,274,323,376,433,494,559,628,701,778,
%T 859,944,1033,1126,1223,1324,1429,1538,1651,1768,1889,2014,2143,2276,
%U 2413,2554,2699,2848,3001,3158,3319,3484,3653,3826,4003,4184,4369,4558
%N a(n) = 2*n^2 + 3*n - 1.
%C a(n) is the position of the row in Pascal's triangle (A007318) in which three consecutive entries appear in the ratio n: n+1: n+2. (Even valid for n = 0 if you allow for a position of -1 to have value 0.) The solution is unique for each n.
%C The row numbers are given by A060626.
%C This sequence plus 1 (i.e., a(n) = 2*n^2 + 3*n) is the sequence A014106. - _Howard A. Landman_, Mar 28 2004
%C If Y and Z are a 2-blocks of a 2n-set X then, for n>=2, a(n-1) is the number of (2n-2)-subsets of X intersecting Y. - _Milan Janjic_, Nov 18 2007
%C One might prepend an initial -1: "-1, 4, 13, 26, 43, ..." - _Vladimir Joseph Stephan Orlovsky_, Oct 25 2008 (This would require too many other changes. - _N. J. A. Sloane_, Mar 27 2014)
%H Bruno Berselli, <a href="/A091823/b091823.txt">Table of n, a(n) for n = 1..1000</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011, p. 21. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.
%H Leo Tavares, <a href="/A091823/a091823.jpg">Illustration: Square/Triangular Union</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = n + 4*binomial(2+n, n), with offset 0. - _Zerinvary Lajos_, May 12 2006
%F G.f.: x*(4 + x - x^2)/(1 - x)^3. - _Vincenzo Librandi_, Mar 28 2014
%F a(n) = A000290(n+1) + 2*A000217(n) - 2. - _Leo Tavares_, Aug 31 2023
%e Entries in the ratio 1:2:3 appear in row 14 of Pascal's triangle (A007318) starting at position 4 (1001, 2002, 3003). Entries in the ratio 2:3:4 appear in row 34 of Pascal's triangle starting at position 13 (927983760, 1391975640, 1855967520); and so on (row 62, pos. 26; row 98, pos. 43; ...).
%p A091823:=n->2*n^2 + 3*n - 1; seq(A091823(n), n=1..100); # _Wesley Ivan Hurt_, Mar 27 2014
%t Table[2 n^2 + 3 n - 1, {n, 50}] (* _Bruno Berselli_, Mar 28 2014 *)
%o (Perl) #!/usr/bin/perl $a = 1; while (1) { $k = $a*(2*$a + 3) - 1; print "$k,"; $a ++; }
%o (Magma) [2*n^2+3*n-1: n in [1..50]]; // _Bruno Berselli_, Mar 28 2014
%o (PARI) a(n)=2*n^2+3*n-1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A007318, A014106, A060626.
%Y Cf. A000290, A000217.
%K nonn,easy
%O 1,1
%A _Howard A. Landman_, Mar 08 2004
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