A091823 a(n) = 2*n^2 + 3*n - 1.

4, 13, 26, 43, 64, 89, 118, 151, 188, 229, 274, 323, 376, 433, 494, 559, 628, 701, 778, 859, 944, 1033, 1126, 1223, 1324, 1429, 1538, 1651, 1768, 1889, 2014, 2143, 2276, 2413, 2554, 2699, 2848, 3001, 3158, 3319, 3484, 3653, 3826, 4003, 4184, 4369, 4558, 4751

Offset: 1

Author

Howard A. Landman, Mar 08 2004

Keywords

• nonn
• easy

Comments

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.

The row numbers are given by A060626.

This sequence plus 1 (i.e., a(n) = 2*n^2 + 3*n) is the sequence A014106. - Howard A. Landman, Mar 28 2004

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

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)

Examples

			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; ...).
		

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011, p. 21. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Milan Janjic, Two Enumerative Functions. [Broken link]
• Leo Tavares, Illustration: Square/Triangular Union
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A007318, A014106, A060626.

Cf. A000290, A000217.

Programs

Magma

[2*n^2+3*n-1: n in [1..50]]; // Bruno Berselli, Mar 28 2014

Maple

A091823:=n->2*n^2 + 3*n - 1; seq(A091823(n), n=1..100); # Wesley Ivan Hurt, Mar 27 2014

Mathematica

Table[2 n^2 + 3 n - 1, {n, 50}] (* Bruno Berselli, Mar 28 2014 *)

PARI

a(n)=2*n^2+3*n-1 \\ Charles R Greathouse IV, Sep 24 2015

Perl

#!/usr/bin/perl $a = 1; while (1) { $k = $a*(2*$a + 3) - 1; print "$k,"; $a ++; }

Formula

a(n) = n + 4*binomial(2+n, n), with offset 0. - Zerinvary Lajos, May 12 2006

G.f.: x*(4 + x - x^2)/(1 - x)^3. - Vincenzo Librandi, Mar 28 2014

a(n) = A000290(n+1) + 2*A000217(n) - 2. - Leo Tavares, Aug 31 2023

E.g.f.: 1 + exp(x)*(2*x^2 + 5*x - 1). - Stefano Spezia, Jun 16 2024