

A091823


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


17



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
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OFFSET

1,1


COMMENTS

a(n) = position in row of 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 2blocks of a 2nset X then, for n>=2, a(n1) is the number of (2n2)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)


LINKS

Bruno Berselli, Table of n, a(n) for n = 1..1000
GuoNiu Han, Enumeration of Standard Puzzles
GuoNiu Han, Enumeration of Standard Puzzles, p. 21. [Cached copy]
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


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


EXAMPLE

Entries in the ratio 1:2:3 appear in row 14 starting at position 4 (1001, 2002, 3003); entries in the ratio 2:3:4 appear in row 34 starting at position 13 (927983760, 1391975640, 1855967520); and so on (row 62 pos 26, row 98 pos 43, ...).


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 *)


PROG

(Perl) #!/usr/bin/perl $a = 1; while (1) { $k = $a*(2*$a + 3)  1; print "$k, "; $a ++; }
(MAGMA) [2*n^2+3*n1: n in [1..50]]; // Bruno Berselli, Mar 28 2014
(PARI) a(n)=2*n^2+3*n1 \\ Charles R Greathouse IV, Sep 24 2015


CROSSREFS

Cf. A007318, A014106, A060626.
Sequence in context: A056708 A307271 A092484 * A024834 A143867 A024809
Adjacent sequences: A091820 A091821 A091822 * A091824 A091825 A091826


KEYWORD

nonn,easy


AUTHOR

Howard A. Landman, Mar 08 2004


STATUS

approved



