A062730 Rows of Pascal's triangle which contain 3 terms in arithmetic progression.

7, 12, 14, 19, 21, 23, 32, 34, 45, 47, 60, 62, 77, 79, 96, 98, 117, 119, 140, 142, 165, 167, 192, 194, 221, 223, 252, 254, 285, 287, 320, 322, 357, 359, 396, 398, 437, 439, 480, 482, 525, 527, 572, 574, 621, 623, 672, 674, 725, 727, 780, 782, 837, 839

Offset: 1

Author

Erich Friedman, Jul 11 2001

Keywords

• nice
• nonn

Comments

Except for n=19, all n < 1000 have the form k^2-2 or k^2-4. When n=k^2-2, the three terms in AP are consecutive binomial coefficients C(n,k(k-1)/2-2), C(n,k(k-1)/2-1), and C(n,k(k-1)/2). When n=k^2-4, the three terms in AP differ by two: C(n,k(k-1)/2-4), C(n,k(k-1)/2-2), and C(n,k(k-1)/2). When n=19, the three terms in AP are C(19,4), C(19,6), and C(19,7). [From T. D. Noe, Mar 23 2009]

Examples

			12 is in the list since the 12th row of Pascal's triangle starts 1 12 (66) 220 (495) 792 (924) and 66 495 924 are in arithmetic progression.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A034868, A007318.

Programs

Haskell

-- import Data.List (intersect)
a062730 n = a062730_list !! (n-1)
a062730_list =  filter f $ [3..] where
   f x = not $ all null $ zipWith
         (\us (v:vs) -> map (v -) us `intersect` map (subtract v) vs)
         (tail $ init $ inits bns) (tail $ init $ tails bns)
         where bns = a034868_row x
-- Reinhard Zumkeller, Jun 10 2013

Mathematica

kmax = 30; row[n_] := Table[Binomial[n, k], {k, 0, Floor[n/2]}]; Reap[Do[r = row[n]; If[ (r /. {_, a_, _, b_, _, c_, _} /; b - a == c - b -> {}) == {}, Print[n]; Sow[n]], {n, Union[{19}, Range[2, kmax]^2 - 2, Range[2, kmax]^2 - 4]}]][[2, 1]] (* Jean-François Alcover, Jul 11 2012, after T. D. Noe *)

Formula

G.f.: (-5x^8+3x^7+7x^6-3x^5+5x^4-5x^3-12x^2+5x+7)/[(1-x)(1-x^2)^2] (conjectured). - Ralf Stephan, May 08 2004

a(n)=(n^2+8*n+8)/4 for n>4 and even; a(n)=(n^2+10*n+9)/4 for n>4 and odd (conjectured). - Colin Barker, Aug 29 2013

Extensions

More terms from Naohiro Nomoto, Oct 01 2001

Offset corrected by Reinhard Zumkeller, Jun 10 2013