OFFSET
1,1
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]
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..100
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
Erich Friedman, Jul 11 2001
EXTENSIONS
More terms from Naohiro Nomoto, Oct 01 2001
Offset corrected by Reinhard Zumkeller, Jun 10 2013
STATUS
approved