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A062730 Rows of Pascal's triangle which contain 3 terms in arithmetic progression. 5

%I

%S 7,12,14,19,21,23,32,34,45,47,60,62,77,79,96,98,117,119,140,142,165,

%T 167,192,194,221,223,252,254,285,287,320,322,357,359,396,398,437,439,

%U 480,482,525,527,572,574,621,623,672,674,725,727,780,782,837,839

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

%C 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]

%H Reinhard Zumkeller, <a href="/A062730/b062730.txt">Table of n, a(n) for n = 1..100</a>

%F 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

%F 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

%e 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.

%t 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-Fran├žois Alcover_, Jul 11 2012, after _T. D. Noe_ *)

%o (Haskell)

%o -- import Data.List (intersect)

%o a062730 n = a062730_list !! (n-1)

%o a062730_list = filter f $ [3..] where

%o f x = not $ all null $ zipWith

%o (\us (v:vs) -> map (v -) us `intersect` map (subtract v) vs)

%o (tail $ init $ inits bns) (tail $ init $ tails bns)

%o where bns = a034868_row x

%o -- _Reinhard Zumkeller_, Jun 10 2013

%Y Cf. A034868, A007318.

%K nice,nonn

%O 1,1

%A _Erich Friedman_, Jul 11 2001

%E More terms from _Naohiro Nomoto_, Oct 01 2001

%E Offset corrected by _Reinhard Zumkeller_, Jun 10 2013

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Last modified June 23 11:02 EDT 2021. Contains 345397 sequences. (Running on oeis4.)