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A169835 Perfect squares that are a product of two triangular numbers. 2
1, 9, 36, 100, 225, 441, 784, 900, 1225, 1296, 2025, 3025, 4356, 6084, 7056, 8281, 11025, 14400, 18496, 23409, 29241, 32400, 36100, 41616, 44100, 53361, 64009, 76176, 88209, 90000, 105625, 108900, 123201, 142884, 164836, 189225, 216225, 246016, 278784, 298116 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Includes (except for 0) A000537 and 3/2*x*(x+1) for x in A132596. - Robert Israel, Jan 16 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Erich Friedman, What's Special About This Number? (See entry 7056.)

MAPLE

N:= 10^6: # to get all terms <= N

A:= select(issqr, {seq(seq(a*(a+1)*b*(b+1)/4,

    b = a .. floor(sqrt(1/4+4*N/a/(a+1))-1/2)), a=1..floor(sqrt(4*N)))});

# if using Maple 11 or earlier, uncomment the next line

# sort(convert(A, list)); # Robert Israel, Jan 16 2015

MATHEMATICA

M = 10^6; (* to get all terms <= M *)

A = Union[Select[Flatten[Table[Table[(1/4) a (a+1) b (b+1), {b, a, Floor[ Sqrt[1/4 + 4M/(a (a+1))] - 1/2]}], {a, 1, Floor[Sqrt[4M]]}]], IntegerQ[ Sqrt[#]]&]] (* Jean-Fran├žois Alcover, Mar 09 2019, after Robert Israel *)

PROG

(PARI) istriangular(n)=issquare(8*n+1) \\ now one can use ispolygonal(n, 3)

isok(n) = {if (issquare(n), fordiv(n, d, if (d > sqrtint(n), break); if (istriangular(d) && istriangular(n/d), return (1)); ); ); return (0); } \\ Michel Marcus, Jul 24 2013

(Haskell)

a169835 n = a169835_list !! (n-1)

a169835_list = f [] (tail a000217_list) (tail a000290_list) where

   f ts us'@(u:us) vs'@(v:vs)

     | u <= v = f (u : ts) us vs'

     | any p $ map (divMod v) ts = v : f ts us' vs

     | otherwise = f ts us' vs

     where p (q, r) = r == 0 && a010054 q == 1

-- Reinhard Zumkeller, Mar 03 2015

CROSSREFS

Superset of A000537. Cf. A000217, A132596, A169836.

Cf. A000290, A010054.

Sequence in context: A085037 A231678 A231682 * A231686 A231688 A000537

Adjacent sequences:  A169832 A169833 A169834 * A169836 A169837 A169838

KEYWORD

nonn

AUTHOR

R. J. Mathar, May 30 2010

EXTENSIONS

Corrected (missing terms inserted) by R. J. Mathar, Jun 04 2010

STATUS

approved

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)