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A230363
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Factorials representable as b^2 + triangular(c).
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0
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1, 1, 2, 6, 24, 120, 362880, 3628800, 39916800, 479001600, 6227020800, 1307674368000, 121645100408832000
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OFFSET
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1,3
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COMMENTS
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Numbers n such that n! is representable as a sum of a square and a triangular number: 0, 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 15, 19, ... .
1! = 1/2*1*(1+1), 3! = 1/2*3*(3+1) and 5! = 1/2*15*(15+1)/2 are triangular terms of the sequence. Next such term, if it exists is greater than 10000!. [Farideh Firoozbakht, Oct 18 2013]
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LINKS
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FORMULA
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EXAMPLE
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13! = 66708^2+1/2*59616(59616+1) = 78693^2+1/2*8298(8298+1), so 13! = 6227020800 is in the sequence. What is the next term of the sequence which has more than one representation of the form b^2 + triangular(c)? [Farideh Firoozbakht, Oct 18 2013]
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PROG
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(Python)
import math
f=1
for n in range(1, 1000000):
f *= n
t = b = 0
while t<=f:
x = f-t
a = int(math.sqrt(x))
if a*a==x:
print str(f)+', ',
break
b += 1
t = b*(b+1)/2
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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