OFFSET
1,1
REFERENCES
Gelca, R. and Andreescu, T. (2007). Putnam and beyond. New York: Springer. Page 278, Question 807.
FORMULA
The formula ((n^2 - 3)^2)/4 + n^2, where n is odd, generates the subsequence 2, 18, 146, 578, 1602, 3602, 7058, 12546, 20738, 32402, 48402 ... .
EXAMPLE
1) k^2 + 2 = 0^2 + 2 = 2 = 1^2 + 1^2 = a^2 + b^2
2) k^2 + 2 = 4^2 + 2 = 18 = 3^2 + 3^2 = a^2 + b^2
3) k^2 + 2 = 12^2 + 2 = 146 = 5^2 + 11^2 = a^2 + b^2
4) k^2 + 2 = 24^2 + 2 = 578 = 7^2 + 23^2 = a^2 + b^2 (also, 578 = 17^2 + 17^2)
5) k^2 + 2 = 40^2 + 2 = 1602 = 9^2 + 39^2 = a^2 + b^2
PROG
(Python)
# A list of squares of integers. Currently contains 0 and 1; will be added elements below when calling 'procedure':
squaresList = [0, 1]
# We define a function that, given k**2+2, finds if it is a sum of two of the squares stored in the list:
def find_if_sum (a_var):
for a_squared in range (0, len(squaresList)):
for b_squared in range (a_squared, len(squaresList)):
if (a_var == squaresList[a_squared] + squaresList[b_squared]):
print (str(a_var) + ", " + "a_squared: " + str(squaresList[a_squared]) + " and b_squared: " + str(squaresList[b_squared]), sep = '', )
# We call this function on the base cases.
my_var = 0**2 + 2 # Here we directly compute k**2 + 2 for k = 0
find_if_sum (my_var) # This is the call to the function
my_var = 1**2 + 2 # Here we directly compute k**2 + 2 for k = 1
find_if_sum (my_var) # This is the call to the function
def procedure ():
for k in range (2, 1501): # Check all integers up to 1500, including
my_var = k**2
squaresList.append(my_var)
my_var += 2 # Add 2 to the square of k
find_if_sum (my_var) # This is the call to the function
procedure() # Call the procedure
(Magma) a:=[]; for m in [0..500] do for k in [0..m+1] do if IsSquare(m^2+2-k^2) then Append(~a, m^2+2); break; end if; end for; end for; a; // Marius A. Burtea, Nov 07 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Miriam Briskman, Nov 06 2019
EXTENSIONS
-
STATUS
approved