Triangular Square Number

$N\_k\; =\; \{1\; \backslash over\; 32\}\; \backslash left(\; \backslash left(\; 1\; +\; \backslash sqrt\{2\}\; \backslash right)^\{2k\}\; -\; \backslash left(\; 1\; -\; \backslash sqrt\{2\}\; \backslash right)^\{2k\}\; \backslash right)^2\; .$

$n(n-1)/2\; =\; m^2.$

$(2n-1)^2=2m^2+1,$

$k^2=2m^2+1$

$s(N)\; =\; \backslash sqrt\{N\},$

$t(N)\; =\; \backslash lfloor\; \backslash sqrt\{2\; N\}\; \backslash rfloor.$

$t(N\_k)\; =\; \{1\; \backslash over\; 4\}\; \backslash left\backslash left(\; 1\; +\; \backslash sqrt\{2\}\; \backslash right)^k\; +\; \backslash left(\; 1\; -\; \backslash sqrt\{2\}\; \backslash right)^k\; \backslash right)^2\; -\; \backslash left(\; 1\; +\; (-1)^k\; \backslash right)^2\; \backslash right$.

External references

• Triangular numbers that are also square. From Interactive Mathematics Miscellany and Puzzles.

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