# 5151

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## Number

$5151$ (five thousand, one hundred and fifty-one) is:

$51 \times 101$

The $7$th hexamorphic number after $1$, $45$, $66$, $1225$, $1326$, $4950$:
$5151 = H_{51}$

The $51$st hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$:
$5151 = \ds \sum_{k \mathop = 1}^{51} \paren {4 k - 3} = 51 \paren {2 \times 51 - 1}$

The $101$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4560$, $4656$, $4753$, $4851$, $4950$, $5050$:
$5151 = \ds \sum_{k \mathop = 1}^{101} k = \dfrac {101 \times \paren {101 + 1} } 2$