非递归公式

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由于算术几何序列有明确的公式，我们可以建立非递归函数来计算简单但有用的结果，例如在任一 tick 上玩家的高度，或者在初始速度与持续时间的基础上计算跳跃的距离。

定义：

${\textstyle v_{0}}$ is the player's initial speed (speed on $t_{0}$ , before jumping)
${\textstyle t}$ is the number of ticks considered (ex: t=12 on flat ground, see Jump Duration)
${\textstyle J}$ is the "jump bonus" (0.3274 for sprintjump, 0.291924 for strafed sprintjump, 0.1 for 45° no-sprint jump...)
${\textstyle M}$ is the movement multiplier after jumping (1.3 for 45° sprint, 1.274 for normal sprint, 1.0 for no-sprint 45°...)

Vertical Movement (jump) [1.8]

Vertical speed after jumping ( $t\geq 6$ )

{\textstyle {\textrm {V}}_{Y}(t)=4\times 0.98^{t-5}-3.92}

Relative height after jumping ( $t\geq 6$ )

{\textstyle {\textrm {Y}}_{rel}(t)={\underset {\textrm {jumppeak}}{\underbrace {197.4-217\times 0.98^{5}} }}+200(0.98-0.98^{t-4})-3.92(t-5)}

For ${\textstyle t<6}$ , see below.

Vertical Movement (jump) [1.9+]

Vertical speed after jumping ( $t\geq 1$ )

{\textstyle {\textrm {V}}_{Y}(t)=0.42\times 0.98^{t-1}+4\times 0.98^{t}-3.92}

Relative height after jumping ( $t\geq 0$ )

{\textstyle {\textrm {Y}}_{rel}(t)=217\times (1-0.98^{t})-3.92t}

Horizontal Movement (instant jump)

Assuming the player was airborne before jumping.

Horizontal speed after sprintjumping ( $t\geq 2$ )

{\textstyle {\textrm {V}}_{H}(v_{0},t)={\frac {0.02M}{0.09}}+0.6\times 0.91^{t}\times \left(v_{0}+{\frac {J}{0.91}}-{\frac {0.02M}{0.6\times 0.91\times 0.09}}\right)}

Sprintjump distance ( $t\geq 2$ )

{\textstyle {\textrm {Dist}}(v_{0},t)=1.91v_{0}+J+{\frac {0.02M}{0.09}}(t-2)+{\frac {0.6\times 0.91^{2}}{0.09}}\times (1-0.91^{t-2})\times \left(v_{0}+{\frac {J}{0.91}}-{\frac {0.02M}{0.6\times 0.91\times 0.09}}\right)}

Note: These formulas are accurate for most values of $v_{0}$ , but some negative values can wind up activating the speed threshold and reset the player's speed at some point, thus rendering these formulas inaccurate.