201 lines
8.8 KiB
Markdown
201 lines
8.8 KiB
Markdown
---
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title: ∆ᵏ-constant Sequence
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::sequence algebra::polynomial
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tags:
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- algebra
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- sequence
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---
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## Overview
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Let $(a_n)$ be a sequence. We define the **$k$th differences of $(a_n)$** recursively:
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* The $0$th differences of $(a_n)$ is $(a_n)$.
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* The $k$th differences of $(a_n)$ is the sequence given by subtracting consecutive terms of the $(k-1)st$ differences of $(a_n)$.
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A sequence is said to be **$\Delta^k$-constant** if the $k$th differences are constant.
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> The closed formula for a sequence will be a degree $k$ polynomial if and only if the sequence is $\Delta^k$-constant.
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This is the discrete analogue to (continuous) derivatives of polynomials.
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%%ANKI
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Basic
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What are the $0$th differences of $(a_n)_{n \geq 0}$?
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Back: $(a_n)_{n \geq 0}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109096-->
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END%%
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%%ANKI
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Basic
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What are the $1$st differences of $(a_n)_{n \geq 0}$?
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Back: $(b_n)_{n \geq 1}$ where $b_n = a_n - a_{n - 1}$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109118-->
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END%%
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%%ANKI
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Basic
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How do we refer to the $1$st differences of $(a_n)$ more naturally?
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Back: As "the differences of $(a_n)$."
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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What is the base case of the recursive definition of the $k$th differences of $(a_n)$?
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Back: $k = 0$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109133-->
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END%%
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%%ANKI
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Basic
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What is the recurrence of the recursive definition of the $(k + 1)$st differences of $(a_n)$?
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Back: The $(k + 1)$st differences is the differences of the $k$th differences.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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What does it mean for $(a_n)$ to be $\Delta^k$-constant?
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Back: The $k$th differences of $(a_n)$ is constant.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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What can be said about the closed formula of a $\Delta^k$-constant sequence?
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Back: It is a polynomial with degree $k$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Cloze
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We say a sequence is {$\Delta^k$-constant} when the {$k$th differences is constant}.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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$\Delta^k$-constant sequences are a discrete analogue to what calculus concept?
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Back: Derivatives.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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Tags: calculus
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<!--ID: 1713580109197-->
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END%%
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%%ANKI
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Basic
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What kind of mathematical expression do $\Delta^k$-constant sequences relate to?
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Back: Polynomials.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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Tags: calculus
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<!--ID: 1713580109203-->
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END%%
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%%ANKI
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Basic
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How can we prove every quadratic sequence, say $(a_n)$, has arithmetic differences?
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Back: By showing $a_{n+1} - a_n$ is linear.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109225-->
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END%%
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%%ANKI
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Basic
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How can we prove every cubic sequence, say $(a_n)$, has quadratic differences?
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Back: By showing $a_{n+1} - a_n$ is quadratic.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109232-->
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END%%
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%%ANKI
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Cloze
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The closed formula for a sequence will be a {degree $k$ polynomial} if and only if the $k$th differences {is constant}.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109237-->
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END%%
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%%ANKI
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Basic
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How are arithmetic sequences defined in terms of $\Delta^k$ polynomials?
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Back: A sequence is arithmetic if and only if it is $\Delta^1$-constant.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109244-->
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END%%
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%%ANKI
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Basic
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How are geometric sequences defined in terms of $\Delta^k$ polynomials?
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Back: N/A
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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*Why* can't a geometric sequence be $\Delta^k$-constant for some $k \geq 0$?
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Back: Because the closed formula of a geometric sequence is not a polynomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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Suppose $(a_n)$ is a $\Delta^2$-constant sequence. What general form describes its closed formula?
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Back: $an^2 + bn + c$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109261-->
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END%%
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%%ANKI
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Basic
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Suppose $(a_n)$ is a $\Delta^3$-constant sequence. What general form describes its closed formula?
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Back: $an^3 + bn^2 + cn + d$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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Suppose $(a_n)$ is a $\Delta^k$-constant sequence. Where is $k$ repeated in $(a_n)$'s closed formula?
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Back: At the largest degree of the polynomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109270-->
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END%%
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%%ANKI
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Basic
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What are the terms in a $\Delta^0$-constant sequences?
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Back: $c, c, c, \ldots$ for some constant $c$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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How do $\Delta^k$-constant sequences relate to polynomial fitting?
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Back: The closed formula of a $\Delta^k$-constant sequence is a polynomial we can fit.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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%%ANKI
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Basic
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How many data points of a $\Delta^k$-constant sequence are needed to polynomial fit its closed formula?
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Back: $k + 1$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
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## Bibliography
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). |