Coffeescript, 25 bytes
k=(n)->((n*n-1)<<(n-1))+1
The formula for generating Kangaroo numbers is this:
Explanation of formula:
The number of 1's in K(n)'s final sum is 2^(n - 1) + 1.
The number of n's in K(n)'s final sum is 2^(n - 1), so the sum of all the n's is n * 2^(n - 1).
The number...
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Using the the ten inferences of the Natural Deduction System prove DeMorgan's laws.
The Rules of Natural Deduction
Negation Introduction: {(P → Q), (P → ¬Q)} ⊢ ¬P
Negation Elimination: {(¬P → Q), (¬P → ¬Q)} ⊢ P
And Introduction: {P, Q} ⊢ P ʌ Q
And Elimination: P ʌ Q ⊢ {P, Q}
Or Int...
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Ungolf batch
This question wants you to golf batch programms. In Batch you can use
set x=averylongbatchcommand
%x% arg1
which expands to
set x=averylongbatchcommand
averylongbatchcommand arg1
Your task is it to expand these statements.
As you can also use set for arithmetic purposes like...0
In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is typically written as ∨ or +.
"A or B" is true if A is true, or if B is true, or if both A and B are true.
In logic, or by itself means the inclusive or, distinguished from an exclusive or, which is false when both of its arguments are true, while an "or" is true in that case.
An operand of a disjunction is called a disjunct.
Related concepts...
I just noticed that the title text (displayed on mouse hover) of the up- and downvote buttons are currently "This answer is useful" and "This answer is not useful". While they're useful for most SE sites, they don't make sense on PPCG. Is there any way we can change these texts?
ffmpeg, 110 113 116 117 119 108 bytes
ffmpeg -lavfi color=#0055a4:49x98[r];color=white:49x98[w];color=#ef4135:49x98[b];[r][w][b]hstack=3 -t 1 .png
The current version of the command will abort with an error BUT will output a single image ".png", same as shown below.
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or n-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.
The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "Cayley" in Cayley–Klein model refers to the English geometer Arthur Cayley.
The Beltrami–Klein model is analogous to the...
