This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(3) = 163: 1 2 3 2 5 4 3 4 6 a(4) = 9850: 1 2 3 4 2 8 5 6 3 5 9 7 4 6 7 10
a(3) = 420:
1 5 6
5 3 4
6 4 2
a(4) = 41451:
1 5 8 10
5 4 9 7
8 9 3 6
10 7 6 2
The pattern of the ladders. Example for the first three:
.
|_|
| | ___ 1 missing
|_| rung
| |
|_| | | ___ 2 missing
| | ___ 1 missing |_| rungs
|_| rung | |
|_| | | | | ___ 3 missing
| | ___ 1 missing | | ___ 2 missing | | rungs
|_| rung |_| rungs |_|
| | | | | |
.
The 2 rung The 3 rung The 4 rung
ladder ladder ladder
.
Constructing the compound ladder:
First, we take the smallest ladder with two rungs. Then we select the next smallest one, which has three rungs. We place its bottom rung in line with the empty place on the first ladder. So we obtain a climbable three-rung ladder assembly. Next, we observe the first missing rung at level 4, to which we try the four rung ladder with success because no rungs clash. The lowest empty place is now at rung 6, to which we try the five-rung ladder. This however will clash with a rung of the assembly. So we fit the next smallest available one with six rungs, which fits well. Any failed ladder should always be tried at later stages where it may fit properly.
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Exploded view of the assembly: Front view:
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|_| |_|
| | | |
|_| |_|
| | | |
| | | |
|_| |_|
|_| | | | |
| | | | | |
|_| | | | |
| | |_| |_|
| | | | | |
|_| | | | |
| | | | | |
|_| | | |_| | | |_|
| | | | | | |_| ___ No clashing ___ |_|
|_| |_| ___ Clashing |_| | | rungs |_|
| | | | rungs | | | | | |
| | | | | | | | ___ Next ladder ___ | |
|_| | | |_| | | to be tried |_|
|_| | | | | |_| | | | | here |_|
| | | | |_| | | | | |_| (including |_|
|_| | | | | |_| | | | | the 5 rung |_|
| | |_| | | |_| one) |_|
|_| | | | | 5 rung |_| | | | | 6 rung |_|
| | |_| does | | |_| does |_|
|_| | | 4 rung not |_| | | 4 rung fit |_|
| | fit | | | |
3 rung 3 rung
2 rung 2 rung n = 4 ladders
assembled with
8 climbable
rungs achieved
seq(n)={my(M=Map(), K=Map(), a=vector(n), b=0); for(n=1, #a, while(mapisdefined(M,b), b++); my(f=1, k=1); while(f, k++; if(!mapisdefined(K,k), f=0; my(s=b); for(i=0, k-2, s += k-i; if(mapisdefined(M,s),f=1;break)); if(!f, for(i=2, k, mapput(M,s,1); s-=i); mapput(M,s,1)))); a[n]=k; mapput(K,k,1)); a} \\ Andrew Howroyd, Oct 18 2024
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