A377171 Number of rungs, k, in deficient ladders to be assembled together in that order, to make a ladder that can be climbed to some height. Details are in the Comments.

2, 3, 4, 6, 7, 10, 18, 13, 15, 24, 45, 56, 65, 81, 89, 114, 179, 29, 161, 281, 302, 362, 383, 424, 356, 541, 735, 455, 1042, 1173, 1007, 1347, 750, 1586, 1501, 2040, 2727, 2435, 2491, 3211, 3911, 3407, 5, 4211, 4753, 5273, 5351, 6051, 7191, 7907, 8541, 9200, 10053, 13552, 10377

Offset: 1

Author

Tamas Sandor Nagy, Oct 18 2024

Keywords

• nonn

Comments

Each distinct ladder has k rungs and k*(k-1)/2 missing rungs. A ladder has one rung at the bottom and one at the top. The missing rungs of the ladders have a pattern. The number of missing rungs between rungs from bottom to top are (1), (2,1), (3,2,1), and so on.

The rule is to select the smallest suitable ladder and place its bottom rung in line with the lowest missing rung of the ladder assembly such that none of its rungs is in line with any other rung of the assembly. No ladder is to be turned upside down.

Examples

			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.
.
Exploded view of the assembly:                                    Front view:
.
                                               |_|                     |_|
                                               | |                     | |
                                               |_|                     |_|
                                               | |                     | |
                                               | |                     | |
                                               |_|                     |_|
                    |_|                        | |                     | |
                    | |                        | |                     | |
                    |_|                        | |                     | |
                    | |                        |_|                     |_|
                    | |                        | |                     | |
                    |_|                        | |                     | |
                    | |                        | |                     | |
              |_|   | |                  |_|   | |                     |_|
              | |   | |                  | |   |_| ___ 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
		

Crossrefs

Cf. A351153 (rung positions of the ladders from bottom up).

Programs

PARI

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

Extensions

a(11) onwards from Andrew Howroyd, Oct 18 2024