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.
A213730(9)=22, and from that branches 24 and 25 (as both A011371(24)=A011371(25)=22) and while 24 is a leaf (in A055938) the other branch 25 further branches to two leaves (as both A011371(28)=A011371(29)=25). When we construct a binary tree from this in such a fashion that the lesser numbers go to the left, we obtain: ........... ...28...29. .....\./... ..24..25... ...\ /..... ....22..... ........... and the binary tree ........ ...\./.. ....*... .\./.... ..*..... ........ is located as A014486(2) in the normal encoding order of binary trees, thus a(9)=2.
Illustration how the growing beanstalk-tree produces the first four terms of this sequence. In this variant, the lesser numbers come to the right hand side: .......... ...\1/.... Coded by A014486(A218779(1)) = A014486(1) = 2 (binary 10). .......... .......... .....\2/.. ...\1/.... Coded by A014486(A218779(2)) = A014486(2) = 10 (bin. 1010). .......... .......... .\3/ \2/.. ...\1/.... Coded by A014486(A218779(3)) = A014486(6) = 50 (110010). .......... .......... ..\4/..... .\3/.\2/.. ...\1/.... Coded by A014486(A218779(4)) = A014486(16) = 210 (11010010). .......... Thus the first four terms of this sequence are 2, 10, 50 and 210.
Illustration how the growing beanstalk-tree produces the first four terms of this sequence. In this "compact" variant, each successive pair of numbers ((2,3), (4,5), (6,7), etc.) adds a new bud (\/) to the beanstalk, with the lesser numbers coming to the right hand side: ---------- ..3...2... ...\./.... Coded by A014486(A218783(1)) = A014486(1) = 2 (binary 10). ....1..... ---------- 5...4..... .\./...... ..3...2... ...\./.... Coded by A014486(A218783(2)) = A014486(3) = 12 (bin. 1100). ....1..... ---------- ..7...6... ...\./.... 5...4..... .\./...... ..3...2... ...\./.... Coded by A014486(A218783(3)) = A014486(7) = 52 (110100). ....1..... ---------- 9...8..... .\./...... ..7...6... ...\./.... 5...4..... .\./...... ..3...2... ...\./.... Coded by A014486(A218783(4)) = A014486(18) = 216 (11011000). ....1..... ---------- Thus the first four terms of this sequence are 2, 12, 52 and 216.
The first four tendrils of the beanstalk sprout at 2, 5, 6 and 9, (the first four nonzero terms of A213730) which are all leaves (i.e., in A055938), thus the first four terms of this sequence are all 0's. The next term A213730(5)=10, which is not leaf, but branches to two leaf-branches (12 and 13, as with both we have: 12-A000120(12)=10 and 13-A000120(13)=10, and both 12 and 13 are found from A055938, so the tendril at 10 is a binary tree of one internal vertex (and two leaves), i.e., \/, thus a(5)=1.
As we must traverse to 4 in A218778-tree (see the example there) by first taking the left branch (car) from the root, resulting bit 1 as the least significant bit of the code, then by taking the right branch (cdr) from 3 to get to 4, resulting bit 0 as the second rightmost bit of the code, which when capped with an extra termination-one, results binary code 101, 5 in decimal, thus a(4)=5.
As we can traverse to 4 in A218776-tree (see the example there) by taking first the right branch (cdr) from the root, resulting bit 0 as the least significant bit of the code, then by taking the left branch (car) from 3 to get to 4, resulting bit 1 as the second rightmost bit of the code, which when capped with an extra termination-one, results binary code 110, 6 in decimal, thus a(4)=6.
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