One of the open problems in combinatorics is enumeration of *meanders*.

Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand.

Since my interest is in some particular kind of these, I will not give the definition in the general case.

These special ones first appeared in a paper by Dergachev and Kirillov "Index of Lie algebras of seaweed type".

Our meanders, unlike the general ones, can be described by pairs of compositions with equal sum. For example, 4+3+4=5+4+2 gives

Call this *good*: it consists of a single interval.

Here is another one, corresponding to 5+3+4=3+5+2+2:

This is *bad* for two reasons: it is not connected, and it contains a non-interval (a cycle).

And here is a general (non-special) meander:

One thus obtains a sequence of numbers $a(n)$, with the $n$th number equal to the number of pairs of compositions of $n$ which encode good special meanders.

There is a recurrence which was used by Martin Plechsmid to write a C program for calculating these numbers up to several dozens. This recurrence was first observed by D. I. Panyushev in "Inductive formulas for the index of seaweed Lie algebras". In short, it is this: suppose given $a_1+a_2+...+a_k=b_1+b_2+...+b_l$ with $a_1>b_1$, let $d=a_1-b_1$ and let $r$ be the smallest nonnegative residue of $b_1$ modulo $d$. Then, the meander corresponding to $a_1+...+a_k=b_1+...+b_l$ is good if and only if good is the one corresponding to $$ (d-r)+r+a_2+...+a_k=b_2+...+b_l $$ (in case $r$ is zero it must be discarded).

Here are the first 109 of the $a(n)$ $$ \begin{array}{rr} 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 14 \\ 5 & 34 \\ 6 & 68 \\ 7 & 150 \\ 8 & 296 \\ 9 & 586 \\ 10 & 1140 \\ 11 & 2182 \\ 12 & 4130 \\ 13 & 7678 \\ 14 & 14368 \\ 15 & 26068 \\ 16 & 48248 \\ 17 & 86572 \\ 18 & 158146 \\ 19 & 281410 \\ 20 & 509442 \\ 21 & 901014 \\ 22 & 1618544 \\ 23 & 2852464 \\ 24 & 5089580 \\ 25 & 8948694 \\ 26 & 15884762 \\ 27 & 27882762 \\ 28 & 49291952 \\ 29 & 86435358 \\ 30 & 152316976 \\ 31 & 266907560 \\ 32 & 469232204 \\ 33 & 821844316 \\ 34 & 1442300988 \\ 35 & 2525295380 \\ 36 & 4426185044 \\ 37 & 7747801190 \\ 38 & 13567867834 \\ 39 & 23745303556 \\ 40 & 41557384062 \\ 41 & 72719208250 \\ 42 & 127217086618 \\ 43 & 222583616898 \\ 44 & 389294870960 \\ 45 & 681055011606 \\ 46 & 1190969037432 \\ 47 & 2083373141104 \\ 48 & 3642902097800 \\ 49 & 6372107745996 \\ 50 & 11141529268952 \\ 51 & 19487429053968 \\ 52 & 34072922408612 \\ 53 & 59593329333396 \\ 54 & 104196373919102 \\ 55 & 182231390839818 \\ 56 & 318626594336984 \\ 57 & 557234724403516 \\ 58 & 974322146677446 \\ 59 & 1703911578212510 \\ 60 & 2979322259711846 \\ 61 & 5210178551385178 \\ 62 & 9110218849890570 \\ 63 & 15931493988982610 \\ 64 & 27857226815017550 \\ 65 & 48714664885792786 \\ 66 & 85181547848196296 \\ 67 & 148957642280387814 \\ 68 & 260466741606092886 \\ 69 & 455476513314908856 \\ 70 & 796449892337843592 \\ 71 & 1392738017582218730 \\ 72 & 2435366204142605618 \\ 73 & 4258661790160448532 \\ 74 & 7446802851854147060 \\ 75 & 13021983359977438148 \\ 76 & 22770642365898383396 \\ 77 & 39818175179779177288 \\ 78 & 69627468187243597212 \\ 79 & 121754707865610780058 \\ 80 & 212904995880239240452 \\ 81 & 372297678714099281570 \\ 82 & 651015082378846020240 \\ 83 & 1138400336916804107622 \\ 84 & 1990655971732690098044 \\ 85 & 3480966121609384598862 \\ 86 & 6086972571117750179712 \\ 87 & 10643994672706675232516 \\ 88 & 18612574887590099590878 \\ 89 & 32546894313258870888576 \\ 90 & 56913010148917999572282 \\ 91 & 99520946619696571486034 \\ 92 & 174027003616514242824506 \\ 93 & 304312275771039225380736 \\ 94 & 532134871943053489298230 \\ 95 & 930517331785546952850436 \\ 96 & 1627147003755362124099240 \\ 97 & 2845309257636125017979486 \\ 98 & 4975444158602380143431196 \\ 99 & 8700305413681034236853658 \\ 100 & 15213772506611898836430634 \\ 101 & 26603545989601689878278594 \\ 102 & 46520243198912300314978832 \\ 103 & 81347565971655249247976872 \\ 104 & 142248282928801736884996866 \\ 105 & 248742274995715373879042070 \\ 106 & 434962771573005719770576034 \\ 107 & 760597063369550445571334010 \\ 108 & 1330016842349701088401439208 \\ 109 & 2325732108141510145312701272 \end{array} $$ The obvious question is what can be said about this sequence - for example, its asymptotics.