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A347433 Irregular triangle read by rows: T(n,k) is the difference between the total arch lengths of a semi-meander multiplied by its number of exterior arches and total arch lengths of the semi-meanders with n + 1 top arches generated by the exterior arch splitting algorithm on the given semi-meander. 0

%I

%S 4,4,4,10,4,11,4,12,20,4,13,22,4,14,24,34,4,15,26,37,4,16,28,40,52,4,

%T 17,30,43,56,4,18,32,46,60,74,4,19,34,49,64,79,4,20,36,52,68,84,100,4,

%U 21,38,55,72,89,106,4,22,40,58,76,94,112,130,4,23,42,61,80,99

%N Irregular triangle read by rows: T(n,k) is the difference between the total arch lengths of a semi-meander multiplied by its number of exterior arches and total arch lengths of the semi-meanders with n + 1 top arches generated by the exterior arch splitting algorithm on the given semi-meander.

%F For n >= 2 and k = 2..floor((n+2)/2), T(n,k) = 4 + (n+2)*(k-2).

%e n = number of top arches, k = number of exterior top arches:

%e n\k 2 3 4 5 6

%e 2: 4

%e 3: 4

%e 4: 4 10

%e 5: 4 11

%e 6: 4 12 20

%e 7: 4 13 22

%e 8: 4 14 24 34

%e 9: 4 15 26 37

%e 10: 4 16 28 40 52

%e Length of each arch = 1 + number of arches covered:

%e Top arches of a given semi-meander: Arch splitting generated

%e n = 5, k = 2 semi-meanders (6 top arches):

%e 1 1 = 2 exterior arches /\

%e /\ //\\

%e /\ //\\ ///\\\

%e //\\ ///\\\ /\ /\ ////\\\\

%e 21 321 = 9 length of top arches 1 1 4321 = 12 length of top arches

%e /\

%e //\\ /\

%e ///\\\ //\\ /\

%e 321 21 1 = 10 length of top arches

%e T(5,2) = 4 + (5+2)(2-2) = 4 --------------------------- 4 = (12+10) - (2 * 9);

%e Top arches of given semi meander:

%e n = 5, k = 3 /\

%e 1 1 1 = 3 exterior arches / \

%e /\ /\ / \

%e /\ //\\ //\\ //\ /\\

%e 1 21 21 = 7 length top arches /\ ///\\//\\\

%e 1 521 21 = 12 length of top arches

%e /\

%e /\ //\\

%e //\\ /\ ///\\\

%e 21 1 321 = 10 length of top arches

%e /\

%e / \

%e / /\\

%e //\//\\\ /\ /\

%e 41 21 1 1 = 10 length of top arches

%e T(5,3) = 4 + (5+2)(3-2) = 11 --------------------- 11 = (12+10+10) - (3 * 7).

%Y Cf. A345747.

%K nonn,tabf

%O 2,1

%A _Roger Ford_, Sep 01 2021

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Last modified January 20 03:25 EST 2022. Contains 350467 sequences. (Running on oeis4.)