cp's OEIS Frontend

A227452 Irregular table where each row lists the partitions occurring on the main trunk of the Bulgarian Solitaire game tree (from the top to the root) for deck of n(n+1)/2 cards. Nonordered partitions are encoded in the runlengths of binary expansion of each term, in the manner explained in A129594.

%I A227452 #26 Sep 09 2017 19:35:19
%S A227452 0,1,5,7,6,18,61,8,11,58,28,25,77,246,66,55,36,237,226,35,46,116,197,
%T A227452 115,102,306,985,265,445,200,155,946,905,285,220,145,475,786,925,140,
%U A227452 185,465,395,826,460,409,1229,3942,1062,1782,1602,823,612,3789,3622,1142
%N A227452 Irregular table where each row lists the partitions occurring on the main trunk of the Bulgarian Solitaire game tree (from the top to the root) for deck of n(n+1)/2 cards. Nonordered partitions are encoded in the runlengths of binary expansion of each term, in the manner explained in A129594.
%C A227452 The terms for row n are computed as A227451(n), A226062(A227451(n)), A226062(A226062(A227451(n))), etc. until a term that is a fixed point of A226062 is reached (A037481(n)), which will be the last term of row n.
%C A227452 Row n has A002061(n) = 1,1,3,7,13,21,... terms.
%D A227452 Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.
%H A227452 Antti Karttunen, <a href="/A227452/b227452.txt">Rows 0-31 of table, flattened</a>
%F A227452 For n < 2, a(n) = n, and for n>=2, if A226062(a(n-1)) = a(n-1) [in other words, when a(n-1) is one of the terms of A037481] then a(n) = A227451(A227177(n+1)), otherwise a(n) = A226062(a(n-1)).
%F A227452 Alternatively, a(n) = value of the A227179(n)-th iteration of the function A226062, starting from the initial value A227451(A227177(n)). [See the other Scheme-definition in the Program section]
%e A227452 Rows 0 - 5 of the table are:
%e A227452 0
%e A227452 1
%e A227452 5, 7, 6
%e A227452 18, 61, 8, 11, 58, 28, 25
%e A227452 77, 246, 66, 55, 36, 237, 226, 35, 46, 116, 197, 115, 102
%e A227452 306, 985, 265, 445, 200, 155, 946, 905, 285, 220, 145, 475, 786, 925, 140, 185, 465, 395, 826, 460, 409
%o A227452 (Scheme);; with _Antti Karttunen_'s IntSeq-library for memoizing definec-macro
%o A227452 ;; Compare with the other definition for A218616:
%o A227452 (definec (A227452 n) (cond ((< n 2) n) ((A226062 (A227452 (- n 1))) => (lambda (next) (if (= next (A227452 (- n 1))) (A227451 (A227177 (+ 1 n))) next)))))
%o A227452 ;; Alternative implementation using nested cached closures for function iteration:
%o A227452 (define (A227452 n) ((compose-A226062-to-n-th-power (A227179 n)) (A227451 (A227177 n))))
%o A227452 (definec (compose-A226062-to-n-th-power n) (cond ((zero? n) (lambda (x) x)) (else (lambda (x) (A226062 ((compose-A226062-to-n-th-power (- n 1)) x))))))
%Y A227452 Left edge A227451. Right edge: A037481. Cf. A227147 (can be computed from this sequence).
%K A227452 nonn,base,tabf
%O A227452 0,3
%A A227452 _Antti Karttunen_, Jul 12 2013