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.

0, 1, 5, 7, 6, 18, 61, 8, 11, 58, 28, 25, 77, 246, 66, 55, 36, 237, 226, 35, 46, 116, 197, 115, 102, 306, 985, 265, 445, 200, 155, 946, 905, 285, 220, 145, 475, 786, 925, 140, 185, 465, 395, 826, 460, 409, 1229, 3942, 1062, 1782, 1602, 823, 612, 3789, 3622, 1142

Offset

0,3

Comments

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.

Row n has A002061(n) = 1,1,3,7,13,21,... terms.

References

Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.

Links

Antti Karttunen, Rows 0-31 of table, flattened

Formula

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)).

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]

Example

Rows 0 - 5 of the table are:
0
1
5, 7, 6
18, 61, 8, 11, 58, 28, 25
77, 246, 66, 55, 36, 237, 226, 35, 46, 116, 197, 115, 102
306, 985, 265, 445, 200, 155, 946, 905, 285, 220, 145, 475, 786, 925, 140, 185, 465, 395, 826, 460, 409

Prog

(Scheme);; with Antti Karttunen's IntSeq-library for memoizing definec-macro
;; Compare with the other definition for A218616:
(definec (A227452 n) (cond ((< n 2) n) ((A226062 (A227452 (- n 1))) => (lambda (next) (if (= next (A227452 (- n 1))) (A227451 (A227177 (+ 1 n))) next)))))
;; Alternative implementation using nested cached closures for function iteration:
(define (A227452 n) ((compose-A226062-to-n-th-power (A227179 n)) (A227451 (A227177 n))))
(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))))))

Crossrefs

Left edge A227451. Right edge: A037481. Cf. A227147 (can be computed from this sequence).

Sequence in context: A200097 A314371 A217678 * A138306 A197257 A217175

Adjacent sequences: A227449 A227450 A227451 * A227453 A227454 A227455

Keyword

nonn,base,tabf

Author

Antti Karttunen, Jul 12 2013

Status

approved