

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.


5



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
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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. 455467, W. W. Norton & Company, 2001.


LINKS

Antti Karttunen, Rows 031 of table, flattened


FORMULA

For n < 2, a(n) = n, and for n>=2, if A226062(a(n1)) = a(n1) [in other words, when a(n1) is one of the terms of A037481] then a(n) = A227451(A227177(n+1)), otherwise a(n) = A226062(a(n1)).
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 Schemedefinition 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 IntSeqlibrary for memoizing definecmacro
;; 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) ((composeA226062tonthpower (A227179 n)) (A227451 (A227177 n))))
(definec (composeA226062tonthpower n) (cond ((zero? n) (lambda (x) x)) (else (lambda (x) (A226062 ((composeA226062tonthpower ( 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



