A227147 Irregular table: palindromic subsections from the rows of array A227141 related to main trunks of game trees drawn for Bulgarian solitaire.

1, 1, 3, 1, 2, 4, 3, 2, 3, 4, 2, 3, 5, 4, 4, 3, 4, 5, 4, 3, 4, 4, 5, 3, 4, 6, 5, 5, 5, 4, 5, 6, 5, 5, 4, 5, 5, 6, 5, 4, 5, 5, 5, 6, 4, 5, 7, 6, 6, 6, 6, 5, 6, 7, 6, 6, 6, 5, 6, 6, 7, 6, 6, 5, 6, 6, 6, 7, 6, 5, 6, 6, 6, 6, 7, 5, 6, 8, 7, 7, 7, 7, 7, 6, 7, 8, 7

Offset

1,3

Comments

Each row n contains A002061(n) terms and is palindromic.

Apart from the last term, each term on row n gives the largest summand in the partitions encountered on the main trunk of the Bulgarian solitaire tree computed for the deck of n(n+1)/2 cards; from row 2 onward, the last term of row k is one less than the largest summand in the unordered triangular partition {1+2+...+k} that is at the root of each game tree of the deck of the same size. The function f(n) = A227185(A227452(n)) would correctly give the largest summand sizes also for those cases.

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, The rows 1..31 of the table, flattened

Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).

Formula

a(n) = A227141(A227177(n),A227181(n)). [As a sequence. Each row n is a subsequence from the section [n,n^2] of the n-th row of ordinary table A227141.]

;; The following two formulas use the table A227452:

a(n) = A227185(A227452(n)) - ([n>1] * (A227177(n+1) - A227177(n))). [Where the expression [n>1] is an instance of Iverson brackets]

a(n) = n when n<2, otherwise a(n) = A005811(A227452(n-1)).

For all n, a(n) = a(A227182(n)). [This is just a claim that each row is symmetric.]

Example

Rows 1-6 of the table are:
1
1, 3, 1
2, 4, 3, 2, 3, 4, 2
3, 5, 4, 4, 3, 4, 5, 4, 3, 4, 4, 5, 3
4, 6, 5, 5, 5, 4, 5, 6, 5, 5, 4, 5, 5, 6, 5, 4, 5, 5, 5, 6, 4
5, 7, 6, 6, 6, 6, 5, 6, 7, 6, 6, 6, 5, 6, 6, 7, 6, 6, 5, 6, 6, 6, 7, 6, 5, 6, 6, 6, 6, 7, 5

Prog

(Scheme): (define (A227147 n) (A227141bi (A227177 n) (A227181 n))) ;; A227141bi given in A227141.
;; Two alternative definitions employing the table A227452:
(define (A227147v2 n) (- (A227185 (A227452 n)) (* (if (> n 1) 1 0) (- (A227177 (+ n 1)) (A227177 n)))))
(define (A227147v3 n) (if (< n 2) n (A005811 (A227452 (- n 1)))))

Crossrefs

Cf. A227141, A227452, A227185, A227181, A227182.

Sequence in context: A139457 A209301 A049992 * A074585 A183312 A108038

Adjacent sequences: A227144 A227145 A227146 * A227148 A227149 A227150

Keyword

nonn,tabf

Author

Antti Karttunen, Jul 03 2013

Status

approved