login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A227147 Irregular table: palindromic subsections from the rows of array A227141 related to main trunks of game trees drawn for Bulgarian solitaire. 8
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 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 6 04:15 EDT 2020. Contains 334859 sequences. (Running on oeis4.)