A227181 -id:A227181 - cp's OEIS frontend

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

Antti Karttunen, Jul 03 2013

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.

			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

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

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).

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

(define (A227147 n) (A227141bi (A227177 n) (A227181 n))) ;; A227141bi given in A227141.

(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)))))

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.]

A227177 n occurs n^2 - n + 1 times.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Antti Karttunen, Jul 03 2013

nonn

a(n) is the least integer k such that A006527(k) >= n, which implies that each n occurs A002061(n) times.

Antti Karttunen, Table of n, a(n) for n = 0..9951

Cf. A000196, A002061, A006527, A227179, A227181, A227147.

Flatten[Map[ConstantArray[#,(#-2) (#-1)+1]-1&,Range[7]]] (* Peter J. C. Moses, Jul 14 2013 *)
Flatten[Table[#,{#^2-#+1}]&/@Range[0,7]] (* Harvey P. Dale, Sep 25 2013 *)

vec(N)=concat(vector(N, i, vector(i^2-i+1, j, i))) \\ Jinyuan Wang, Dec 01 2018

from sympy import integer_nthroot
def A227177(n): return (m:=integer_nthroot(k:=3*n,3)[0])+(k>m*(m**2+2)) # Chai Wah Wu, Nov 07 2024

a(k + (j^3-j^2+5*j)/3) = j for all j>=0, k=0..(j^2-j). - Jinyuan Wang, Nov 24 2018

a(n) = m+1 if 3n>m*(m^2+2) and a(n) = m otherwise where m=floor((3n)^(1/3)). - Chai Wah Wu, Nov 07 2024

A227179 After initial 0, integers from 0 to n(n-1) followed by integers from 0 to n(n+1) and so on.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30

Offset: 0

Antti Karttunen, Jul 03 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..9951

Cf. A002378, A006527, A227177, A227181, A227182.

Flatten[(Range[(#-2) (#-1)+1]-1&)[Range[7]]] (* Peter J. C. Moses, Jul 11 2013 *)

from sympy import integer_nthroot
def A227179(n): return n-1-(f:=lambda x:x*(x**2+2))((m:=integer_nthroot(k:=3*n,3)[0])-(k<=f(m)))//3 # Chai Wah Wu, Nov 07 2024

(define (A227179 n) (- n (+ 1 (A006527 (- (A227177 n) 1)))))

a(n) = n - (1 + A006527(A227177(n)-1)).

cp's OEIS Frontend

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

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A227177 n occurs n^2 - n + 1 times.

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A227179 After initial 0, integers from 0 to n(n-1) followed by integers from 0 to n(n+1) and so on.

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