A227451 Number whose binary expansion encodes via runlengths the partition that is at the top of the main trunk of Bulgarian solitaire game tree drawn for the deck with n(n+1)/2 cards.

0, 1, 5, 18, 77, 306, 1229, 4914, 19661, 78642, 314573, 1258290, 5033165, 20132658, 80530637, 322122546, 1288490189, 5153960754, 20615843021, 82463372082, 329853488333, 1319413953330, 5277655813325, 21110623253298, 84442493013197, 337769972052786, 1351079888211149

Offset: 0

Author

Antti Karttunen, Jul 12 2013

Keywords

• nonn
• base
• easy

Comments

The terms have particular patterns in their binary expansion, which encodes for an "almost triangular partition" when runlength encoding of unordered partitions are used (please see A129594 for how that encoding works). These are obtained from the perfectly triangular partitions shown in A037481 by inserting 1 to the front of the partition and decrementing the last summand (the largest) by one:

n a(n) same in binary run lengths unordered partition

0 0 0 [] {}

1 1 1 [1] {1}

2 5 101 [1,1,1] {1+1+1}

3 18 10010 [1,2,1,1] {1+1+2+2}

4 77 1001101 [1,2,2,1,1] {1+1+2+3+3}

5 306 100110010 [1,2,2,2,1,1] {1+1+2+3+4+4}

6 1229 10011001101 [1,2,2,2,2,1,1] {1+1+2+3+4+5+5}

7 4914 1001100110010 [1,2,2,2,2,2,1,1] {1+1+2+3+4+5+6+6}

8 19661 100110011001101 [1,2,2,2,2,2,2,1,1] {1+1+2+3+4+5+6+7+7}

9 78642 10011001100110010 [1,2,2,2,2,2,2,2,1,1] {1+1+2+3+4+5+6+7+8+8}

These partitions occur at the tops of the main trunks of the game trees constructed for decks consisting of 1+2+3+...+k cards. See A037481 for the encoding of the roots of the main trunks of the same trees.

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, Table of n, a(n) for n = 0..1000
• Wikipedia, Bulgarian solitaire
• Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Crossrefs

Cf. A037481, A182512.

The left edge of the table A227452.

Programs

Mathematica

LinearRecurrence[{4,1,-4},{0,1,5,18,77},40] (* Harvey P. Dale, Sep 22 2016 *)

PARI

a(n)=if(n<1,0,if(n==1,1,(3*4^n+7*(-1)^n-5)/10)) \\ Ralf Stephan

Formula

a(0)=0, a(1)=1, for n>=2, a(n) = A053645(2*A037481(n)) + (1 - (n mod 2)). [Follows from the "insert 1 and decrement the largest part by one" operation on triangular partitions]

Alternatively:

a(0)=0, a(1)=1, and for n>=2, if n is even, then a(n) = 1 + (4*A182512((n-2)/2)) + 2^(2*(n-1)), and if n is odd, then a(n) = 2 + (16*A182512((n-3)/2)) + 2^(2*(n-1)).

From Ralf Stephan, Jul 20 2013: (Start)

a(n) = (1/10) * (3*4^n + 7*(-1)^n - 5).

a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3), n>3.

G.f.: (4*x^4 - 3*x^3 + x^2 + x)/((1-x)*(1+x)*(1-4*x)). (End)