A227451 -id:A227451 - cp's OEIS frontend

A037481 Base 4 digits are, in order, the first n terms of the periodic sequence with initial period 1,2.

0, 1, 6, 25, 102, 409, 1638, 6553, 26214, 104857, 419430, 1677721, 6710886, 26843545, 107374182, 429496729, 1717986918, 6871947673, 27487790694, 109951162777, 439804651110, 1759218604441, 7036874417766, 28147497671065

Offset: 0

Clark Kimberling

The terms have a particular pattern in their binary expansion, which encodes for a "triangular partition" when runlength encoding of unordered partitions are used (please see A129594 for how that encoding works).
  n   a(n)    same in binary           run lengths   unordered partition
    0                 0                    []                    {}
    1                 1                   [1]                   {1}
    6               110                 [2,1]                 {1+2}
   25             11001               [2,2,1]               {1+2+3}
  102           1100110             [2,2,2,1]             {1+2+3+4}
  409         110011001           [2,2,2,2,1]           {1+2+3+4+5}
 1638       11001100110         [2,2,2,2,2,1]         {1+2+3+4+5+6}
 6553     1100110011001       [2,2,2,2,2,2,1]       {1+2+3+4+5+6+7}
26214   110011001100110     [2,2,2,2,2,2,2,1]     {1+2+3+4+5+6+7+8}
104857 11001100110011001   [2,2,2,2,2,2,2,2,1]   {1+2+3+4+5+6+7+8+9}
These partitions are the only fixed points of "Bulgarian Solitaire" operation (see Gardner reference or Wikipedia page), and thus the terms of this sequence give the fixed points for A226062 which implements that operation (using the same encoding for partitions). This also implies that these partitions are the roots of the game trees constructed for decks consisting of 1+2+3+...+k cards. See A227451 for the encoding of the corresponding tops of the main trunks of the same trees. - Antti Karttunen, Jul 12 2013

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

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

Cf. A014985, A015521, A227451.
Cf. A037487 (decimal digits 1,2).
The right edge of the table A227452. The fixed points of A226062.

I:=[0, 1, 6]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 21 2012

LinearRecurrence[{4,1,-4},{0,1,6},40] (* Vincenzo Librandi, Jun 21 2012 *)
Module[{nn=30,ps},ps=PadRight[{},nn,{1,2}];Table[FromDigits[Take[ps,n],4],{n,0,nn}]] (* Harvey P. Dale, Jul 18 2013 *)

concat(0, Vec(x*(2*x+1)/((x-1)*(x+1)*(4*x-1)) + O(x^100))) \\ Colin Barker, Apr 30 2014

a(n) = 2<<(2*n) \ 5; \\ Kevin Ryde, Jun 24 2023

def A037481(n): return (1<<(n<<1|1))//5 # Chai Wah Wu, Jun 28 2023

(define (A037481 n) (/ (- (/ (+ (expt 4 (1+ n)) (expt -1 n)) 5) 1) 2)) ;; Using Ralf Stephan's direct formula - Antti Karttunen, Jul 12 2013

a(n) = ((4^(n+1) - (-1)^(n+1))/5 - 1)/2. - Ralf Stephan
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - Vincenzo Librandi, Jun 21 2012
a(n) = A226062(A129594(A227451(n))). [See page 465 in Gardner's book] - Antti Karttunen, Jul 12 2013
G.f.: x*(2*x+1) / ((x-1)*(x+1)*(4*x-1)). - Colin Barker, Apr 30 2014

A243353 Permutation of natural numbers which maps between the partitions as encoded in A227739 (binary based system, zero-based) to A112798 (prime-index based system, one-based).

Original entry on oeis.org

1, 2, 4, 3, 9, 8, 6, 5, 25, 18, 16, 27, 15, 12, 10, 7, 49, 50, 36, 75, 81, 32, 54, 125, 35, 30, 24, 45, 21, 20, 14, 11, 121, 98, 100, 147, 225, 72, 150, 245, 625, 162, 64, 243, 375, 108, 250, 343, 77, 70, 60, 105, 135, 48, 90, 175, 55, 42, 40, 63, 33, 28, 22, 13, 169, 242, 196, 363, 441, 200, 294, 605, 1225, 450, 144

Offset: 0

Antti Karttunen, Jun 05 2014

nonn

Note the indexing: the domain includes zero, but the range starts from one.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers

A243354 gives the inverse mapping.

Cf. A227739, A112798, A075157, A241909, A005940, A003188, A226062, A242424.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Table[f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, 74}] (* Michael De Vlieger, May 09 2017 *)

from sympy import prime
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n)) # Indranil Ghosh, May 07 2017

(define (A243353 n) (A005940 (+ 1 (A003188 n))))

a(n) = A005940(1+A003188(n)).
a(n) = A241909(1+A075157(n)). [With A075157's original starting offset]
For all n >= 0, A243354(a(n)) = n.
A227183(n) = A056239(a(n)). [Maps between the corresponding sums ...]
A227184(n) = A003963(a(n)). [... and products of parts of each partition].
For n >= 0, a(A037481(n)) = A002110(n). [Also "triangular partitions", the fixed points of Bulgarian solitaire, A226062 & A242424].
For n >= 1, a(A227451(n+1)) = 4*A243054(n).

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.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 12 2013

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.

			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

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, Rows 0-31 of table, flattened

Left edge A227451. Right edge: A037481. Cf. A227147 (can be computed from this sequence).

;; with Antti Karttunen's IntSeq-library for memoizing definec-macro
;; 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) ((compose-A226062-to-n-th-power (A227179 n)) (A227451 (A227177 n))))
(definec (compose-A226062-to-n-th-power n) (cond ((zero? n) (lambda (x) x)) (else (lambda (x) (A226062 ((compose-A226062-to-n-th-power (- n 1)) x))))))

For n < 2, a(n) = n, and for n>=2, if A226062(a(n-1)) = a(n-1) [in other words, when a(n-1) is one of the terms of A037481] then a(n) = A227451(A227177(n+1)), otherwise a(n) = A226062(a(n-1)).

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 Scheme-definition in the Program section]

A227753 Numbers which do not occur in A226062; numbers which encode Garden of Eden partitions in Bulgarian Solitaire in runlengths of their binary representation.

Original entry on oeis.org

5, 10, 18, 20, 21, 22, 26, 37, 41, 42, 43, 45, 53, 69, 73, 74, 75, 77, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 101, 105, 106, 107, 109, 117, 138, 146, 148, 149, 150, 154, 162, 164, 165, 166, 168, 169, 170, 171, 172, 173, 174, 178, 180, 181, 182, 186, 202

Offset: 1

Antti Karttunen, Jul 26 2013

nonn

Positions of zeros in A227752.

A225794 gives the sizes of the corresponding partitions.

Antti Karttunen, Table of n, a(n) for n = 1..86

Cf. A226062, A227752, A225794, A123975.

After its first two initial terms, all the terms of A227451 can be found in this sequence.

cp's OEIS Frontend

A037481 Base 4 digits are, in order, the first n terms of the periodic sequence with initial period 1,2.

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A243353 Permutation of natural numbers which maps between the partitions as encoded in A227739 (binary based system, zero-based) to A112798 (prime-index based system, one-based).

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A227753 Numbers which do not occur in A226062; numbers which encode Garden of Eden partitions in Bulgarian Solitaire in runlengths of their binary representation.

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