A037481 -id:A037481 - cp's OEIS frontend

A140725 Inverse binomial transform of (0 followed by A037481).

0, 1, 4, 10, 34, 94, 298, 862, 2650, 7822, 23722, 70654, 212986, 636910, 1914826, 5736286, 17225242, 51642958, 154994410, 464852158, 1394818618, 4183931566, 12552843274, 37656432670, 112973492314, 338912088334, 1016753042218

Offset: 0

Paul Curtz, Jul 12 2008

nonn

From Sean A. Irvine, Jun 07 2025: (Start)
For n>=1, the number of walks of length n-1 starting at vertex 1 (or, by symmetry, vertex 4) in the graph K_{1,1,3}:
    1---2
   /|\ /
  0 | X
   \|/ \
    4---3. (End)

Sean A. Irvine, Walks on Graphs.

Cf. A083421 (bin. transform of (0 followed by A037481)).

Join[{0},LinearRecurrence[{1,6},{1,4},26]] (* or *) a[0]=0;a[n_]:= ((-2)^n+4*3^n)/10;Array[a,27,0] (* James C. McMahon, Jul 13 2025 *)

a(n)= (-1)^n*A091003(n), n>0.
a(n+1)-3*a(n) = (-1)^(n+1)*A000079(n-1), n>0.
|a(n+1)-3*a(n)| = A011782(n).
From R. J. Mathar, Jul 14 2008: (Start)
O.g.f.: (1+3*x)*x / ((1+2*x)*(1-3*x)).
a(n) = ((-2)^n+4*3^n)/10, n>0. (End)
a(n) = a(n-1)+6*a(n-2) for n>2, a(0)=0, a(1)=1, a(2)=4. - Philippe Deléham, Nov 17 2013
a(n) + a(n+1) = A140796(n). - Philippe Deléham, Nov 17 2013
a(n+1) = sum_{k=0..n} A108561(n,k)*(-3)^k. - Philippe Deléham, Nov 17 2013

Edited and extended by R. J. Mathar, Jul 14 2008

A140684 A037481 mod 10.

Original entry on oeis.org

0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, 2

Offset: 0

Paul Curtz, Jul 11 2008

The sequence is periodic with length 10 and contains all 10 digits.

The first differences 1, 5, -1, -3, 7, -1, -5, 1, 3, -7 are also periodic with period length 10.

a(n) = A037481(n) mod 10. a(n)= a(n-10).
a(n)+a(n+5)= A010734(n).
O.g.f.: (7x^4-3x^3-x^2+5x+1)x/((1-x)(1+x)(x^4-x^3+x^2-x+1)). - R. J. Mathar, Jul 14 2008

Edited by R. J. Mathar, Jul 14 2008

A226062 a(n) = Bulgarian solitaire operation applied to the partition encoded in the runlengths of binary expansion of n.

Original entry on oeis.org

0, 1, 3, 2, 13, 7, 6, 6, 11, 29, 15, 58, 9, 14, 4, 14, 19, 27, 61, 54, 245, 31, 122, 52, 27, 25, 30, 50, 25, 12, 12, 30, 35, 23, 59, 46, 237, 125, 118, 44, 235, 501, 63, 1002, 233, 250, 116, 40, 51, 19, 57, 38, 229, 62, 114, 36, 59, 17, 28, 34, 57, 8, 28, 62

Offset: 0

Antti Karttunen, Jul 06 2013

For this sequence the partitions are encoded in the binary expansion of n in the same way as in A129594.
In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them. The question originally posed was: on what condition the resulting partitions will eventually reach a fixed point, that is, a collection of piles that will be unchanged by the operation. See Martin Gardner reference and the Wikipedia-page.
A037481 gives the fixed points of this sequence, which are numbers that encode triangular partitions: 1 + 2 + 3 + ... + n.
A227752(n) tells how many times n occurs in this sequence, and A227753 gives the terms that do not occur here.
Of further interest: among each A000041(n) numbers j_i: j1, j2, ..., jk for which A227183(j_i)=n, how many cycles occur and what is the size of the largest one? (Both are 1 when n is in A000217, as then the fixed points are the only cycles.) Cf. A185700, A188160.
Also, A123975 answers how many Garden of Eden partitions there are for the deck of size n in Bulgarian Solitaire, corresponding to values that do not occur as the terms of this sequence.

			5 has binary expansion "101", whose runlengths are [1,1,1], which are converted to nonordered partition {1+1+1}.
6 has binary expansion "110", whose runlengths are [1,2] (we scan the runs of bits from right to left), which are converted to nonordered partition {1+2}.
7 has binary expansion "111", whose list of runlengths is [3], which is converted to partition {3}.
In "Bulgarian Operation" we subtract one from each part (with 1-parts vanishing), and then add a new part of the same size as there originally were parts, so that the total sum stays same.
Thus starting from a partition encoded by 5, {1,1,1} the operation works as 1-1, 1-1, 1-1 (all three 1's vanish) but appends part 3 as there originally were three parts, thus we get a new partition {3}. Thus a(5)=7.
From the partition {3} -> 3-1 and 1, which gives a new partition {1,2}, so a(7)=6.
For partition {1+2} -> 1-1 and 2-1, thus the first part vanishes, and the second is now 1, to which we add the new part 2, as there were two parts originally, thus {1+2} stays as {1+2}, and we have reached a fixed point, a(6)=6.

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, Table of n, a(n) for n = 0..8191
Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
Antti Karttunen, Python-program for computing this and related sequences.
Wikipedia, Bulgarian solitaire

Cf. A227147, A227183, A227452.
Cf. A037481 (gives the fixed points).
Cf. A227752 (how many times n occurs here).
Cf. A227753 (numbers that do not occur here).
Cf. A129594 (conjugates the partitions encoded with the same system).
Cf. also A123975, A074909, A185700, A071762, A075157, A075158, A243353, A243354, A242424, A243051.

Other identities:
A227183(a(n)) = A227183(n). [This operation doesn't change the total sum of the partition.]
a(n) = A243354(A242424(A243353(n))).
a(n) = A075158(A243051(1+A075157(n))-1).

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).

A242422 Numbers in whose prime factorization the indices of primes sum to a triangular number.

Original entry on oeis.org

1, 2, 5, 6, 8, 13, 21, 22, 25, 27, 28, 29, 30, 36, 40, 46, 47, 48, 57, 64, 73, 76, 85, 86, 91, 102, 107, 117, 121, 123, 130, 136, 142, 147, 151, 154, 156, 164, 165, 175, 185, 189, 196, 197, 198, 201, 206, 208, 210, 217, 220, 222, 225, 243, 250, 252, 257, 264, 268, 270, 279, 280, 296, 298, 299, 300

Offset: 1

Antti Karttunen, May 16 2014

nonn

Numbers k such that A010054(A056239(k)) is one, or equally, that A002262(A056239(k)) is zero.
In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them. The question originally posed was: on what condition the resulting partitions will eventually reach a fixed point, that is, a collection of piles that will be unchanged by the operation. See Martin Gardner reference and the Wikipedia-page.
This sequence answers the question when we implement the operation on the partition list A112798: These are all such numbers that starting iterating A242424 from them leads eventually to a fixed point, which will be one of the primorial numbers, A002110.
Contains the same terms as rows of A215366 indexed with triangular numbers (A000217: 0, 1, 3, 6, ...), although not in the same order. {1}, {2}, {5, 6, 8}, {13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64}, etc.
Heinz numbers of integer partitions of triangular numbers. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). - Gus Wiseman, Nov 13 2018

			1 is present as it has an empty factorization, for which the sum of prime indices is zero, and zero is also a triangular number.
2 = p_1 is present as 1 is a triangular number.
6 = p_1 * p_2 is present, as 1+2 = 3 is a triangular number.
300 = 2*2*3*5*5 = p_1 * p_1 * p_2 * p_3 * p_3 is present, as 1+1+2+3+3 = 10 is a triangular number.
Any primorial number p_1 * p_2 * p_3 * ... * p_n is present, as 1+2+3+...+n is by definition a triangular number.
The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (), (1), (3), (2,1), (1,1,1), (6), (4,2), (5,1), (3,3), (2,2,2), (4,1,1), (10), (3,2,1), (2,2,1,1), (3,1,1,1), (9,1), (15), (2,1,1,1,1), (8,2), (1,1,1,1,1,1), (21), (8,1,1), (7,3), (14,1), (6,4). - _Gus Wiseman_, Nov 13 2018

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, Table of n, a(n) for n = 1..2455
Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
Wikipedia, Bulgarian solitaire

Complement: A242423.
A002110 (primorial numbers) is a subsequence.
Cf. A112798, A215366, A242424, A000217, A002262, A010054, A056239, A226062, A037481.
Cf. A066723, A321470, A321471, A321472.

triQ[n_]:=Module[{k,i},For[k=n;i=1,k>0,i++,k-=i];k==0];
Select[Range[100],triQ[Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]]&] (* Gus Wiseman, Nov 13 2018 *)

A243354 Permutation of natural numbers which maps between the partitions as encoded in A112798 (prime-index based system, one-based) to A227739 (binary based system, zero-based).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 5, 4, 14, 31, 13, 63, 30, 12, 10, 127, 9, 255, 29, 28, 62, 511, 26, 8, 126, 11, 61, 1023, 25, 2047, 21, 60, 254, 24, 18, 4095, 510, 124, 58, 8191, 57, 16383, 125, 27, 1022, 32767, 53, 16, 17, 252, 253, 65535, 22, 56, 122, 508, 2046, 131071

Offset: 1

Antti Karttunen, Jun 05 2014

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A243353.

Cf. A227739, A112798, A006068, A156552, A075158, A241909.

(define (A243354 n) (A006068 (A156552 n)))

a(n) = A006068(A156552(n)).
a(n) = A075158(A241909(n)-1). [With A075158's original starting offset].
For all n >= 1, A243353(a(n)) = n.
A056239(n) = A227183(a(n)).
A003963(n) = A227184(a(n)).
A037481(n) = a(A002110(n)).

A108981 a(n) = 3a(n-1) + 4a(n-2), a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 19, 77, 307, 1229, 4915, 19661, 78643, 314573, 1258291, 5033165, 20132659, 80530637, 322122547, 1288490189, 5153960755, 20615843021, 82463372083, 329853488333, 1319413953331, 5277655813325, 21110623253299

Offset: 0

Philippe Deléham, Jul 23 2005

The Hankel transform of this sequence is [1,-6,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Apr 15 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-2, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = charpoly(A,2). - Milan Janjic, Jan 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A003948, A037481.

I:=[1,5]; [n le 2 select I[n] else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 17 2014

LinearRecurrence[{3,4},{1,5},30] (* Harvey P. Dale, Feb 16 2014 *)

Vec((1+2*x)/(1+x)/(1-4*x)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

def A108981(n): return ((4<<(m:=n+1<<1))|2)//5-((1<>1 # Chai Wah Wu, Apr 22 2025

Inverse binomial transform of A003948.
a(2n) = 4a(2n-1) - 1; a(2n+1) = 4a(2n) + 1.
O.g.f.: (1+2*x)/((1+x)(1-4*x)). - R. J. Mathar, Apr 02 2008
Sum_{k=0..n} a(k) = A037481(n+1). - Philippe Deléham, Apr 15 2008

Corrected by T. D. Noe, Nov 07 2006

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 14 2008

A247281 a(n) = 4^n -(-1)^n.

Original entry on oeis.org

0, 5, 15, 65, 255, 1025, 4095, 16385, 65535, 262145, 1048575, 4194305, 16777215, 67108865, 268435455, 1073741825, 4294967295, 17179869185, 68719476735, 274877906945, 1099511627775, 4398046511105, 17592186044415

Offset: 0

Paul Curtz, Sep 11 2014

OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A015521, A037481, A084623.

LinearRecurrence[{3, 4}, {0, 5}, 23] (* Jean-François Alcover, May 22 2016 *)

concat(0, Vec(-5*x / ((x+1)*(4*x-1)) + O(x^100))) \\ Colin Barker, Sep 11 2014

vector(100,n,4^(n-1)+(-1)^n) \\ Derek Orr, Sep 11 2014

def A247281(n): return (1<<(n<<1))+(1 if n&1 else -1) # Chai Wah Wu, Jun 28 2023

a(n) = 5*A015521(n).
a(n+1) = 10*A037481(n) + 5.
a(n+1) = 4*a(n) + 5*(-1)^n.
a(n) = 3*a(n-1) + 4*a(n-2). - Colin Barker, Sep 11 2014
G.f.: -5*x / ((x+1)*(4*x-1)). - Colin Barker, Sep 11 2014

Typos in data fixed by Colin Barker, Sep 11 2014

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.

Original entry on oeis.org

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

Antti Karttunen, Jul 12 2013

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                    []                    {}
    1                 1                   [1]                   {1}
    5               101               [1,1,1]               {1+1+1}
   18             10010             [1,2,1,1]             {1+1+2+2}
   77           1001101           [1,2,2,1,1]           {1+1+2+3+3}
  306         100110010         [1,2,2,2,1,1]         {1+1+2+3+4+4}
 1229       10011001101       [1,2,2,2,2,1,1]       {1+1+2+3+4+5+5}
 4914     1001100110010     [1,2,2,2,2,2,1,1]     {1+1+2+3+4+5+6+6}
19661   100110011001101   [1,2,2,2,2,2,2,1,1]   {1+1+2+3+4+5+6+7+7}
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.

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, 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. A037481, A182512.

The left edge of the table A227452.

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

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

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)

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]

cp's OEIS Frontend

A140725 Inverse binomial transform of (0 followed by A037481).

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A140684 A037481 mod 10.

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A226062 a(n) = Bulgarian solitaire operation applied to the partition encoded in the runlengths of binary expansion of n.

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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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A242422 Numbers in whose prime factorization the indices of primes sum to a triangular number.

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A243354 Permutation of natural numbers which maps between the partitions as encoded in A112798 (prime-index based system, one-based) to A227739 (binary based system, zero-based).

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A108981 a(n) = 3a(n-1) + 4a(n-2), a(0) = 1, a(1) = 5.

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A247281 a(n) = 4^n -(-1)^n.

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