A373556 -id:A373556 - cp's OEIS frontend

A373854 Row sums of A373556.

1, 5, 6, 7, 12, 8, 13, 14, 9, 14, 15, 16, 22, 10, 15, 16, 17, 23, 18, 24, 25, 11, 16, 17, 18, 24, 19, 25, 26, 20, 26, 27, 28, 35, 12, 17, 18, 19, 25, 20, 26, 27, 21, 27, 28, 29, 36, 22, 28, 29, 30, 37, 31, 38, 39, 13, 18, 19, 20, 26, 21, 27, 28, 22, 28, 29, 30

Offset: 1

Paolo Xausa, Jun 19 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A373346, A373556, A373853.

Join[{1}, Map[Total[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 2000, 2], DigitCount[#, 2, 1] == IntegerExponent[#, 2] + 1 &]]]

A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 01 2024

A Schreier set is a subset of the positive integers with cardinality less than or equal to the minimum element in the set.
Each term k of A371176 can be put into a one-to-one correspondence with a Schreier set by interpreting the 1-based position of the ones in the binary expansion of k (where position 1 corresponds to the least significant bit) as the elements of the corresponding Schreier set (see A371176 and Bird link).
See A373359 for the elements in each set arranged in increasing order.
The number of sets having maximum element m is A000045(m).

			Triangle begins:
                                   Corresponding Schreier
   n  A371176(n)  bin(A371176(n))  set (this sequence)
  -------------------------------------------------------
   1      1              1         {1}
   2      2             10         {2}
   3      4            100         {3}
   4      6            110         {3, 2}
   5      8           1000         {4}
   6     10           1010         {4, 2}
   7     12           1100         {4, 3}
   8     16          10000         {5}
   9     18          10010         {5, 2}
  10     20          10100         {5, 3}
  11     24          11000         {5, 4}
  12     28          11100         {5, 4, 3}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.

Cf. A000045, A371176, A373359, A373556, A373557.

Cf. A007895 (conjectured row lengths), A072649 (first column), A373346 (row sums), A373347.

Join[{{1}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 200, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]]]

T(n,k) = A373557(n,k) - 1.

A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 09 2024

A strong Schreier set is a subset of the positive integers with cardinality less than the minimum element in the set (see Chu link).
Each term k of 2*A371176 can be put into a one-to-one correspondence with a strong Schreier set by interpreting the 1-based position of the ones in the binary expansion of k (where position 1 corresponds to the least significant bit) as the elements of the corresponding strong Schreier set.
Arranging the elements in each set in decreasing order results in the sets being listed in lexicographical order (see example). Cf. A373579 for the elements arranged in increasing order.
The number of sets having maximum element m is A000045(m-1).

			Triangle begins:
                                        Corresponding
   n  A371176(2*n)  bin(A371176(2*n))   strong Schreier set
                                        (this sequence)
  ---------------------------------------------------------
   1        2               10          {2}
   2        4              100          {3}
   3        8             1000          {4}
   4       12             1100          {4, 3}
   5       16            10000          {5}       Sets are
   6       20            10100          {5, 3}    lexicographically
   7       24            11000          {5, 4}    ordered
   8       32           100000          {6}
   9       36           100100          {6, 3}
  10       40           101000          {6, 4}
  11       48           110000          {6, 5}
  12       56           111000          {6, 5, 4}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Hùng Việt Chu, The Fibonacci Sequence and Schreier-Zeckendorf Sets, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.

Subsequence of A373345.

Cf. A000045, A007895 (conjectured row lengths), A371176, A373556, A373579, A373853 (row sums).

Join[{{2}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[4, 400, 4], DigitCount[#, 2, 1] < IntegerExponent[#, 2] + 1 &]]]

T(n,k) = A373345(n,k) + 1.

A355489 Numbers k such that A000120(k) = A007814(k) + 2.

Original entry on oeis.org

3, 5, 9, 14, 17, 22, 26, 33, 38, 42, 50, 60, 65, 70, 74, 82, 92, 98, 108, 116, 129, 134, 138, 146, 156, 162, 172, 180, 194, 204, 212, 228, 248, 257, 262, 266, 274, 284, 290, 300, 308, 322, 332, 340, 356, 376, 386, 396, 404, 420, 440, 452, 472, 488, 513, 518

Offset: 1

Mikhail Kurkov, Jul 04 2022 [verification needed]

Each term k, doubled, can be put into a one-to-one correspondence with a maximal Schreier set (a subset of the positive integers with cardinality equal to the minimum element in the set) by interpreting the 1-based position of the ones in the binary expansion of 2*k (where position 1 corresponds to the least significant bit) as the elements of the corresponding maximal Schreier set. See A373556 for more information. Cf. also A371176. - Paolo Xausa, Jun 13 2024

Cf. A000045, A000120, A007814, A010056, A025480, A048679, A072649, A371176, A373556.

Select[Range[500], DigitCount[#, 2, 1] == IntegerExponent[#, 2] + 2 &] (* Amiram Eldar, Jul 04 2022 *)

r=quadgen(5);
A355489_upto(nMax)={my(v1,v2,v3,v4); v1=vector(nMax,i,0); v1[1]=1; for(i=1,nMax-1,v1[i+1]=v1[i\r+1]+1); v2=vector(nMax,i,0); v2[1]=1; for(i=2,nMax,v2[i]=v1[i]-v1[i-1]); v3=vector(nMax,i,0); for(i=1,3,v3[i]=2^(i-1)); for(i=4,nMax,v3[i]=if(v2[i-1]==1,5,2*v3[i-fibonacci(v1[i-1]+1)]-if(v2[i]==1,1,0))); v4=vector(nMax,i,0); v4[1]=3; for(i=2,nMax,v4[i]=v4[i-1]+v3[i]); v4}

isok(k) = hammingweight(k) == valuation(k, 2) + 2; \\ Michel Marcus, Jul 06 2022
(Python 3.10+)
from itertools import count, islice
def A355489_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n.bit_count()==(n&-n).bit_length()+1,count(max(startvalue,1)))
A355480_list = list(islice(A355489_gen(),30)) # Chai Wah Wu, Jul 15 2022

a(n) = a(n-1) + b(n) for n > 1 with a(1) = 3 where b(n) = {2^(n-1) if n < 4; 5 if c(n-1) = 1; otherwise 2*b(n - A000045(A072649(n-1) + 1)) - [c(n) = 1]} and where c(n) = A010056(n).
A025480(a(n)-1) = A048679(n) for n > 0.
a(A000045(n)) = 2^(n-1) + 1 for n > 1.

A373558 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in increasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 10 2024

See A373556 (where elements in each set are listed in decreasing order) for more information.

			Triangle begins:
                                           Corresponding
   n  2*A355489(n-1)  bin(2*A355489(n-1))  maximal Schreier set
                                           (this sequence)
  ---------------------------------------------------------------
   1                                       {1}
   2         6                 110         {2, 3}
   3        10                1010         {2, 4}
   4        18               10010         {2, 5}
   5        28               11100         {3, 4, 4}
   6        34              100010         {2, 6}
   7        44              101100         {3, 4, 6}
   8        52              110100         {3, 5, 6}
   9        66             1000010         {2, 7}
  10        76             1001100         {3, 4, 7}
  11        84             1010100         {3, 5, 7}
  12       100             1100100         {3, 6, 7}
  13       120             1111000         {4, 5, 6, 7}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10003 (rows 1..1892 of the triangle, flattend).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Hùng Việt Chu, The Fibonacci Sequence and Schreier-Zeckendorf Sets, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.

Subsequence of A373359.

Cf. A143299 (conjectured row lengths), A355489, A373556, A373579, A373854 (row sums).

Join[{{1}}, Map[PositionIndex[Reverse[IntegerDigits[#, 2]]][1] &, Select[Range[2, 500, 2], DigitCount[#, 2, 1] == IntegerExponent[#, 2] + 1 &]]]

A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

Original entry on oeis.org

2, 1, 3, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 7, 9, 1, 2, 3, 6, 11, 14, 1, 2, 3, 6, 10, 17, 22, 1, 2, 3, 5, 10, 17, 26, 35, 1, 2, 3, 5, 10, 16, 28, 40, 56, 1, 2, 3, 5, 8, 16, 26, 45, 62, 90, 1, 2, 3, 5, 8, 16, 26, 43, 71, 97, 145, 1, 2, 3, 5, 8, 13, 26, 42, 71, 111, 153, 234

Offset: 1

Paolo Xausa, Jun 21 2024

			The array begins:
  k\n|  1  2  3  4   5   6   7   8   9   10  ...
  ----------------------------------------------
   1 |  2, 3, 4, 6,  9, 14, 22, 35, 56,  90, ... = A001611 (from n = 2).
   2 |  1, 2, 4, 7, 11, 17, 26, 40, 62,  97, ...
   3 |  1, 2, 4, 6, 10, 17, 28, 45, 71, 111, ...
   4 |  1, 2, 3, 6, 10, 16, 26, 43, 71, 116, ...
   5 |  1, 2, 3, 5, 10, 16, 26, 42, 68, 111, ...
   6 |  1, 2, 3, 5,  8, 16, 26, 42, 68, 110, ...
   7 |  1, 2, 3, 5,  8, 13, 26, 42, 68, 110, ...
   8 |  1, 2, 3, 5,  8, 13, 21, 42, 68, 110, ...
   9 |  1, 2, 3, 5,  8, 13, 21, 34, 68, 110, ...
  10 |  1, 2, 3, 5,  8, 13, 21, 34, 55, 110, ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
Hung Viet Chu and Zachary Louis Vasseur, Weighted Schreier-type Sets and the Fibonacci Sequence, arXiv:2405.19352 [math.CO], 2024. See p. 2, Table 1 and Theorem 1.2.

Cf. A000045, A001611, A373345, A373556, A373557.

A373889[k_, n_] := Which[n < k, Fibonacci[n+1], k == 1, Fibonacci[n-k+2] + 1, True, 2*Sum[Binomial[n-k, i]*Fibonacci[k-i], {i, 0, k-2}] + 2*Binomial[n-k, k-1] + Sum[Binomial[j, n-j], {j, n-k}]];
Table[A373889[k-n+1, n], {k, 15}, {n, k}]

T(k,n) = A000045(n-k+2) + 1, for k = 1 and n >= k;
T(k,n) = 2*(Sum_{i=0..k-2} binomial(n-k,i)*A000045(k-i)) + 2*binomial(n-k,k-1) + Sum_{j=1..n-k} binomial(j,n-j), for k >= 2 and n >= k;
T(k,n) = A000045(n+1) otherwise.
T(n,n) = 2*A000045(n).

cp's OEIS Frontend

A373854 Row sums of A373556.

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A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

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A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).

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A355489 Numbers k such that A000120(k) = A007814(k) + 2.

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A373558 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in increasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

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A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

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