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A326031 Weight of the set-system with BII-number n. 88
0, 1, 1, 2, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 5, 2, 3, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 6, 7, 4, 5, 5, 6, 6, 7, 7, 8, 5, 6, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 6, 7, 4, 5, 5, 6, 6, 7, 7, 8, 5, 6, 6, 7, 7, 8, 8, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets of positive integers has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18. The weight of a set-system is the sum of sizes of its elements (sometimes called its edges).

LINKS

Table of n, a(n) for n=0..87.

FORMULA

a(2^x + ... + 2^z) = w(x + 1) + ... + w(z + 1), where x...z are distinct nonnegative integers and w = A000120. For example, a(6) = a(2^2 + 2^1) = w(3) + w(2) = 3.

EXAMPLE

The sequence of set-systems together with their BII-numbers begins:

   0: {}

   1: {{1}}

   2: {{2}}

   3: {{1},{2}}

   4: {{1,2}}

   5: {{1},{1,2}}

   6: {{2},{1,2}}

   7: {{1},{2},{1,2}}

   8: {{3}}

   9: {{1},{3}}

  10: {{2},{3}}

  11: {{1},{2},{3}}

  12: {{1,2},{3}}

  13: {{1},{1,2},{3}}

  14: {{2},{1,2},{3}}

  15: {{1},{2},{1,2},{3}}

  16: {{1,3}}

  17: {{1},{1,3}}

  18: {{2},{1,3}}

  19: {{1},{2},{1,3}}

  20: {{1,2},{1,3}}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Table[Length[Join@@bpe/@bpe[n]], {n, 0, 100}]

CROSSREFS

Cf. A000120, A029931, A048793, A061775, A070939, A072639, A116549, A302242, A305830, A326701, A326702, A326703, A326704.

Sequence in context: A071456 A071505 A071508 * A322997 A085561 A260651

Adjacent sequences:  A326028 A326029 A326030 * A326032 A326033 A326034

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 20 2019

STATUS

approved

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Last modified April 17 21:53 EDT 2021. Contains 343071 sequences. (Running on oeis4.)