

A326031


Weight of the setsystem with BIInumber 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
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OFFSET

0,4


COMMENTS

A binary index of n is any position of a 1 in its reversed binary expansion. We define the setsystem with BIInumber 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 BIInumber. 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 BIInumber of {{2},{1,3}} is 18. The weight of a setsystem 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 setsystems together with their BIInumbers 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



