This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A225620 #93 Apr 24 2024 14:53:17
%S A225620 1,2,3,4,6,7,8,10,12,14,15,16,20,24,26,28,30,31,32,36,40,42,48,52,56,
%T A225620 58,60,62,63,64,72,80,84,96,100,104,106,112,116,120,122,124,126,127,
%U A225620 128,136,144,160,164,168,170,192,200,208,212,224,228,232,234,240,244,248,250,252,254,255
%N A225620 Indices of partitions in the table of compositions of A228351.
%C A225620 Also triangle read by rows in which T(n,k) is the decimal representation of a binary number whose mirror represents the k-th partition of n according with the list of juxtaposed reverse-lexicographically ordered partitions of the positive integers (A026792).
%C A225620 In order to construct this sequence as a triangle we use the following rules:
%C A225620 - In the list of A026792 we replace each part of size j of the k-th partition of n by concatenation of j - 1 zeros and only one 1.
%C A225620 - Then replace this new set of parts by the concatenation of its parts.
%C A225620 - Then replace this string by its mirror version which is a binary number.
%C A225620 T(n,k) is the decimal value of this binary number, which represents the k-th partition of n (see example).
%C A225620 The partitions of n are represented by a subsequence with A000041(n) integers starting with 2^(n-1) and ending with 2^n - 1, n >= 1. The odd numbers of the sequence are in A000225.
%C A225620 First differs from A065609 at a(23).
%C A225620 Conjecture: this sequence is a sorted version of b(n) where b(2^k) = 2^k for k >= 0, b(n) = A080100(n)*(2*b(A053645(n)) + 1) otherwise. - _Mikhail Kurkov_, Oct 21 2023
%F A225620 Conjecture: a(A000070(m) - k) = 2^m - A228354(k) for m > 0, 0 < k <= A000041(m). - _Mikhail Kurkov_, Oct 20 2023
%e A225620 T(6,8) = 58 because 58 in base 2 is 111010 whose mirror is 010111 which is the concatenation of 01, 01, 1, 1, whose number of digits are 2, 2, 1, 1, which are also the 8th partition of 6.
%e A225620 Illustration of initial terms:
%e A225620 The sequence represents a table of partitions (see below):
%e A225620 --------------------------------------------------------
%e A225620 . Binary Partitions
%e A225620 n k T(n,k) number Mirror Diagram (A026792)
%e A225620 . 1 2 3 4 5 6
%e A225620 --------------------------------------------------------
%e A225620 . _
%e A225620 1 1 1 1 1 | 1,
%e A225620 . _ _
%e A225620 1 1 2 10 01 _ | 2,
%e A225620 2 2 3 11 11 | | 1,1,
%e A225620 . _ _ _
%e A225620 3 1 4 100 001 _ _ | 3,
%e A225620 3 2 6 110 011 _ | | 2,1,
%e A225620 3 3 7 111 111 | | | 1,1,1,
%e A225620 . _ _ _ _
%e A225620 4 1 8 1000 0001 _ _ | 4,
%e A225620 4 2 10 1010 0101 _ _|_ | 2,2,
%e A225620 4 3 12 1100 0011 _ _ | | 3,1,
%e A225620 4 4 14 1110 0111 _ | | | 2,1,1,
%e A225620 4 5 15 1111 1111 | | | | 1,1,1,1,
%e A225620 . _ _ _ _ _
%e A225620 5 1 16 10000 00001 _ _ _ | 5,
%e A225620 5 2 20 10100 00101 _ _ _|_ | 3,2,
%e A225620 5 3 24 11000 00011 _ _ | | 4,1,
%e A225620 5 4 26 11010 01011 _ _|_ | | 2,2,1,
%e A225620 5 5 28 11100 00111 _ _ | | | 3,1,1,
%e A225620 5 6 30 11110 01111 _ | | | | 2,1,1,1,
%e A225620 5 7 31 11111 11111 | | | | | 1,1,1,1,1,
%e A225620 . _ _ _ _ _ _
%e A225620 6 1 32 100000 000001 _ _ _ | 6
%e A225620 6 2 36 100100 001001 _ _ _|_ | 3,3,
%e A225620 6 3 40 101000 000101 _ _ | | 4,2,
%e A225620 6 4 42 101010 010101 _ _|_ _|_ | 2,2,2,
%e A225620 6 5 48 110000 000011 _ _ _ | | 5,1,
%e A225620 6 6 52 110100 001011 _ _ _|_ | | 3,2,1,
%e A225620 6 7 56 111000 000111 _ _ | | | 4,1,1,
%e A225620 6 8 58 111010 010111 _ _|_ | | | 2,2,1,1,
%e A225620 6 9 60 111100 001111 _ _ | | | | 3,1,1,1,
%e A225620 6 10 62 111110 011111 _ | | | | | 2,1,1,1,1,
%e A225620 6 11 63 111111 111111 | | | | | | 1,1,1,1,1,1,
%e A225620 .
%e A225620 Triangle begins:
%e A225620 1;
%e A225620 2, 3;
%e A225620 4, 6, 7;
%e A225620 8, 10, 12, 14, 15;
%e A225620 16, 20, 24, 26, 28, 30, 31;
%e A225620 32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63;
%e A225620 ...
%e A225620 From _Gus Wiseman_, Apr 01 2020: (Start)
%e A225620 Using the encoding of A066099, this sequence ranks all finite nonempty multisets, as follows.
%e A225620 1: {1}
%e A225620 2: {2}
%e A225620 3: {1,1}
%e A225620 4: {3}
%e A225620 6: {1,2}
%e A225620 7: {1,1,1}
%e A225620 8: {4}
%e A225620 10: {2,2}
%e A225620 12: {1,3}
%e A225620 14: {1,1,2}
%e A225620 15: {1,1,1,1}
%e A225620 16: {5}
%e A225620 20: {2,3}
%e A225620 24: {1,4}
%e A225620 26: {1,2,2}
%e A225620 28: {1,1,3}
%e A225620 30: {1,1,1,2}
%e A225620 31: {1,1,1,1,1}
%e A225620 (End)
%t A225620 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A225620 Select[Range[0,100],LessEqual@@stc[#]&] (* _Gus Wiseman_, Apr 01 2020 *)
%Y A225620 Column 1 is A000079. Row n has length A000041(n). Right border gives A000225.
%Y A225620 Cf. A026792, A065609, A135010, A141285, A186114, A194446, A194546, A206437, A207779, A211978, A225600, A225610, A228351.
%Y A225620 The case covering an initial interval is A333379 or A333380.
%Y A225620 All of the following pertain to compositions in the order of A066099.
%Y A225620 - The weakly increasing version is this sequence.
%Y A225620 - The weakly decreasing version is A114994.
%Y A225620 - The strictly increasing version is A333255.
%Y A225620 - The strictly decreasing version is A333256.
%Y A225620 - The unequal version is A233564.
%Y A225620 - The equal version is A272919.
%Y A225620 - The case covering an initial interval is A333217.
%Y A225620 - Initial intervals are ranked by A164894.
%Y A225620 - Reversed initial intervals are ranked by A246534.
%Y A225620 Cf. A000120, A029931, A048793, A070939, A124766, A233249, A333219, A333220.
%K A225620 nonn,tabf
%O A225620 1,2
%A A225620 _Omar E. Pol_, Aug 03 2013