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A114034 Let f(n) be the number of sequences of 1's and 2's which sum to n. Sequence contains the string of sequences.

%I A114034 #19 Jan 14 2024 16:10:22
%S A114034 1,2,11,12,21,111,22,112,121,211,1111,122,212,221,1112,1121,1211,2111,
%T A114034 11111,222,1122,1212,1221,2112,2121,2211,11112,11121,11211,12111,
%U A114034 21111,111111,1222,2122,2212,2221,11122,11212,11221,12112,12121,12211,21112,21121,21211,22111,111112,111121,111211,112111,121111,211111,1111111
%N A114034 Let f(n) be the number of sequences of 1's and 2's which sum to n. Sequence contains the string of sequences.
%C A114034 Number of sequences of ones and twos that sum to n are Fibonacci(n+1). The maximum number of terms in a sequence is n. (111111 n times). Following is the triangle of the frequency of sequences of each size:
%C A114034 1
%C A114034 1 1
%C A114034 0 2 1
%C A114034 0 1 3 1
%C A114034 0 0 3 4 1
%C A114034 0 0 1 6 5 1
%C A114034 ...
%C A114034 This is a vertical Pascal's triangle and the horizontal sum gives the Fibonacci numbers.
%C A114034 Each row of the irregular triangle provides a list of increasing positive integers of only 1s and 2s that sum up to n (see Example section). - _Stefano Spezia_, Jan 14 2024
%H A114034 N. Karimilla Bi, Amritanshu Prasad, and P. Giftson Santhosh, <a href="https://arxiv.org/abs/1702.06684">Residues modulo powers of two in the Young-Fibonacci lattice</a>, arXiv:1702.06684 [math.CO], 2017. See Figure 1.
%e A114034 The irregular triangle begins:
%e A114034   n
%e A114034   1:   1;                                          f(1) = 1.
%e A114034   2:   2,  11;                                     f(2) = 2.
%e A114034   3:  12,  21, 111;                                f(3) = 3.
%e A114034   4:  22, 112, 121,  211, 1111;                    f(4) = 5.
%e A114034   5: 122, 212, 221, 1112, 1121, 1211, 2111, 11111; f(5) = 8.
%e A114034 ...
%t A114034 row[n_] := Select[Range[(10^n-1)/9], SubsetQ[{1,2}, DeleteDuplicates[digits = IntegerDigits[#]]] && Total[digits]==n &]; Array[row,7]//Flatten (* _Stefano Spezia_, Jan 14 2024 *)
%Y A114034 Cf. A000045, A242614.
%K A114034 nonn,base,tabf
%O A114034 1,2
%A A114034 _Amarnath Murthy_, Nov 13 2005
%E A114034 More terms from Terryjames Morris (trm5002(AT)psu.edu), Mar 09 2007
%E A114034 Duplicate term removed by _Stefano Spezia_, Jan 14 2024