OFFSET
1,2
COMMENTS
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:
1
1 1
0 2 1
0 1 3 1
0 0 3 4 1
0 0 1 6 5 1
...
This is a vertical Pascal's triangle and the horizontal sum gives the Fibonacci numbers.
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
LINKS
N. Karimilla Bi, Amritanshu Prasad, and P. Giftson Santhosh, Residues modulo powers of two in the Young-Fibonacci lattice, arXiv:1702.06684 [math.CO], 2017. See Figure 1.
EXAMPLE
The irregular triangle begins:
n
1: 1; f(1) = 1.
2: 2, 11; f(2) = 2.
3: 12, 21, 111; f(3) = 3.
4: 22, 112, 121, 211, 1111; f(4) = 5.
5: 122, 212, 221, 1112, 1121, 1211, 2111, 11111; f(5) = 8.
...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,base,tabf
AUTHOR
Amarnath Murthy, Nov 13 2005
EXTENSIONS
More terms from Terryjames Morris (trm5002(AT)psu.edu), Mar 09 2007
Duplicate term removed by Stefano Spezia, Jan 14 2024
STATUS
approved