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.

1, 2, 11, 12, 21, 111, 22, 112, 121, 211, 1111, 122, 212, 221, 1112, 1121, 1211, 2111, 11111, 222, 1122, 1212, 1221, 2112, 2121, 2211, 11112, 11121, 11211, 12111, 21111, 111111, 1222, 2122, 2212, 2221, 11122, 11212, 11221, 12112, 12121, 12211, 21112, 21121, 21211, 22111, 111112, 111121, 111211, 112111, 121111, 211111, 1111111

Offset: 1

Author

Amarnath Murthy, Nov 13 2005

Keywords

• nonn
• base
• tabf

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

Examples

			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.
...
		

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.

Crossrefs

Cf. A000045, A242614.

Programs

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 *)

Extensions

More terms from Terryjames Morris (trm5002(AT)psu.edu), Mar 09 2007

Duplicate term removed by Stefano Spezia, Jan 14 2024