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

1,2

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

Table of n, a(n) for n=1..53.

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

Cf. A000045, A242614.

Sequence in context: A063112 A038113 A089600 * A136970 A136967 A213975

Adjacent sequences: A114031 A114032 A114033 * A114035 A114036 A114037

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