1,2

As an interspersion (and dispersion), the array is, as a sequence, a permutation of the positive integers. Column k consists of the numbers m such that the least summand in the tribonacci representation of m is T(1,k). For example, column 1 consists of numbers with least summand 1. Except for initial terms in some cases, (column 1)=A003265, (row 1)=(A000073), (row 2)=(A001590). This array arises from tribonacci representations in much the same way that the Wythoff array, A035513, arises from Fibonacci (or Zeckendorf) representations.

T(1,1)=1, T(1,2)=2, T(1,3)=4, T(1,k)=T(1,k-1)+T(1,k-2)+T(1,k-3) for k>3. Row 1 is the tribonacci basis; write B(k)=T(1,k). Each row satisfies the recurrence T(n,k)=T(n,k-1)+T(n,k-2)+T(n,k-3). T(n,1) is least number not in an earlier row. If T(n,1) has tribonacci representation B(k(1))+B(k(2))+...+B(k(m)), then T(n,2) = B(k(2))+B(k(3))+...+B(k(m+1)) and T(n,3) = B(k(3))+B(k(4))+...+B(k(m+2)). (Continued shifting of indices gives the other terms in row n, also.)

Northwest corner:

1 2 4 7 13 24 44 81

3 6 11 20 37 68 125 230

5 9 17 36 57 105 193 355

8 15 28 51 94 173 318 585

Cf. A035513.

Adjacent sequences: A136172 A136173 A136174 this_sequence A136176 A136177 A136178

Sequence in context: A064578 A057027 A090894 this_sequence A129258 A104650 A083179

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Dec 18 2007