0,2

Every Fibonacci sequence with positive terms occurs as some row of Wythoff's array (A035513), so a(n) is always defined. There appear to be two possible offsets for the sequence of sums within row a(n); either T(n,j-1) + T(n,j+1) = T(a(n),j-3) for j>=3 or T(n,j-1) + T(n,j+1) = T(a(n),j-1) for j>=1. The first case seems to occur whenever n has a Zeckendorf representation which ends in 1, 10100, 101000100, 1010001000100, 10100010001000100, etc. (each successive ending is obtained by changing the lefthand 1 to 10100). The values of these endings are 1,11,79,545,3739,25631,175681 ... and equal F(i)*F(i+1) + F(i+2)^2 for i = 0,2,4,6,... where F(i) is the i-th Fibonacci number. These values also appear in the table A127561 at a(1,0), a(1,1), a(2,3), a(5,8), ..., a(F(2n-1),F(2n)) for n = 0,1,2,3....

The Zeckendorf representation of n is the unique binary sequence ...,b(4),b(3),b(2) for which n = sum_{i>=2} b(i)F(i) and two consecutive b's cannot both be 1. For example, the Zeckendorf representation of 100 is 1000010100, since 100 = 89+8+3 = F(11)+F(6)+F(4).

Conjecture: If the Zeckendorf representation of n ends in 1, then a(n) = 15 + H(n-H(n))*29 + (n-H(n) - H(n-H(n)))*18, where H(n) is Hofstadter's G sequence A005206. Otherwise, a(n) = 1 + H(H(n))*7 + (H(n) - H(H(n)))*4 unless the Zeckendorf representation of n has one of the 0-endings listed in the first comment line, in which case a(n) = a(n+1) - 11.

a(2)=8 because the sequence of sums T(2,j-1)+T(2,j+1) begins with 6+16=22=T(8,0) and 10+26=36=T(8,1). a(1)=15 because the sequence of sums T(1,j-1)+T(1,j+1) begins with 4+11=15, 7+18=25, 11+29=40=T(15,0), and 18+47=65=T(15,1).

T[i_, j_]:=i*Fibonacci[j+1]+Fibonacci[j+2]*Floor[(i+1)(1+Sqrt[5])/2]; U[i_, j_]:=T[i, j-1]+T[i, j+1]; Tpair[i_, j_]:={T[i, j], T[i, j+1]}; Upair[i_, j_]:={U[i, j], U[i, j+1]}; a[n_]:=a[n]=Module[{v}, For[v=0, True, v++, If[Upair[n, 1]==Tpair[v, 0]||Upair[n, 3]==Tpair[v, 0], Return[v]]]]

Cf. A035513, A005206, A003622, A127561.

Sequence in context: A103241 A094501 A090636 this_sequence A139725 A009929 A123542

Adjacent sequences: A126889 A126890 A126891 this_sequence A126893 A126894 A126895

nonn

Kenneth J Ramsey (Ramseykk2(AT)aol.com), Jan 13 2007

Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Feb 09 2007