All linear second-order sequences are a linear combination of these two sequences.
1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 8, 13, 13, 21, 21, 34, 34, 55, 55, 89, 89, 144, 144, 233, 233, 377, 377, 610, 610, 987, 987, 1597, 1597, 2584, 2584, 4181, 4181, 6765, 6765, 10946, 10946, 17711, 17711, 28657, 28657, 46368, 46368, 75025, 75025, 121393, 121393
1
Note that the two sequences are identical except for being shifted by one.
T. D. Noe, Plot of 150 pairs
T. D. Noe, Table of 150 pairs
Eric W. Weisstein, MathWorld: Linear Recurrence Equation
(Mma) nn = 2; t = IdentityMatrix[nn]; Do[AppendTo[t, Sum[t[[k - i]], {i, nn}]], {k, nn + 1, nn + 60/nn}]; t = Drop[Flatten[t], nn^2]; t
nonn,tabl
T. D. Noe, Jan 15 2016