An oscillating Fibonacci sequence?

[Creighton Dement]

Hi, 

The sequence 

( see http://www.crowdog.de/OsciFib.html )

1vesrokseq: 0, 1, 0, -1, 2, 1, -2, -1, 4, -5, -10, -5, -4, 3, 12, 1, -2,
-1, 14, -5, -10, 5, 16, -1, -8, 15, 32, 21, 26, 19, 16, 5, -10, -5, 18,
-21, -38, -23, -24, -9, 6, 37, 2, -3, -44, -3, -2, 41, -8, -15, 26, 61,
36, 45, 28, 19, -8, -45, 4, 17, -38, 39, 62, 39, 38, 13, -14, -67, -14,
-13, 42, -41, -70, -39, -36, -1, 38, -39, -78, -39, -38, 3, 46, -33,
-70, -19, -4, 63, -28, -53, 8, 45, -38, -85, -30, -21, 44, -73, -126,
-101, -128, -129, -156, -183, -236, -315, -446, -655, -994, -1541,
-2426, -3857, -6172, -9917, -15976, -25779, -41640, -67303, -108826,
-176011, -284718, -460609, -745206, -1205693, -1950776, -3156345,
-5106996, -8263215, -13370084, -21633171, -35003126

with indices:  

1vesrokseq: 0[1], 1[2], 0[3], -1[4], 2[5], 1[6], -2[7], -1[8], 4[9],
-5[10], -10[11], -5[12], -4[13], 3[14], 12[15], 1[16], -2[17], -1[18],
14[19], -5[20], -10[21], 5[22], 16[23], -1[24], -8[25], 15[26], 32[27],
21[28], 26[29], 19[30], 16[31], 5[32], -10[33], -5[34], 18[35], -21[36],
-38[37], -23[38], -24[39], -9[40], 6[41], 37[42], 2[43], -3[44],
-44[45], -3[46], -2[47], 41[48], -8[49], -15[50], 26[51], 61[52],
36[53], 45[54], 28[55], 19[56], -8[57], -45[58], 4[59], 17[60], -38[61],
39[62], 62[63], 39[64], 38[65], 13[66], -14[67], -67[68], -14[69],
-13[70], 42[71], -41[72], -70[73], -39[74], -36[75], -1[76], 38[77],
-39[78], -78[79], -39[80], -38[81], 3[82], 46[83], -33[84], -70[85],
-19[86], -4[87], 63[88], -28[89], -53[90], 8[91], 45[92], -38[93],
-85[94], -30[95], -21[96], 44[97], -73[98], -126[99], -101[100],
-128[101], -129[102], -156[103], -183[104], -236[105], -315[106],
-446[107], -655[108], -994[109], -1541[110], -2426[111], -3857[112],
-6172[113], -9917[114], -15976[115], -25779[116], -41640[117],
-67303[118], -108826[119], -176011[120], -284718[121], -460609[122],
-745206[123], -1205693[124], -1950776[125], -3156345[126],
-5106996[127], -8263215[128], -13370084[129], -21633171[130]

has me quite confused. The sequence seems, more or less,  to oscillate
around 0 up to the 99th term. If any of the first 99 terms are included
in a SuperSeeker search, apparently, nothing is returned.  However, on
the 100th term (which happens to have the value 101) the sequence begins
decreasing monotonely and a generating function is found:

 Report on [
-101,-128,-129,-156,-183,-236,-315,-446,-655,-994,-1541,-2426]: Many
tests are carried out, but only potentially useful information (if any)
is reported here.

Guesss suggests that the generating function F(x) may satisfy the
following algebraic or differential equation:

-126*x^3+53*x^2+175*x-101+(x^4-x^3-2*x^2+3*x-1)*F(x) = 0

If this is correct the next 6 numbers in the sequence are:

[-3857, -6172, -9917, -15976, -25779, -41640]

*****

As the terms get greater and greater, one sees that each terms is
approximated by the two preceeding terms and a(n+1)/a(n) appears to
approach the golden ratio phi. 

Is there a way to show that the first  hundred terms are indeed a part
of the latter subsequence?

I cannot exclude this being due to program error on my part or to 
something I overlooked in the "rok symmetry", for example.

Sincerely, 
Creighton