Fibonacci-Related Numbers

My diurnal age today is \( \textbf{28047} \) and this number has the interesting property that it's part of a Fibonacci sequence with seed numbers of 3 and 9. This leads to the following sequence of numbers:

3, 9, 12, 21, 33, 54, 87, 141, 228, 369, 597, 966, 1563, 2529, 4092, 6621, 10713, 17334, 28047, ...

These initial numbers form part of OEIS A022379. 

The thought occurred to me that given two seed digits between 0 and 9, there ought to be a limited set of numbers from 10 to let's say 40000 that are generated by two single digit seeds. I got Gemini to write a Python program (permalink) to investigate this and it turns out that there are 651 numbers with this property. Here they are:

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 120, 121, 123, 124, 125, 126, 128, 129, 131, 133, 134, 136, 137, 138, 139, 141, 142, 144, 146, 147, 149, 150, 152, 154, 155, 157, 159, 160, 162, 163, 165, 167, 168, 170, 173, 175, 176, 178, 180, 181, 183, 186, 188, 189, 191, 194, 196, 199, 201, 202, 204, 207, 209, 212, 215, 217, 220, 222, 223, 225, 228, 230, 233, 236, 238, 241, 243, 246, 249, 251, 254, 257, 259, 262, 264, 267, 270, 272, 275, 280, 283, 285, 288, 291, 293, 296, 301, 304, 306, 309, 314, 317, 322, 325, 327, 330, 335, 338, 343, 348, 351, 356, 359, 361, 364, 369, 372, 377, 382, 385, 390, 393, 398, 403, 406, 411, 416, 419, 424, 427, 432, 437, 440, 445, 453, 458, 461, 466, 471, 474, 479, 487, 492, 495, 500, 508, 513, 521, 526, 529, 534, 542, 547, 555, 563, 568, 576, 581, 584, 589, 597, 602, 610, 618, 623, 631, 636, 644, 652, 657, 665, 673, 678, 686, 691, 699, 707, 712, 720, 733, 741, 746, 754, 762, 767, 775, 788, 796, 801, 809, 822, 830, 843, 851, 856, 864, 877, 885, 898, 911, 919, 932, 940, 945, 953, 966, 974, 987, 1000, 1008, 1021, 1029, 1042, 1055, 1063, 1076, 1089, 1097, 1110, 1118, 1131, 1144, 1152, 1165, 1186, 1199, 1207, 1220, 1233, 1241, 1254, 1275, 1288, 1296, 1309, 1330, 1343, 1364, 1377, 1385, 1398, 1419, 1432, 1453, 1474, 1487, 1508, 1521, 1529, 1542, 1563, 1576, 1597, 1618, 1631, 1652, 1665, 1686, 1707, 1720, 1741, 1762, 1775, 1796, 1809, 1830, 1851, 1864, 1885, 1919, 1940, 1953, 1974, 1995, 2008, 2029, 2063, 2084, 2097, 2118, 2152, 2173, 2207, 2228, 2241, 2262, 2296, 2317, 2351, 2385, 2406, 2440, 2461, 2474, 2495, 2529, 2550, 2584, 2618, 2639, 2673, 2694, 2728, 2762, 2783, 2817, 2851, 2872, 2906, 2927, 2961, 2995, 3016, 3050, 3105, 3139, 3160, 3194, 3228, 3249, 3283, 3338, 3372, 3393, 3427, 3482, 3516, 3571, 3605, 3626, 3660, 3715, 3749, 3804, 3859, 3893, 3948, 3982, 4003, 4037, 4092, 4126, 4181, 4236, 4270, 4325, 4359, 4414, 4469, 4503, 4558, 4613, 4647, 4702, 4736, 4791, 4846, 4880, 4935, 5024, 5079, 5113, 5168, 5223, 5257, 5312, 5401, 5456, 5490, 5545, 5634, 5689, 5778, 5833, 5867, 5922, 6011, 6066, 6155, 6244, 6299, 6388, 6443, 6477, 6532, 6621, 6676, 6765, 6854, 6909, 6998, 7053, 7142, 7231, 7286, 7375, 7464, 7519, 7608, 7663, 7752, 7841, 7896, 7985, 8129, 8218, 8273, 8362, 8451, 8506, 8595, 8739, 8828, 8883, 8972, 9116, 9205, 9349, 9438, 9493, 9582, 9726, 9815, 9959, 10103, 10192, 10336, 10425, 10480, 10569, 10713, 10802, 10946, 11090, 11179, 11323, 11412, 11556, 11700, 11789, 11933, 12077, 12166, 12310, 12399, 12543, 12687, 12776, 12920, 13153, 13297, 13386, 13530, 13674, 13763, 13907, 14140, 14284, 14373, 14517, 14750, 14894, 15127, 15271, 15360, 15504, 15737, 15881, 16114, 16347, 16491, 16724, 16868, 16957, 17101, 17334, 17478, 17711, 17944, 18088, 18321, 18465, 18698, 18931, 19075, 19308, 19541, 19685, 19918, 20062, 20295, 20528, 20672, 20905, 21282, 21515, 21659, 21892, 22125, 22269, 22502, 22879, 23112, 23256, 23489, 23866, 24099, 24476, 24709, 24853, 25086, 25463, 25696, 26073, 26450, 26683, 27060, 27293, 27437, 27670, 28047, 28280, 28657, 29034, 29267, 29644, 29877, 30254, 30631, 30864, 31241, 31618, 31851, 32228, 32461, 32838, 33215, 33448, 33825, 34435, 34812, 35045, 35422, 35799, 36032, 36409, 37019, 37396, 37629, 38006, 38616, 38993, 39603, 39980

The following code (permalink) can be used to find the seeds of any number in this list. It's not impressive code I'm sure but it seems to do the job. Let's try with the last number in the previous list, 39980.

c=39980 L=[] b=round(c/((1+sqrt(5))/2)) L.append(c);L.append(b) a=0 while b>=a and c>b: a=c-b if a <b: c=b b=a L.append(a) L.reverse() if len(str(L[1]))==2: a=L[1]-L[0] b=L[0] print("Seed numbers for Fibonacci sequence are", a,"and",b) X=[a,b] L.remove(L[0]) print(X+L) else: print("Seed numbers for Fibonacci sequence are", L[0],"and",L[1]) print(L)

Seed numbers for Fibonacci sequence are 9 and 4

[9, 4, 13, 17, 30, 47, 77, 124, 201, 325, 526, 851, 1377, 2228, 3605, 5833, 9438, 15271, 24709, 39980]

Interestingly, 39980 is listed in the OEIS as OEIS A022132 but the seeds are listed as 4 and 13 but this is equivalent of course to seeds of 9 and 4. Zero can be a seed number in the first position but it can't then be followed by another zero and so there are 10 x 9 = 90 distinct pairs of single digit seeds. 

Two different pairs of single digit seeds can lead to the same sequence. For example:

• seeds of 1 and 3 \( \rightarrow\) 1, 3, 4, 7, 11, 18, 29, ...
• seeds of 2 and 1 \( \rightarrow \) 2, 1, 3, 4, 7, 11, 18, 29, ...

Figure 1 shows a plot of these Fibonacci-related numbers and it's interesting that they form a definite exponential curve with the density of points noticeably thinning out as the numbers become larger.

Figure 1

ADDENDUM:

After completing this post, I discovered that I'd already covered this topic in a post from Thursday, 12th June 2025 titled Fibonacci Numbers Derived From Single Digits. However, in that post I only considered seed digits where the first was smaller than the second and so I discounted a number like 59 because it requires seed digits of 7 and 3 such that the sequence 7, 3, 10, 13, 23, 36, 59 is generated.