Investigating constant C=0.1688... in A081881 comment

[Gerald McGarvey]

sizes ...
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Date: Mon, 28 Jan 2008 10:38:32 +0100
From: Rainer Rosenthal <r.rosenthal at web.de>
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Subject: Re: Investigating constant C=0.1688... in A081881 comment
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Gerald McGarvey wrote:
> An observation about A081881: for n=2 to 17 either
>    a(n) = floor(exp(1)*a(n-1)) - 1 
> or  
>    a(n) = floor(exp(1)*a(n-1))   (for n=2,4,16)

Here are more terms for n = 2, 4, 16, ...

2, 4, 16, 24, 26, 38, 46, 50, 52, 58, 62, 63, 65, 70, 73, 78, 79,
85, 86, 90, 101, 121, 126, 133, 138, 140, 148, 152, 155, 159,
182, 186, 189, 210, 211, 212, 229, 237, 238, 249, 256, 263,
294, 299, 301, 304, 331, 342, 358, 377, 405, 412, 413, 425,
461, 463, 480, 487, 492, 499, 514, 516, 517, 530, 535, 537,
540, 542, 547, 554, 563, 565, 570, 586, 589, 590, 596, 618,
626, 629, 631, 633, 637, 648, 659, 693, 696, 698, 699, 701,
708, 736, 738, 741, 742, 755, 756, 757, 760, 763, 773, 777,
779, 785, 786, 788, 793, 795, 802, 803, 809, 815, 821, 824,
830, 834, 849, 854, 855, 864, 865, 876, 882, 894, 903, 930,
931, 937, 938, 941, 942, 945, 947, 949, 953, 954, 955, 959,
978, 983, 993, 994, 995, 1001, 1005, 1019, 1029, 1034, 1038,
1073, 1075, 1085, 1092, 1107, 1134, 1135, 1144, 1149, 1162,
1178, 1187, 1190, 1193, 1197, 1202, 1203, 1226, 1241, 1243,
1255, 1261, 1262, 1273, 1280, 1287, 1296, 1320, 1324, 1327,
1330, 1361, 1367, 1390, 1394, 1416, 1423, 1425, 1431, 1433,
1437, 1441, 1442, 1447, 1457, 1462, 1473, 1475, 1476, 1497,
1505, 1507, 1510, 1512, 1564, 1576, 1583, 1587, 1616, 1620,
1622, 1623, 1627, 1632, 1652, 1655, 1670, 1674, 1677, 1697,
1701, 1717, 1722, 1725, 1726, 1733, 1734, 1739, 1741, 1755,
1757, 1769, 1773, 1778, 1782, 1785, 1796, 1824, 1838, 1845,
1863, 1866, 1884, 1889, 1891, 1921, 1928, 1934, 1942, 1943,
1954, 1968, 1969, 1975, 1989, 1992, 1994, 1995, 1996, 1997,
2007, 2017, 2023, 2039, 2048, 2049, 2051, 2052, 2061, 2063,
2067, 2077, 2078, 2088, 2122, 2129, 2159, 2174, 2183, 2192,
2206, 2207, 2219, 2220, 2223, 2229, 2235, 2236, 2251, 2258,
2260, 2269, 2272, 2283, 2286, 2292, 2297, 2319, 2321, 2360,
2376, 2385, 2389, 2391, 2400, 2404, 2406, 2410, 2411, 2412,
2419, 2425, 2433, 2437, 2439, 2454, 2461, 2470, 2473, 2477,
2488, 2493, 2507, 2512, 2517, 2525, 2530, 2544, 2547, 2553,
2554, 2560, 2591, 2592, 2617, 2624, 2630, 2631, 2644, 2645,
2646, 2667, 2668, 2670, 2679, 2682, 2687, 2695, 2703, 2705,
2710, 2711, 2728, 2731, 2740, 2741, 2745, 2752, 2753, 2759,
2764, 2765, 2798, 2801, 2809, 2816, 2817, 2818, 2829, 2838,
2841, 2850, 2855, 2880, 2883, 2887, 2892, 2893, 2913, 2921,
2926, 2927, 2928, 2939, 2944, 2947, 2948, 2954, 2979, 2988,
3008, 3019, 3028, 3046, 3047, 3058, 3060, 3077, 3094, 3103,
3112, 3121, 3123, 3139, 3143, 3144, 3148, 3170, 3171, 3173,
3183, 3186, 3195, 3198, 3204, 3213, 3214, 3220, 3224, 3228,
3229, 3232, 3246, 3259, 3261, 3262, 3274, 3284, 3285, 3291,
3298, 3304, 3305, 3310, 3336, 3342, 3349, 3353, 3354, 3356,
3365, 3366, 3388, 3398, 3404, 3412, 3427, 3436, 3439, 3449,
3451, 3455, 3475, 3481, 3486, 3491, 3500, 3502, 3507, 3512,
3543, 3549, 3554, 3569, 3575, 3584, 3601, 3607, 3610, 3632,
3635, 3644, 3654, 3658, 3664, 3673, 3675, 3683, 3685, 3689,
3690, 3699, 3704, 3709, 3713, 3717, 3722, 3723, 3726, 3737,
3738, 3741, 3751, 3758, 3766, 3772, 3778, 3779, 3791, 3794,
3802, 3820, 3821, 3832, 3836, 3837, 3845, 3862, 3871, 3884,
3889, 3901, 3906, 3912, 3915, 3916, 3919, 3924, 3930, 3931,
3937, 3938, 3941, 3952, 3954, 3955, 3957, 3969, 3971, 3984,
3999, 4005, 4009, 4010, 4012, 4014, 4015, 4029, 4041, 4049,
4057, 4063, 4075, 4079, 4082, 4091, 4094, 4107, 4134, 4159,
4163, 4165, 4168, 4174, 4214, 4215, 4216, 4219, 4224, 4227,
4234, 4253, 4254, 4269, 4274, 4280, 4283, 4297, 4323, 4325,
4332, 4343, 4359, 4365, 4377, 4379, 4380, 4390, 4399, 4427,
4432, 4434, 4435, 4444, 4450, 4462, 4472, 4475, 4477, 4485,
4492, 4510, 4518, 4540, 4546, 4558, 4567, 4569, 4582, 4601,
4605, 4614, 4619, 4632, 4634, 4641, 4646, 4656, 4661, 4664,
4669, 4671, 4673, 4674, 4675, 4677, 4685, 4700, 4717, 4730,
4737, 4740, 4746, 4750, 4752, 4753, 4756, 4757, 4758, 4760,
4789, 4794, 4796, 4803, 4821, 4834, 4838, 4842, 4850, 4853,
4857, 4860, 4866, 4867, 4870, 4881, 4884, 4896, 4899, 4907,
4917, 4921, 4927, 4934, 4941, 4946, 4951, 4965, 4986, 4988,
4997, 5014, 5020, 5021, 5022, 5028, 5040, 5058, 5071, 5091,
5104, 5105, 5117, 5122, 5131, 5132, 5147, 5153, 5159, 5162,
5171, 5174, 5176, 5180, 5196, 5203, 5204, 5210, 5221, 5232,
5243, 5246, 5251, 5263, 5265, 5267, 5268, 5278, 5279, 5288,
5307, 5313, 5329, 5341, 5359, 5368, 5370, 5376, 5378, 5386,
5393, 5398, 5399, 5407, 5409, 5420, 5424, 5443, 5447, 5460,
5466, 5476, 5478, 5492, 5503, 5513, 5515, 5521, 5534, 5535,
5539, 5544, 5545, 5559, 5567, 5576, 5581, 5585, 5598, 5613,
5615, 5617, 5623, 5630, 5633, 5648, 5666, 5669, 5671, 5672,
5688, 5696, 5700, 5703, 5711, 5713, 5719, 5750, 5767, 5768,
5775, 5777, 5784, 5788, 5792, 5799, 5801, 5835, 5846, 5869,
5872, 5874, 5887, 5894, 5896, 5922, 5926, 5952, 5965, 5973,
5975, 5976, 5978, 5980, 5985, 5993, 5996, 5999, 6015, 6021,
6029, 6034, 6037, 6038, 6048, 6053, 6064, 6067, 6074, 6081,
6082, 6083, 6084, 6105, 6111, 6118, 6122, 6126, 6133, 6140,
6154, 6157, 6159, 6162, 6164, 6168, 6178, 6180, 6184, 6190,
6199, 6217, 6226, 6233, 6248, 6257, 6269, 6283, 6287, 6291,
6296, 6321, 6330, 6350, 6351, 6354, 6355, 6358, 6359, 6366,
6367, 6371, 6373, 6383, 6385, 6386, 6392, 6397, 6402, 6407,
6413, 6414, 6416, 6429, 6431, 6445, 6454, 6456, 6459, 6471,
6474, 6476, 6482, 6493, 6494, 6495, 6496, 6501, 6502, 6509,
6515, 6529, 6531, 6537, 6556, 6562, 6577, 6585, 6586, 6592,
6616, 6617, 6620, 6624, 6628, 6638, 6650, 6654, 6670, 6674,
6683, 6685, 6688, 6690, 6695, 6697, 6698, 6699, 6716, 6731,
6743, 6746, 6764, 6767, 6775, 6779, 6783, 6797, 6806, 6809,
6810, 6817, 6825, 6828, 6829, 6831, 6844, 6850, 6861, 6863,
6870, 6873, 6874, 6879, 6882, 6894, 6897, 6899, 6901, 6902,
6907, 6909, 6929, 6933, 6940, 6942, 6946, 6949, 6951, 6954,
6963, 6972, 6974, 6976, 6977, 6980, 6986, 6989, 7001, 7012,
7015, 7031, 7033, 7035, 7059, 7063, 7066, 7074, 7076, 7086,
7091, 7095, 7104, 7120, 7142, 7157, 7169, 7181, 7182, 7195,
7200, 7216, 7217, 7225, 7226, 7240, 7252, 7268, 7289, 7293,
7294, 7301, 7317, 7332, 7336, 7370, 7382, 7384, 7385, 7392,
7403, 7407, 7416, 7422, 7437, 7438, 7440, 7446, 7456, 7500,
7507, 7518, 7526, 7529, 7534, 7536, 7543, 7554, 7559, 7568,
7570, 7571, 7578, 7583, 7590, 7594, 7623, 7627, 7633, 7640,
7649, 7655, 7656, 7667, 7672, 7673, 7679, 7681, 7699, 7703,
7730, 7735, 7739, 7744, 7749, 7752, 7754, 7756, 7757, 7759,
7768, 7769, 7771, 7783, 7792, 7793, 7795, 7808, 7809, 7826,
7828, 7840, 7865, 7866, 7875, 7879, 7891, 7893, 7917, 7923,
7934, 7938, 7941, 7949, 7952, 7962, 7964, 7975, 7990, 7998,
8007, 8021, 8022, 8024, 8026, 8027, 8031, 8041, 8043, 8046,
8049, 8051, 8052, 8063, 8065, 8073, 8084, 8089, 8095, 8149,
8150, 8167, 8169, 8171, 8172, 8176, 8180, 8190, 8193, 8202,
8203, 8206, 8214, 8228, 8232, 8259, 8262, 8269, 8281, 8284,
8287, 8296, 8315, 8316, 8323, 8325, 8352, 8356, 8377, 8381,
8389, 8398, 8409, 8411, 8421, 8441, 8449, 8451, 8457, 8458,
8461, 8462, 8465, 8482, 8483, 8489, 8504, 8518, 8520, 8527,
8531, 8547, 8549, 8555, 8560, 8569, 8580, 8582, 8585, 8588,
8589, 8601, 8605, 8619, 8624, 8632, 8638, 8645, 8657, 8662,
8667, 8669, 8673, 8676, 8679, 8681, 8683, 8686, 8689, 8694,
8700, 8704, 8705, 8716, 8725, 8744, 8745, 8751, 8757, 8760,
8762, 8768, 8769, 8778, 8783, 8787, 8790, 8800, 8803, 8806,
8809, 8810, 8811, 8817, 8820, 8835, 8842, 8844, 8849, 8852,
8859, 8870, 8898, 8901, 8910, 8916, 8930, 8936, 8946, 8957,
8961, 8963, 8967, 8969, 8989, 8993, 9000, 9002, 9009, 9011,
9023, 9030, 9035, 9036, 9037, 9048, 9063, 9064, 9075, 9086,
9087, 9104, 9129, 9137, 9138, 9145, 9146, 9147, 9157, 9159,
9164, 9166, 9172, 9186, 9189, 9209, 9210, 9219, 9223, 9227,
9238, 9242, 9245, 9252, 9257, 9271, 9281, 9284, 9290, 9295,
9296, 9298, 9300, 9326, 9328, 9334, 9339, 9343, 9346, 9349,
9351, 9356, 9364, 9366, 9367, 9371, 9375, 9392, 9401, 9403,
9410, 9412, 9425, 9427, 9435, 9436, 9439, 9444, 9449, 9460,
9461, 9467, 9486, 9488, 9493, 9502, 9505, 9514, 9515, 9565,
9572, 9573, 9588, 9593, 9598, 9612, 9617, 9627, 9628, 9632,
9655, 9664, 9694, 9705, 9716, 9721, 9723, 9724, 9741, 9756,
9760, 9766, 9773, 9777, 9788, 9790, 9800, 9806, 9810, 9814,
9820, 9823, 9826, 9841, 9843, 9846, 9859, 9881, 9888, 9892,
9894, 9897, 9902, 9918, 9925, 9926, 9931, 9935, 9954, 9958,
9961, 9964, 9966, 9973, 9981

(based on my list of the first 10000 entries of A081881)

Cheers,
Rainer Rosenthal
r.rosenthal at web.de