Growth of sequences
This is a description of the rates of growth of sequences: bounded, not primitive recursive, and many things in-between like iterated logarithmic, tetrational, and polynomial. Note that not all sequences are included—sequences like A124625 with alternating growth do not fall into the classification below.
- 1 Bounded
- 2 Subpolynomial
- 3 Polynomial
- 4 Sub-exponential but superpolynomial
- 5 Exponential
- 6 Superexponential
- 6.1 Superexponential but sub-doubly-exponential
- 6.2 Doubly-exponential
- 6.3 Elementary, but super-doubly-exponential
- 6.4 Nonelementary, but sub-tetrational
- 6.5 Iterated exponential and super-iterated exponential
- 6.6 Uncomputable
- 7 See also
- 8 Notes
- 9 External links
- 10 Cite this page as
Sequences for which there is an with for all . By definition this includes all sequences with (because these are the digits of a constant, and the digits have a fixed range depending only on the base used), but there are many other examples.
- A000002, A002376, A002828, A008683, A039702, A039703, A039704, A039705, A039706, A051034, A073058, A080066, A084301, A086937, A086965
- Base: A000030, A000455, A001073, A001191, A001355, A002993, A002994, A003076, A003893, A004426, A004427, A004428, A004429, A004430, A007376, A007652, A008564, A008565, A008566, A008567, A008568, A008569, A008570, A008571, A008572, A008573, A008905, A008952, A008960, A008963, A023961, A030132, A030588, A030604, A031007, A031045, A031253, A031269, A031312, A031347, A033988, A033989, A038194, A054054, A054055
- A086966, A086967, A091664, A095048, A104608, A104609, A104610, A104611, A104612, A104613, A104614, A104615, A104616, A104617, A104618, A104619, A105198, A106276, A106277, A106278, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A136690, A136691, A136692, A136693, A136694, A136695, A136696, A136697, A136698, A136699, A136700, A136701, A136702, A144539, A144540, A144541, A144542, A144543, A144544, A158799, A215732, A225224, A227781, A227782, A227783, A227784, A266798, A269591, A269592, A290566, A359141
Sequences for which there is a such that for all (or for all for eventually periodic).
- A000035, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A010880, A010881, A010882, A010883, A010884, A010885, A010886, A010887 A010875, A038605, A131027, A133880, A165472, A175723, A203571, A242627
The native denominator in the generating functions of these is , which makes them traceable through the index to linear recurrences. If 1 is also a root of the numerator of the generating function, this may lower the degree of the denominator polynomial, and this simple recurrence becomes some other product of cyclotomic polynomials in .
Sequences for which there is a such that for all (or for all for eventually constant).
- Eventually constant: A000007, A035613, A021040, A186684
- Constant: A000004, A000012, A007395, A010701, A010709, A010716, A010722, A010727, A010731, A010734, A010692, A010850, A010851, A010852, A010853, A010854, A010855, A010856, A010857, A010858, A010859, A010860, A010861, A010862, A010863, A010864, A010865, A010866, A010867, A010868, A010869, A010870, A010871
These are in the index to linear recurrences in the order 01, (1) category.
Unbounded but sub-iterated logarithmic
Sequences for which there are with with if and otherwise.
Super-iterated logarithmic but sub-doubly logarithmic
Super-doubly logarithmic but sublogarithmic
- A099563 (inverse gamma growth, ~ log n/log log n)
- A000193, A000195, A000523, A003434, A004233, A029837, A038605, A055778, A063511, A070939, A070941, A084320, A102572, A133775, A190796
Superlogarithmic but subpolynomial
- A010553 (exp(sqrt(log n)) or so)
Sequences with polynomial growth: those for which for some and all large
This includes sequences with for some and large enough n. For example, sequences which grow as Listed below are some sequences with for all and large enough n.
- A000006, A000194, A000267, A000196, A000703, A000934, A001650, A001670, A002024, A003057, A003059, A005145
Exponent phi - 1
These are sequences are sublinear yet grow faster than for any : for all and large enough n.
Sequences where each are called arithmetic sequences. For example: A085959, in which = 37. This includes all first-degree polynomials, which are found in the index to linear recurrences under order 02, (-2,1). It also includes non-polynomials of linear growth like A212445.
A strictly increasing sequence has linear growth if and only if its lower density is positive.
All Beatty sequences have linear growth (by definition).
- Beatty sequences: A001950, A003622†, A022844
- 0 < lower density < 1: A000069, A000408, A000452, A001477, A001633, A001637, A001969, A002035, A003052, A003159, A003657, A004709, A005100, A005101, A005114, A005117, A005408, A005835, A005843, A006037, A006285, A007310, A007775, A008365, A008366, A008585, A008905, A010061, A014847, A016777, A016813, A016825, A022342, A023196, A026179, A026225, A030059, A030229, A034683, A047303, A063743, A068782, A078923, A103288, A275616, A016789, A004767, A008586, A008574, A016921, A008588, A008587, A016945, A080653, A007378, A080637, A079905, A336175, A058986, A336909
- Lower density not known to be 1 or smaller than 1, but known positive: A005279
- Density 1: A000027, A000414, A000977, A001690, A002036, A002808, A004169, A004748, A004749, A004750, A004751, A004752, A004753, A004779, A004780, A004781, A007369, A007617, A011539, A183217, A028310*
- A035336, A035337, A036785, A038109, A038838, A039955, A039957, A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045572, A046099, A046101, A046642, A048107, A048109, A049511, A049532, A053224, A053226, A053738, A053754, A054979, A055041, A055047, A056913, A059269, A062289, A064736, A065502, A066498, A066500, A066502, A067259, A067497, A068140, A068403, A068404, A068781, A069059, A072587, A080147, A080148, A081706, A082293, A087248, A088725, A091177, A091178, A092433, A104210, A121707, A121943, A122132, A125134, A129575, A131835, A132141, A133466, A134334, A134860, A143028, A143029, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A145204, A160591, A162643, A162644, A171102, A176995, A187238, A187398, A187516, A190641, A195086, A195087, A197680, A209061, A217394, A217395, A217397, A217398, A217399, A217400, A217401, A217402, A228082, A229300, A229301, A229302, A229303, A229304, A229305, A229306, A229307, A229829, A232705, A232744, A232745, A248150, A252895, A276078, A307958, A308053, A318720, A320628, A320629, A321147, A323024, A325128, A325130, A325424, A326835, A333634, A336064, A336066, A336068, A338539, A338540, A338541, A338542, A340152, A342050, A342051, A358350
* Not a set
† Linear shift of a Beatty sequence
** Proved on the assumption of GRH.
Superlinear but subquadratic
This includes all second-degree polynomials, which are found in the index to linear recurrences under order 03, (3,-3,1).
It also includes non-polynomials of quadratic growth:
Superquadratic but subcubic
Supercubic but subquartic
Sub-exponential but superpolynomial
Sequences for which for all and all large but for which for all and all large
Sequences for which for some and all large
Sequences with rational generating functions with denominator (in lowest terms) which is not a cyclotomic polynomial have exponential growth. Equivalently, linear recurrence relations in which one of the roots of the characteristic polynomial has absolute value greater than 1.
Minimally superincreasing sequences have exponential growth (though some superincreasing sequences have superexponential growth).
Sequences are sorted by base, when possible. Ranges can represent either uncertainty about the value of the base, , or uncertainty about which of these cases hold. See the individual sequence entries for more details.
- 1.1184... to 2: A160675
- 1.3017597 to 1.3017619: A006156
- 2: A002804, A215422
- phi^2 = phi + 1 = 2.61803...: A065885, A007598
- e = 2.718...: A002387, A034386
- 6: A342452
- e^3 = 20.08...: A091463
Superexponential but sub-doubly-exponential
Sequences for which for all and sufficiently large but for all and sufficiently large
- A000312, A000178, A002109, A003266, A050614, A055462, A063439, A113511, A120838, A155877, A161774, A165554, A182856, A200564, A217534, A296653
Sequences for which for some and all large .
- A000278, A000284, A000301, A000371, A002665, A002716, A002813, A002814, A002794, A002795, A006888, A007570, A016088, A039941, A046024, A050548, A052129, A055402, A057194, A064183, A064991, A070177, A087417, A091456, A096580, A100009, A100010, A100011, A100012, A100865, A101361, A103591, A103592, A103593, A103594, A103595, A103596, A103597, A103598, A103599, A103600, A110368, A112961, A114953, A114954, A114957, A115410, A116961, A118017, A123180, A125149, A133026, A135378, A142471, A143684, A153450, A159344, A162634, A162647, A171728, A178981, A185981, A210508, A216151, A333554, A348456
Sequences of the form a(n+1) = a(n)^2 + ...
See the Index page.
- A000058, A000289, A000324, A001042, A001146, A001510, A001543, A001544, A001566, A001696, A001697, A001699, A001999, A002065, A003010, A003095, A003096, A003423, A003487, A004019, A005267, A013589, A014253, A028300, A051179, A056207, A000215, A000283, A007018, A058181, A058182, A062000, A063573, A065035, A067686, A076725, A086851, A092500, A092501, A098152, A099941, A099729, A100523, A100528, A110360, A110383, A113848, A115590, A117805, A118623, A123180, A125046, A126023, A129871, A135927, A143760, A143761, A143762, A143763, A143764, A143765, A143766, A153059, A153060, A153061, A153062, A174864, A186750, A204321, A228931
These are questionable members: a strict definition would not admit them but some might. For example, the last term may be not just squared but multiplied by a constant.
Elementary, but super-doubly-exponential
Sequences for which for all and all large enough but for which for some k, where is the k-times iterated exponential. Informally, these are towers with fixed height (see ELEMENTARY).
Nonelementary, but sub-tetrational
Sequences for which for which for all k and large but for which for all and sufficiently large
No sequences in the OEIS are known to have this behavior, but see Chazelle 2009 for an example of a sequence with behavior
Iterated exponential and super-iterated exponential
Sequences for which for some and all large . Informally, gives the height of the exponential tower.
Primitive recursive, but super-tetrational
Sequences for which for all and all large but for which there is a with for some
Not primitive recursive
Sequences for which for all and large enough
Sequences (like Busy Beavers) which are not computable. In particular, given some formal system, there is an index beyond which it is not possible to compute terms of the sequence, even if they are in principle finite.
- Mohammad R. Salavatipour, Lecture 5: Growth of Functions, CMPUT 204: Algorithms I, Department of Computing Science, University of Alberta.
Cite this page as
Charles R Greathouse IV, Growth_of_sequences. — From the On-Line Encyclopedia of Integer Sequences® (OEIS®) wiki.Available at https://oeis.org/wiki/Growth_of_sequences