794:
31:
1103:
169:
Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.
101:
Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a
985:
258:. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model (Hamkins et al. 2013).
975:
628:
1068:
185:. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a
909:
533:
112:
87:
919:
914:
80:) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description (sequence
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if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.
234:; for others, only some integer sequences are (Hamkins et al. 2013). There is no systematic way to define in
76:
by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the
1055:
877:
383:
717:
664:
538:
924:
669:
388:
553:
Hamkins, Joel David; Linetsky, David; Reitz, Jonas (2013), "Pointwise
Definable Models of Set Theory",
1035:
872:
641:
347:
317:
1015:
882:
515:
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215:, there are definable integer sequences that are not computable, such as sequences that encode the
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203:) in the language of set theory, with one free variable and no parameters, which is true in
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1005:
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17:
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contains all integer sequences, then the set of integer sequences definable in
246:. Similarly, the map from the set of formulas that define integer sequences in
393:
39:
689:
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151:
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163:
90:). The sequence 0, 3, 8, 15, ... is formed according to the formula
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29:
610:
107:
82:
250:
to the integer sequences they define is not definable in
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1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
976:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
297:
Integer sequences that have their own name include:
166:), and so not all integer sequences are computable.
1054:
998:
933:
902:
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801:
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150:> 0. The set of computable integer sequences is
238:itself the set of sequences definable relative to
115:), even though we do not have a formula for the
211:for all other integer sequences. In each such
622:
242:and that set may not even exist in some such
8:
1069:Hypergeometric function of a matrix argument
285:A sequence of positive integers is called a
925:1 + 1/2 + 1/3 + ... (Riemann zeta function)
899:
831:
654:
629:
615:
607:
981:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
568:
534:On-Line Encyclopedia of Integer Sequences
133:if there exists an algorithm that, given
603:. Articles are freely available online.
207:for that integer sequence and false in
226:of ZFC, every sequence of integers in
154:. The set of all integer sequences is
64:An integer sequence may be specified
7:
946:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
273:and be countable and countable in
123:Computable and definable sequences
25:
1064:Generalized hypergeometric series
98:th term: an explicit definition.
1102:
1101:
1074:Lauricella hypergeometric series
792:
1084:Riemann's differential equation
1:
1079:Modular hypergeometric series
920:1/4 + 1/16 + 1/64 + 1/256 + ⋯
470:Regular paperfolding sequence
195:if there exists some formula
94: − 1 for the
27:Ordered list of whole numbers
601:Journal of Integer Sequences
68:by giving a formula for its
1089:Theta hypergeometric series
529:Constant-recursive sequence
222:For some transitive models
57:(i.e., an ordered list) of
1149:
971:Infinite arithmetic series
915:1/2 + 1/4 + 1/8 + 1/16 + ⋯
910:1/2 − 1/4 + 1/8 − 1/16 + ⋯
1097:
790:
556:Journal of Symbolic Logic
230:is definable relative to
384:Highly composite numbers
802:Properties of sequences
127:An integer sequence is
665:Arithmetic progression
539:List of OEIS sequences
475:Rudin–Shapiro sequence
389:Highly totient numbers
42:
1056:Hypergeometric series
670:Geometric progression
318:Binomial coefficients
254:and may not exist in
190:sequence relative to
164:that of the continuum
33:
1133:Arithmetic functions
1036:Trigonometric series
828:Properties of series
675:Harmonic progression
491:Superperfect numbers
399:Hyperperfect numbers
348:Even and odd numbers
219:of computable sets.
1016:Formal power series
579:10.2178/jsl.7801090
516:Wolstenholme number
501:Thue–Morse sequence
480:Semiperfect numbers
308:Baum–Sweet sequence
119:th perfect number.
18:Consecutive numbers
814:Monotonic function
733:Fibonacci sequence
496:Triangular numbers
465:Recamán's sequence
409:Kolakoski sequence
323:Carmichael numbers
281:Complete sequences
78:Fibonacci sequence
43:
36:Fibonacci sequence
1128:Integer sequences
1115:
1114:
1046:Generating series
994:
993:
966:1 − 2 + 4 − 8 + ⋯
961:1 + 2 + 4 + 8 + ⋯
956:1 − 2 + 3 − 4 + ⋯
951:1 + 2 + 3 + 4 + ⋯
941:1 + 1 + 1 + 1 + ⋯
891:
890:
819:Periodic sequence
788:
787:
773:Triangular number
763:Pentagonal number
743:Heptagonal number
728:Complete sequence
650:Integer sequences
449:Practical numbers
439:Partition numbers
359:Fibonacci numbers
338:Deficient numbers
333:Composite numbers
287:complete sequence
38:on a building in
34:Beginning of the
16:(Redirected from
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1105:
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1031:Dirichlet series
900:
832:
796:
768:Polygonal number
748:Hexagonal number
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655:
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404:Juggler sequence
369:Figurate numbers
303:Abundant numbers
179:transitive model
173:Suppose the set
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51:integer sequence
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999:Kinds of series
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896:Explicit series
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809:Cauchy sequence
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738:Figurate number
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709:
700:Powers of three
644:
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552:
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444:Perfect numbers
434:Padovan numbers
429:Natural numbers
424:Motzkin numbers
374:Golomb sequence
328:Catalan numbers
295:
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81:
28:
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1041:Fourier series
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595:External links
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563:(1): 139–156,
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364:Fibonacci word
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269:will exist in
183:ZFC set theory
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103:perfect number
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1006:Taylor series
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695:Powers of two
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680:Square number
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511:Weird numbers
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454:Prime numbers
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419:Lucas numbers
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414:Lucky numbers
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379:Happy numbers
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343:Euler numbers
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1011:Power series
753:Lucas number
705:Powers of 10
685:Cubic number
649:
560:
554:
506:Ulam numbers
313:Bell numbers
296:
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274:
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217:Turing jumps
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105:, (sequence
100:
95:
91:
73:
72:th term, or
69:
65:
63:
50:
44:
878:Conditional
866:Convergence
857:Telescoping
842:Alternating
758:Pell number
459:Pseudoprime
394:Home primes
160:cardinality
156:uncountable
47:mathematics
1122:Categories
903:Convergent
847:Convergent
547:References
146:, for all
130:computable
74:implicitly
66:explicitly
40:Gothenburg
934:Divergent
852:Divergent
714:Advanced
690:Factorial
638:Sequences
570:1105.4597
485:Semiprime
353:Factorial
188:definable
162:equal to
152:countable
1107:Category
873:Absolute
587:43689192
523:See also
293:Examples
59:integers
55:sequence
883:Uniform
487:numbers
461:numbers
355:numbers
111:in the
108:A000396
86:in the
83:A000045
835:Series
642:series
585:
158:(with
778:array
658:Basic
583:S2CID
565:arXiv
177:is a
53:is a
49:, an
718:list
640:and
113:OEIS
88:OEIS
575:doi
261:If
181:of
45:In
1124::
581:,
573:,
561:78
559:,
277:.
61:.
720:)
716:(
630:e
623:t
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275:M
271:M
267:M
263:M
256:M
252:M
248:M
244:M
240:M
236:M
232:M
228:M
224:M
213:M
209:M
205:M
201:x
199:(
197:P
192:M
175:M
148:n
143:n
139:a
135:n
117:n
96:n
92:n
70:n
20:)
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