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Template:Tetration
From OeisWiki
(Redirected from Template:^^)
The following documentation is located at Template:Tetration/doc. [<Edit> Template:Tetration/doc]
[⧼Purge⧽ Template:Tetration]
The {{tetration}} mathematical function template returns the tetration [⧼Purge⧽ Template:Tetration]
b ↑↑ h |
b |
h |
otherwise returns an error message.
Beware: Calculated with the MediaWiki #expr parser function extension, which has a significand precision of 53 bits (52 explicitly stored) (approximately 16 decimal digits, 53 log10 2 ≈ 15.955).[1]
Contents
Usage
- {{tetration|base|height}}
or
- {{^^|base|height}}
Valid arguments
If neither the base nor the height is given, the empty product 1 is returned.
- First argument: base of power tower, as a nonnegative real number;
- Second argument: height of power tower, as a nonnegative integer in [0..32] (defaults to 1).
Examples
Examples with valid input
Returnsb ↑↑ h |
Code Result Code Result {{^^}} 1 {{^^|2}} 2 {{^^|0|0}} 1 {{^^|-1|0}} 1 {{^^|2|0}} 1 {{^^|2|1}} 2 {{^^|2|2}} 4 {{^^|2|3}} 16 {{^^|2|4}} 65536 {{^^|2|5}} INF {{^^|3|3}} 7625597484987 {{^^|3|4}} INF
b |
b = − 1 |
- {{#expr: (-1)^(-1) }} = -1
- {{#expr: (-1.1)^(-1.1) }} = NAN
- {{#expr: 1 / ( (-1.1)^(1.1) ) }} = NAN
-3 | -0.037037037037037 | NAN | NAN | NAN | NAN | NAN | |
-2.5 | NAN | NAN | NAN | NAN | NAN | NAN | |
-2 | 0.25 | NAN | NAN | NAN | NAN | NAN | |
-1.5 | NAN | NAN | NAN | NAN | NAN | NAN | |
-1 | -1 | -1 | -1 | -1 | -1 | -1 | |
-0.5 | NAN | NAN | NAN | NAN | NAN | NAN | |
0 | 1 | 0 | 1 | 0 | 1 | 1 | undefined |
0.1 | 0.79432823472428 | 0.16057272043212 | 0.69091922875998 | 0.20374209674284 | 0.62554405820608 | 0.4229202149354 | 0.39901297826025 |
0.2 | 0.7247796636777 | 0.31145890709838 | 0.60575856902197 | 0.37721845362499 | 0.5449235984186 | 0.46970963528893 | 0.46962192293561 |
0.3 | 0.69684530193595 | 0.43214988609837 | 0.59434417379411 | 0.48891128772632 | 0.55508396180265 | 0.52895696102551 | 0.52895644844295 |
0.4 | 0.69314484315515 | 0.5298707364733 | 0.6153797974711 | 0.56900457469329 | 0.59370446435584 | 0.58504317471451 | 0.58504317197440 |
0.5 | 0.70710678118655 | 0.61254732653607 | 0.65404086004207 | 0.63549784581337 | 0.64371864172287 | 0.64118574451391 | 0.64118574450499 |
0.6 | 0.73602192281783 | 0.68661684945978 | 0.70416577097103 | 0.69788152946118 | 0.700125434673 | 0.69953450919474 | 0.69953450919473 |
0.7 | 0.77905591267045 | 0.75739548555921 | 0.76326958144204 | 0.76167209672267 | 0.76210620802468 | 0.76201343081072 | 0.76201343081072 |
0.8 | 0.83651164207302 | 0.82972398762427 | 0.83098165747098 | 0.83074848273819 | 0.83079170890087 | 0.83078494866324 | 0.83078494866324 |
0.9 | 0.90953257608296 | 0.90861953913804 | 0.90870695076419 | 0.90869858185342 | 0.90869938310125 | 0.90869931309174 | 0.90869931309174 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1.1 | 1.1105342410546 | 1.1116498000258 | 1.111768001503 | 1.1117805265341 | 1.1117818537373 | 1.1117820110418 | 1.11178201104184 |
1.2 | 1.244564747204 | 1.2547181707692 | 1.2570430412015 | 1.2575759820011 | 1.2576981823156 | 1.2577345413765 | 1.25773454137653 |
1.3 | 1.4064566732379 | 1.446293346286 | 1.4614888698918 | 1.4673271088311 | 1.469576402415 | 1.47098896009 | 1.47098896009015 |
1.4 | 1.6016928982022 | 1.7141634748254 | 1.780276017527 | 1.8203220817394 | 1.8450158019786 | 1.886662556747 | 1.88666330624633 |
1.5 | 1.8371173070874 | 2.1062033521489 | 2.3490053186119 | 2.5920257049075 | 2.8604414974606 | INF | |
1.6 | 2.1212505710976 | 2.7101269178649 | 3.5743004801538 | 5.3652029167449 | 12.449305553378 | INF | |
1.7 | 2.4646948994849 | 3.698164974975 | 7.1160358861268 | 43.639804309351 | 11395826597.324 | INF |
0 | 1 | 0 | 1 | 0 | 1 | undefined | |
0.1 | 0.79432823472428 | 0.16057272043212 | 0.69091922875998 | 0.20374209674284 | 0.62554405820608 | 0.39901297826025 | |
0.2 | 0.7247796636777 | 0.31145890709838 | 0.60575856902197 | 0.37721845362499 | 0.5449235984186 | 0.46962192293561 | |
0.3 | 0.69684530193595 | 0.43214988609837 | 0.59434417379411 | 0.48891128772632 | 0.55508396180265 | 0.52895644844295 | |
0.4 | 0.69314484315515 | 0.5298707364733 | 0.6153797974711 | 0.56900457469329 | 0.59370446435584 | 0.58504317197440 | |
0.5 | 0.70710678118655 | 0.61254732653607 | 0.65404086004207 | 0.63549784581337 | 0.64371864172287 | 0.64118574450499 | |
0.6 | 0.73602192281783 | 0.68661684945978 | 0.70416577097103 | 0.69788152946118 | 0.700125434673 | 0.69953450919473 | |
0.7 | 0.77905591267045 | 0.75739548555921 | 0.76326958144204 | 0.76167209672267 | 0.76210620802468 | 0.76201343081072 | |
0.8 | 0.83651164207302 | 0.82972398762427 | 0.83098165747098 | 0.83074848273819 | 0.83079170890087 | 0.83078494866324 | |
0.9 | 0.90953257608296 | 0.90861953913804 | 0.90870695076419 | 0.90869858185342 | 0.90869938310125 | 0.90869931309174 | |
1 | 1 | 1 | 1 | 1 | 1 | 1 | |
1.1 | 1.1105342410546 | 1.1116498000258 | 1.111768001503 | 1.1117805265341 | 1.1117818537373 | 1.11178201104184 | |
1.2 | 1.244564747204 | 1.2547181707692 | 1.2570430412015 | 1.2575759820011 | 1.2576981823156 | 1.25773454137653 | |
1.3 | 1.4064566732379 | 1.446293346286 | 1.4614888698918 | 1.4673271088311 | 1.469576402415 | 1.47098896009015 | |
1.4 | 1.6016928982022 | 1.7141634748254 | 1.780276017527 | 1.8203220817394 | 1.8450158019786 | 1.88666330624633 | |
1.5 | 1.8371173070874 | 2.1062033521489 | 2.3490053186119 | 2.5920257049075 | 2.8604414974606 | ||
1.6 | 2.1212505710976 | 2.7101269178649 | 3.5743004801538 | 5.3652029167449 | 12.449305553378 | ||
1.7 | 2.4646948994849 | 3.698164974975 | 7.1160358861268 | 43.639804309351 | 11395826597.324 |
Examples with invalid input
Returns an error message.
Code Result {{^^|a|0}} Expression error: Unrecognized word "a". {{^^|a|1}} Expression error: Unrecognized word "a". {{^^|2i|3}} Expression error: Unrecognized word "i". {{^^|2i+1|3}} Expression error: Unrecognized word "i". {{^^|2|33}} Tetration error: Second argument (height) must be a nonnegative integer up to 32 {{^^|2|-1}} Tetration error: Second argument (height) must be a nonnegative integer up to 32 {{^^|2|0.5}} Tetration error: Second argument (height) must be a nonnegative integer
Test
The following few results for the tetration function (using {{^^|b|h}} template) fully agree (all 14 decimal digits) with Mathematica's Power @@ Table[b, {h}].
h | {{^^|(1/e)^e|h}} | Power @@ Table[(1/e)^e, {h}] |
---|---|---|
0 | 1 | 1 |
1 | 0.065988035845313 | 0.0659880358453125 |
2 | 0.83579318845359 | 0.835793188453588 |
3 | 0.10311386287497 | 0.103113862874971 |
4 | 0.75556267703185 | |
5 | 0.12824249755517 | |
6 | 0.70567587269525 | |
7 | 0.14686728817137 | |
8 | 0.6708386179955 | 0.670838617995497 |
9 | 0.16145506261086 | |
10 | 0.644757912042 | |
11 | 0.17331687438423 | |
12 | 0.62430009159136 | 0.624300091591359 |
13 | 0.18322807430165 | |
14 | 0.6077051016648 | |
15 | 0.19168272746053 | 0.191682727460533 |
16 | 0.59389800988891 | 0.593898009888908 |
17 | 0.19901358938169 | |
18 | 0.58218033596699 | |
19 | 0.20545459299035 | 0.205454592990346 |
20 | 0.57207596716079 | |
21 | 0.21117592903209 | |
22 | 0.56324775217168 | |
23 | 0.21630493523809 | |
24 | 0.5554493934993 | 0.555449393499302 |
25 | 0.22093914155261 | |
26 | 0.54849624036848 | |
27 | 0.22515474374947 | 0.225154743749465 |
28 | 0.54224679029599 | |
29 | 0.2290122927811 | |
30 | 0.53659054896353 | |
31 | 0.53659054896353 | |
32 | 0.53659054896353 | 0.531439817592522 |