[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a121854 -id:a121854
Displaying 1-10 of 18 results found.
page 1 2
     Sort: relevance | references | number | modified | created      Format: long | short | data
A121381
a(n) = ceiling(n*Pi).
+10
5
0, 4, 7, 10, 13, 16, 19, 22, 26, 29, 32, 35, 38, 41, 44, 48, 51, 54, 57, 60, 63, 66, 70, 73, 76, 79, 82, 85, 88, 92, 95, 98, 101, 104, 107, 110, 114, 117, 120, 123, 126, 129, 132, 136, 139, 142, 145, 148, 151, 154, 158, 161, 164, 167, 170, 173, 176, 180, 183, 186
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
MATHEMATICA
Table[Ceiling[n Pi], {n, 0, 80}] (* Vincenzo Librandi, Feb 22 2013 *)
PROG
(PARI) for(n=0, 50, print1(ceil(n*Pi), ", ")) \\ G. C. Greubel, Oct 28 2017
(Magma) [Ceiling(n*Pi): n in [0..50]]; // G. C. Greubel, Oct 28 2017
CROSSREFS
Essentially the same as A004084. Cf. A022844, A121854, A121855.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 06 2006
STATUS
approved
A121905
a(n) = ceiling(e^(n*Pi)).
+10
4
1, 24, 536, 12392, 286752, 6635624, 153552936, 3553321281, 82226315586, 1902773895293, 44031505860633, 1018919543279305, 23578503968558227, 545622913077172101, 12626092124920479898, 292176517015939695008
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..699 [Offset shifted by Georg Fischer, Sep 02 2022]
MATHEMATICA
Ceiling[E^(Pi Range[0, 20])] (* Vincenzo Librandi, Feb 21 2013 *)
PROG
(PARI) for(n=0, 50, print1(ceil(exp(Pi*n)), ", ")) \\ G. C. Greubel, Nov 06 2017
(Magma) C<i> := ComplexField(); [Ceiling(Exp(Pi(C)*n)): n in [0..50]]; // G. C. Greubel, Nov 06 2017
CROSSREFS
Cf. A062360 (floor).
Cf. A121854, A121855, A121856, A121857, A121904.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 01 2006
EXTENSIONS
Offset changed to 0 by Georg Fischer, Sep 02 2022
STATUS
approved
A121915
a(n) = floor((Pi+e)^(n*Pi)).
+10
4
1, 258, 66801, 17265408, 4462406595, 1153350806021, 298094324981778, 77045272021641916, 19913072619720776032, 5146720243221262934093, 1330218081751512472685763, 343807329988307215923432746, 88860226586342124489251555256, 22966758356328845813340839281381
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..399 [Offset shifted by Georg Fischer, Sep 02 2022]
MATHEMATICA
With[{c=\[Pi]+E}, Floor[c^(\[Pi] Range[0, 20])]] (* Harvey P. Dale, Mar 20 2011 *)
PROG
(PARI) for(n=0, 50, print1(floor((Pi+exp(1))^(n*Pi)), ", ")) \\ G. C. Greubel, Nov 06 2017
(Magma) C<i> := ComplexField(); [Floor((Pi(C)+Exp(1))^(n*Pi(C))): n in [0..50]]; // G. C. Greubel, Nov 06 2017
CROSSREFS
Cf. A121854, A121855, A121856, A121857, A121916, A121917.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 02 2006
EXTENSIONS
Offset changed to 0 by Georg Fischer, Sep 02 2022
STATUS
approved
A331859
The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.
+10
4
3, 5, 5, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Suppose there is a block A of mass n sliding left toward a stationary block B of mass 1, to the left of which is a wall. Assuming the sliding is frictionless and the collisions are elastic, a(n) is the number of collisions between A and B plus the number of collisions between B and the wall. (See Grant Sanderson links for animated examples.)
a(100^n) = A011545(n).
Since arctan(sqrt(1/n)) is approximately sqrt(1/n) for large values of n, a(n) = A121854(n) for most values of n.
Conjecture: The values of n for which a(n) != A121854(n) is a subset of A331903.
Initial phase:
\ | ______________________
\ \| | |
\ | | |
\ \| | |
\ | | |
\ \| <=== | Block A |
\ | _________ | |
\ \| | | | M = n |
\ | | Block B | | |
\ \| | | | | |
\ | | M = 1 | | |
\ \| |_________| |______________________|
\ L----------------------------------------------------------
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \|
\ | ______________________
\ \| | |
\ | | |
\ \| | |
\ | | |
\ \| <=== | |
\ | _________ | |
\ \| | || |
\ | | || |
\ \| | || |
\ | | || |
\ \| |_________||______________________|
\ L----------------------------------------------------------
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \|
\ | ______________________
\ \| | |
\ | | |
\ \| | |
\ | | |
\ \| <== | |
\ | _________ | |
\ \| | | | |
\ | | | | |
\ \|<===>| | | |
\ | | | | |
\ \| |_________| |______________________|
\ L----------------------------------------------------------
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
LINKS
Peter Kagey, Table of n, a(n) for n = 1..10000
Code Golf Stack Exchange, Elastic collisions between blocks
Grant Sanderson, How Pi Connects Colliding Blocks to a Quantum Search Algorithm, Quanta Magazine (2020).
Grant Sanderson, The most unexpected answer to a counting puzzle, 3Blue1Brown video (2019)
Grant Sanderson, Why do colliding blocks compute pi?, 3Blue1Brown video (2019)
FORMULA
a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1.
MATHEMATICA
Table[Ceiling[Pi/ArcTan[Sqrt[1/n]] - 1], {n, 1, 100}]
CROSSREFS
Cf. A121854, A331903, A331904.
KEYWORD
nonn
AUTHOR
Peter Kagey, Jan 29 2020
STATUS
approved
A121900
a(n) = ceiling((Pi - e)*sqrt(n)).
+10
3
0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by Georg Fischer, Sep 02 2022]
MATHEMATICA
Table[Ceiling[(Pi - E) Sqrt[n]], {n, 0, 110}] (* Vincenzo Librandi, Feb 21 2013 *)
PROG
(PARI) for(n=0, 50, print1(ceil((Pi - exp(1))*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017
(Magma) C<i> := ComplexField(); [Ceiling((Pi(C) - Exp(1))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017
CROSSREFS
Cf. A073244 (Pi-e), A121854, A121855, A121856, A121857.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 01 2006
EXTENSIONS
Offset corrected by Mohammad K. Azarian, Nov 20 2008
Offset changed to 0 by Georg Fischer, Sep 02 2022
STATUS
approved
A121904
a(n) = floor(Pi^(n*e)).
+10
3
1, 22, 504, 11328, 254433, 5714356, 128339632, 2882400037, 64736277048, 1453922256329, 32653869265129, 733378399940296, 16471061151498380, 369926160190271626, 8308229975861003525, 186595847388277259847, 4190785566084546949287, 94121513992523815815369
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..699 [Offset shifted by Georg Fischer, Sep 02 2022]
MATHEMATICA
Floor[Pi^(E Range[0, 20])] (* Vincenzo Librandi, Feb 21 2013 *)
PROG
(PARI) for(n=0, 50, print1(floor(Pi^(n*exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017
(Magma) C<i> := ComplexField(); [Floor(Pi(C)^(n*Exp(1))): n in [0..50]]; // G. C. Greubel, Nov 06 2017
CROSSREFS
Cf. A121854, A121855, A121856, A121857, A121903.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 01 2006
EXTENSIONS
Offset changed to 0 by Georg Fischer, Sep 02 2022
STATUS
approved
A121917
a(n) = ceiling((Pi+e)^(n*e)).
+10
3
1, 123, 14952, 1828145, 223535960, 27332807666, 3342112728282, 408656059975458, 49968325108097956, 6109865382293662598, 747082374864324679925, 91349324397617876090444, 11169717488538903806777418, 1365774619533204572560235118
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..399 [Offset shifted by Georg Fischer, Sep 02 2022]
MATHEMATICA
Ceiling[(Pi + E)^(E (Range[0, 20]))] (* Vincenzo Librandi, Feb 21 2013 *)
PROG
(PARI) for(n=0, 50, print1(ceil((Pi+exp(1))^(n*exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017
(Magma) C<i> := ComplexField(); [Ceiling((Pi(C)+Exp(1))^(n*Exp(1))): n in [0..50]]; // G. C. Greubel, Nov 06 2017
CROSSREFS
Cf. A121854, A121855, A121856, A121857, A121915, A121918.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 02 2006
EXTENSIONS
Offset changed to 0 by Georg Fischer, Sep 02 2022
STATUS
approved
A121929
a(n) = ceiling(n*(e^Pi + Pi^e)).
+10
3
0, 46, 92, 137, 183, 228, 274, 320, 365, 411, 456, 502, 548, 593, 639, 684, 730, 776, 821, 867, 912, 958, 1004, 1049, 1095, 1140, 1186, 1232, 1277, 1323, 1368, 1414, 1460, 1505, 1551, 1596, 1642, 1688, 1733, 1779, 1824, 1870, 1916, 1961, 2007, 2052, 2098
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by Georg Fischer, Sep 02 2022]
MAPLE
A121929:=n->ceil((n-1)*(exp(1)^Pi+Pi^exp(1))): seq(A121929(n), n=1..100); # Wesley Ivan Hurt, Jan 21 2017
MATHEMATICA
With[{a = E^Pi + Pi^E}, Ceiling[a Range[0, 80]]] (* Vincenzo Librandi, Feb 21 2013 *)
PROG
(PARI) for(n=0, 50, print1(ceil(n*(Pi^exp(1)+exp(Pi))), ", ")) \\ G. C. Greubel, Nov 06 2017
(Magma) C<i> := ComplexField(); [Ceiling(n*(Pi(C)^Exp(1) + Exp(1)^Pi(C))): n in [0..50]]; // G. C. Greubel, Nov 06 2017
CROSSREFS
Cf. A019314, A121854, A121855, A121856, A121857, A121928, A121930.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 02 2006
EXTENSIONS
Offset corrected by Mohammad K. Azarian, Nov 20 2008
Offset changed to 0 by Georg Fischer, Sep 02 2022
STATUS
approved
A121930
a(n) = floor(n*(e^Pi - Pi^e)).
+10
3
0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 47, 47, 48, 49, 49
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Beatty sequence of A063504. - R. J. Mathar, Aug 11 2012
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..10000
MATHEMATICA
With[{c=E^Pi-Pi^E}, Floor[c*Range[0, 80]]] (* Harvey P. Dale, Jan 06 2012 *)
PROG
(PARI) for(n=0, 50, print1(floor(n*(exp(Pi) - Pi^exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017
(Magma) C<i> := ComplexField(); [Floor(n*(Exp(1)^Pi(C) - Pi(C)^Exp(1) )): n in [0..50]]; // G. C. Greubel, Nov 06 2017
CROSSREFS
Cf. A121854, A121855, A121856, A121857.
KEYWORD
nonn
AUTHOR
Mohammad K. Azarian, Sep 02 2006
STATUS
approved
A121899
a(n) = ceiling((Pi + e)*sqrt(n)).
+10
2
0, 6, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 42, 43, 43, 44, 44, 44, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by Georg Fischer, Sep 02 2022]
MATHEMATICA
Table[Ceiling[(Pi + E) Sqrt[n]], {n, 0, 70}] (* Vincenzo Librandi, Feb 21 2013 *)
PROG
(PARI) for(n=0, 50, print1(ceil((exp(1)+Pi)*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017
(Magma) C<i> := ComplexField(); [Ceiling((Exp(1) + Pi(C))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017
CROSSREFS
Cf. A059742 (Pi+e), A121854, A121855, A121856, A121857.
KEYWORD
nonn,easy
AUTHOR
Mohammad K. Azarian, Sep 01 2006
EXTENSIONS
Offset corrected by Mohammad K. Azarian, Nov 20 2008
Offset changed to 0 by Georg Fischer, Sep 02 2022
STATUS
approved
page 1 2

Search completed in 0.009 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified December 14 23:20 EST 2024. Contains 378766 sequences.
License Agreements, Terms of Use, Privacy Policy