A121854 -id:A121854 - cp's OEIS frontend

A121381 a(n) = ceiling(n*Pi).

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

Offset: 0

Mohammad K. Azarian, Sep 06 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Essentially the same as A004084. Cf. A022844, A121854, A121855.

[Ceiling(n*Pi): n in [0..50]]; // G. C. Greubel, Oct 28 2017

Table[Ceiling[n Pi], {n, 0, 80}] (* Vincenzo Librandi, Feb 22 2013 *)

for(n=0,50, print1(ceil(n*Pi), ", ")) \\ G. C. Greubel, Oct 28 2017

A121905 a(n) = ceiling(e^(n*Pi)).

Original entry on oeis.org

1, 24, 536, 12392, 286752, 6635624, 153552936, 3553321281, 82226315586, 1902773895293, 44031505860633, 1018919543279305, 23578503968558227, 545622913077172101, 12626092124920479898, 292176517015939695008

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..699 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A062360 (floor).

Cf. A121854, A121855, A121856, A121857, A121904.

C := ComplexField(); [Ceiling(Exp(Pi(C)*n)): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

Ceiling[E^(Pi Range[0, 20])] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil(exp(Pi*n)), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121915 a(n) = floor((Pi+e)^(n*Pi)).

Original entry on oeis.org

1, 258, 66801, 17265408, 4462406595, 1153350806021, 298094324981778, 77045272021641916, 19913072619720776032, 5146720243221262934093, 1330218081751512472685763, 343807329988307215923432746, 88860226586342124489251555256, 22966758356328845813340839281381

Offset: 0

Mohammad K. Azarian, Sep 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..399 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A121854, A121855, A121856, A121857, A121916, A121917.

C := ComplexField(); [Floor((Pi(C)+Exp(1))^(n*Pi(C))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

With[{c=\[Pi]+E}, Floor[c^(\[Pi] Range[0, 20])]] (* Harvey P. Dale, Mar 20 2011 *)

PARI

for(n=0,50, print1(floor((Pi+exp(1))^(n*Pi)), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset changed to 0 by Georg Fischer, Sep 02 2022

A331859 The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.

Original entry on oeis.org

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

Offset: 1

Peter Kagey, Jan 29 2020

nonn

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----------------------------------------------------------
  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

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)

Cf. A121854, A331903, A331904.

Table[Ceiling[Pi/ArcTan[Sqrt[1/n]] - 1], {n, 1, 100}]

a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1.

A121900 a(n) = ceiling((Pi - e)*sqrt(n)).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A073244 (Pi-e), A121854, A121855, A121856, A121857.

C := ComplexField(); [Ceiling((Pi(C) - Exp(1))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017

Mathematica

Table[Ceiling[(Pi - E) Sqrt[n]], {n, 0, 110}] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil((Pi - exp(1))*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121904 a(n) = floor(Pi^(n*e)).

Original entry on oeis.org

1, 22, 504, 11328, 254433, 5714356, 128339632, 2882400037, 64736277048, 1453922256329, 32653869265129, 733378399940296, 16471061151498380, 369926160190271626, 8308229975861003525, 186595847388277259847, 4190785566084546949287, 94121513992523815815369

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..699 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A121854, A121855, A121856, A121857, A121903.

C := ComplexField(); [Floor(Pi(C)^(n*Exp(1))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

Floor[Pi^(E Range[0, 20])] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(floor(Pi^(n*exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121917 a(n) = ceiling((Pi+e)^(n*e)).

Original entry on oeis.org

1, 123, 14952, 1828145, 223535960, 27332807666, 3342112728282, 408656059975458, 49968325108097956, 6109865382293662598, 747082374864324679925, 91349324397617876090444, 11169717488538903806777418, 1365774619533204572560235118

Offset: 0

Mohammad K. Azarian, Sep 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..399 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A121854, A121855, A121856, A121857, A121915, A121918.

C := ComplexField(); [Ceiling((Pi(C)+Exp(1))^(n*Exp(1))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

Ceiling[(Pi + E)^(E (Range[0, 20]))] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil((Pi+exp(1))^(n*exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121929 a(n) = ceiling(n*(e^Pi + Pi^e)).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Sep 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A019314, A121854, A121855, A121856, A121857, A121928, A121930.

C := ComplexField(); [Ceiling(n*(Pi(C)^Exp(1) + Exp(1)^Pi(C))): n in [0..50]]; // G. C. Greubel, Nov 06 2017

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 *)

PARI

for(n=0,50, print1(ceil(n*(Pi^exp(1)+exp(Pi))), ", ")) \\ G. C. Greubel, Nov 06 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

A121930 a(n) = floor(n*(e^Pi - Pi^e)).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Sep 02 2006

nonn

Beatty sequence of A063504. - R. J. Mathar, Aug 11 2012

Ivan Panchenko, Table of n, a(n) for n = 0..10000

Cf. A121854, A121855, A121856, A121857.

C := ComplexField(); [Floor(n*(Exp(1)^Pi(C) - Pi(C)^Exp(1) )): n in [0..50]]; // G. C. Greubel, Nov 06 2017

Mathematica

With[{c=E^Pi-Pi^E},Floor[c*Range[0,80]]] (* Harvey P. Dale, Jan 06 2012 *)

PARI

for(n=0,50, print1(floor(n*(exp(Pi) - Pi^exp(1))), ", ")) \\ G. C. Greubel, Nov 06 2017

A121899 a(n) = ceiling((Pi + e)*sqrt(n)).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Sep 01 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..999 [Offset shifted by _Georg Fischer_, Sep 02 2022]

Cf. A059742 (Pi+e), A121854, A121855, A121856, A121857.

C := ComplexField(); [Ceiling((Exp(1) + Pi(C))*Sqrt(n)): n in [0..50]]; // G. C. Greubel, Oct 28 2017

Mathematica

Table[Ceiling[(Pi + E) Sqrt[n]], {n, 0, 70}] (* Vincenzo Librandi, Feb 21 2013 *)

PARI

for(n=0,50, print1(ceil((exp(1)+Pi)*sqrt(n)), ", ")) \\ G. C. Greubel, Oct 28 2017

Extensions

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Offset changed to 0 by Georg Fischer, Sep 02 2022

cp's OEIS Frontend

A121381 a(n) = ceiling(n*Pi).

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Magma

Mathematica

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A121905 a(n) = ceiling(e^(n*Pi)).

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Magma

Mathematica

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A121915 a(n) = floor((Pi+e)^(n*Pi)).

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Magma

Mathematica

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A331859 The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.

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Mathematica

Formula

A121900 a(n) = ceiling((Pi - e)*sqrt(n)).

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Magma

Mathematica

PARI

Extensions

A121904 a(n) = floor(Pi^(n*e)).

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Magma

Mathematica

PARI

Extensions

A121917 a(n) = ceiling((Pi+e)^(n*e)).

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Magma

Mathematica

PARI

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A121929 a(n) = ceiling(n*(e^Pi + Pi^e)).

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