cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A000501 a(n) = floor(cosh(n)).

Original entry on oeis.org

1, 1, 3, 10, 27, 74, 201, 548, 1490, 4051, 11013, 29937, 81377, 221206, 601302, 1634508, 4443055, 12077476, 32829984, 89241150, 242582597, 659407867, 1792456423, 4872401723, 13244561064, 36002449668, 97864804714, 266024120300, 723128532145, 1965667148572
Offset: 0

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Author

Keywords

Crossrefs

Cf. A000471.

Programs

  • Maple
    f := n->floor(evalf(cosh(n)));
  • Mathematica
    Table[Floor[Cosh[n]], {n, 0, 50}] (* T. D. Noe, Jun 20 2012 *)

Extensions

Corrected and extended by Franklin T. Adams-Watters, Apr 26 2006

A309104 a(n) = Sum_{k >= 0} floor(n^(2*k+1) / (2*k+1)!).

Original entry on oeis.org

0, 1, 3, 9, 25, 72, 199, 545, 1487, 4048, 11007, 29930, 81371, 221199, 601295, 1634499, 4443044, 12077466, 32829974, 89241138, 242582585, 659407853, 1792456409, 4872401708, 13244561050, 36002449653, 97864804699, 266024120286, 723128532126, 1965667148555
Offset: 0

Views

Author

Rémy Sigrist, Jul 12 2019

Keywords

Comments

This sequence is inspired by the Maclaurin series for the hyperbolic sine function.

Examples

			For n = 5:
- we have:
  k  5^(2*k+1)/(2*k+1)!
  -  ------------------
  0                   5
  1                  20
  2                  26
  3                  15
  4                   5
  5                   1
  >=6                 0
- hence a(5) = 5 + 20 + 26 + 15 + 5 + 1 = 72.
		

Crossrefs

See A309087 for similar sequences.
Cf. A000471.

Programs

  • Maple
    f:= proc(n) local t,k,v;
      v:= n; t:= n;
      for k from 1 do
        v:= v*n^2/(2*k*(2*k+1));
        if v < 1 then return t fi;
        t:= t + floor(v);
      od
    end proc:
    map(f, [$0..30]); # Robert Israel, Mar 18 2020
  • PARI
    a(n) = { my (v=0, d=n); forstep (k=2, oo, 2, if (d<1, return (v), v += floor(d); d *= n^2/(k*(k+1)))) }

Formula

a(n) ~ sinh(n) as n tends to infinity.
a(n) <= A000471(n).

Extensions

Definition corrected by Robert Israel, Mar 18 2020

A361669 a(n) = floor of sinh(sinh(sinh(...(1)...))) with n iterations.

Original entry on oeis.org

1, 1, 1, 2, 3, 22, 3355531547
Offset: 0

Views

Author

Sylvia Zevi Abrams, Mar 20 2023

Keywords

Comments

a(7) = ~2.67*10^1457288834.
sinh(x) = (e^x - e^-x)/2 is dominated by e^x as x tends to infinity, meaning that the sequence grows tetrationally.

Examples

			a(0) = 1,
a(1) = floor(sinh(1)) = floor(1.175...) = 1,
a(2) = floor(sinh(sinh(1))) = floor(1.465...) = 1,
a(3) = floor(sinh(sinh(sinh(1)))) = floor(2.048...) = 2,
a(4) = floor(sinh(sinh(sinh(sinh(1))))) = floor(3.812...) = 3,
a(5) = floor(sinh(sinh(sinh(sinh(sinh(1)))))) = floor(22.627...) = 22,
a(6) = floor(sinh(sinh(sinh(sinh(sinh(sinh(1))))))) = 3355531547.
		

Crossrefs

Programs

  • Maple
    a:= n-> floor(evalf((sinh@@n)(1), 45)):
    seq(a(n), n=0..6);  # Alois P. Heinz, Mar 20 2023
  • Mathematica
    Table[Floor[Nest[Sinh[n],1,n]],{n,0,7}]
Showing 1-3 of 3 results.