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.

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.

Links

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

A309103 a(n) = Sum_{k >= 0} (-1)^k * floor(n^k / k!).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -1, -2, -1, -3, 0, 1, 0, -2, -1, -2, 2, 1, 1, 2, -2, 2, 0, -2, -3, 0, -1, -2, 0, -2, 3, -8, 1, -4, -3, -4, 1, -2, 1, -3, -2, -2, 2, 2, 3, 3, 2, 0, -5, -2, -3, -5, -2, -4, 3, 4, -2, -2, 4, -7, 3, 5, 3, 5, 0, -1, 1, -8, 6, -3, -1, 8, -5, 0, -6
Offset: 0

Views

Author

Rémy Sigrist, Jul 12 2019

Keywords

Comments

This sequence mimics the Maclaurin series for the function x -> exp(-x).
The series in the name is well defined; for any n > 0, only the first A065027(n) terms are different from zero.

Examples

			For n = 3:
- we have:
  k  floor(3^k / k!)
  -  ---------------
  0                1
  1                3
  2                4
  3                4
  4                3
  5                2
  6                1
  >=7              0
- hence a(3) = 1 - 3 + 4 - 4 + 3 - 2 + 1 = 0.

Crossrefs

See A309087 for similar sequences.
Cf. A065027.

Programs

  • PARI
    a(n) = { my (v=0, d=1, s=+1); for (k=1, oo, if (d<1, return (v), v += s*floor(d); d *= n/k; s = -s)) }

A309105 a(n) = Sum_{k >= 0} floor(n^(2*k) / (2*k)!).

Original entry on oeis.org

1, 1, 3, 9, 25, 71, 198, 543, 1486, 4045, 11007, 29931, 81371, 221197, 601294, 1634497, 4443046, 12077467, 32829975, 89241140, 242582583, 659407855, 1792456409, 4872401706, 13244561047, 36002449653, 97864804698, 266024120284, 723128532126, 1965667148553
Offset: 0

Views

Author

Rémy Sigrist, Jul 12 2019

Keywords

Comments

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

Examples

			For n = 5:
- we have:
  k   5^(2*k)/(2*k)!
  --  --------------
   0               1
   1              12
   2              26
   3              21
   4               9
   5               2
   6               0
- hence a(5) = 1 + 12 + 26 + 21 + 9 + 2 = 71.

Links

Crossrefs

See A309087 for similar sequences.
Cf. A000501.

Programs

  • PARI
    a(n) = { my (v=0, d=1); forstep (k=1, oo, 2, if (d<1, return (v), v += floor(d); d *= n^2/(k*(k+1)))) }

Formula

a(n) ~ cosh(n) as n tends to infinity.
a(n) <= A000501(n).

Extensions

Definition corrected by Rémy Sigrist, Aug 06 2020
Showing 1-3 of 3 results.

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