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.

A000471 a(n) = floor(sinh(n)).

Original entry on oeis.org

0, 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

Programs

Formula

a(n) = floor(e^n/2). - Bruno Berselli, Mar 04 2016

Extensions

More terms from Stefan Steinerberger, Apr 15 2006

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

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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.
		

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

A346169 a(n) = floor(cosh(n+1) - cosh(n)).

Original entry on oeis.org

0, 2, 6, 17, 46, 127, 346, 942, 2561, 6961, 18923, 51440, 139829, 380095, 1033206, 2808546, 7634421, 20752508, 56411165, 153341447, 416825269, 1133048555, 3079945300, 8372159341, 22757888603, 61862355045, 168159315586, 457104411844, 1242538616426
Offset: 0

Views

Author

Treanungkur Mal, Jul 08 2021

Keywords

Examples

			For n = 3, a(3) = 17 because the difference between cosh(4) and cosh(3) is 17.24057..., and floor(17.24057...) = 17.
		

Crossrefs

Cf. A000501.

Programs

  • Mathematica
    a[n_] := Floor[Cosh[n+1] - Cosh[n]]; Array[a, 30,0] (* Amiram Eldar, Jul 12 2021 *)

Extensions

More terms from Jinyuan Wang, Jul 08 2021
Showing 1-3 of 3 results.