A081853 -id:A081853 - cp's OEIS frontend

A081849 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached.

3, 20, 14, 468, 33, 299, 60, 47328, 95, 1218, 138, 25475, 189, 3161, 248, 20830128, 315, 6512, 390, 181138, 473, 11655, 564, 9015167, 663, 18974, 770, 671745, 885, 28853, 1008, 38906570560, 1139, 41676, 1278, 1799888, 1425, 57827, 1580, 110341278, 1743, 77690

Offset: 1

N. J. A. Sloane, Apr 13 2003

k = A001511(n) is the number of steps to reach an integer b(k).

Cf. A001511, A081853, A073524, A074078.

Digits := 100: c := ceil; A081849 := proc(a) local i,t0,t; t0 := a; t := 0; for i from 1 to 100 do if whattype(t0) <> integer then t0 := a*c(t0); t := t+1; else RETURN(t0); fi; od; RETURN('FAIL'); end;

a[n_]:=Module[{b=b0=(2n+1)/2},While[!IntegerQ[b],b=b0*Ceiling[b]]; b]; Array[a,42] (* Stefano Spezia, Jun 26 2024 *)

a(n) = if(n==1,3, my(t=2*n+1, k=1+valuation(n,2)); n*t^(k+1) >>k \ (t-2)); \\ Kevin Ryde, Jun 30 2024

from math import ceil
from fractions import Fraction
def a(n):
  b0 = b = Fraction((2*n+1), 2)
  while b.denominator != 1: b = b0*ceil(b)
  return b.numerator
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Mar 20 2021

a(n) = s*(n*s^k - 1/2) / (s-1) where s = b(0) = (2*n+1)/2 and k = A001511(n). - Kevin Ryde, Jun 30 2024

A057016 Consider recurrence b(0) = (2n+1)/2, b(n) = b(n-1)*floor(b(n-1)); sequence gives first integer reached.

Original entry on oeis.org

5, 105, 18, 550935, 39, 2730, 68, 2789204756584545, 105, 15939, 150, 943242300, 203, 53940, 264, 3714817857497700192049140000201816119835, 333, 137085, 410, 41463649689, 495, 291870, 588, 27194270698105759097850, 689, 550935, 798, 535022394030, 915

Offset: 2

N. J. A. Sloane, Sep 27 2003

nonn

Number of steps to reach an integer is given by A001511. No integer is reached if initial value is 3/2.

J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A081853, A087666.

a(2t) = t(4t+1).

A087675 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*floor(b(n-1)); sequence gives first integer reached.

Original entry on oeis.org

5, 35, 18, 814, 39, 390, 68, 72827, 105, 1449, 150, 31887, 203, 3596, 264, 27852510, 333, 7215, 410, 208464, 495, 12690, 588, 10561998, 689, 20405, 798, 744049, 915, 30744, 1040, 46620858503, 1173, 44091, 1314, 1950450, 1463, 60830, 1620, 121575329, 1785

Offset: 2

N. J. A. Sloane, following a suggestion of Bela Bajnok (bbajnok(AT)gettysburg.edu), Sep 27 2003

Robert Israel, Table of n, a(n) for n = 2..10000
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

A001511 gives number of steps to reach an integer.

Cf. A001511, A081853, A085071, A057016.

f:= proc(n)
  local b0, b;
  b0:= (2*n+1)/2;
  b:= b0;
  do
    b:= b0*floor(b);
    if b::integer then return b fi
  od
end proc:
map(f, [$2..100]); # Robert Israel, Nov 25 2019

f[n_] := Module[{b0, b}, b0 = (2n+1)/2; b = b0; While[True, b = b0*Floor[b]; If[IntegerQ[b], Return[b]]]];
Table[f[n], {n, 2, 100}] (* Jean-François Alcover, Oct 23 2023, after Robert Israel *)

The even-indexed terms are given by A007742.

Offset corrected by Robert Israel, Nov 25 2019

A081854 a(n) = (8n - 3)(4n - 1)(8n^2 - 5n + 1).

Original entry on oeis.org

3, 60, 2093, 13398, 47415, 123728, 268065, 512298, 894443, 1458660, 2255253, 3340670, 4777503, 6634488, 8986505, 11914578, 15505875, 19853708, 25057533, 31222950, 38461703, 46891680, 56636913, 67827578, 80599995, 95096628, 111466085, 129863118, 150448623

Offset: 0

N. J. A. Sloane, Apr 13 2003

J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Value of A081853 when started at b(0) with 2*b(0) == 5 (mod 8).

Table[(8n-3)(4n-1)(8n^2-5n+1),{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{3,60,2093,13398,47415},30] (* Harvey P. Dale, Mar 20 2015 *)

a(n)=(8*n-3)*(4*n-1)*(8*n^2-5*n+1) \\ Charles R Greathouse IV, Oct 21 2022

G.f.: (60 + 1793*x + 3533*x^2 + 755*x^3 + 3*x^4)/(1-x)^5.
a(0)=3, a(1)=60, a(2)=2093, a(3)=13398, a(4)=47415, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Mar 20 2015
E.g.f.: exp(x)*(3 + 57*x + 988*x^2 + 1216*x^3 + 256*x^4). - Stefano Spezia, Jun 26 2024

cp's OEIS Frontend

A081849 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached.

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A057016 Consider recurrence b(0) = (2n+1)/2, b(n) = b(n-1)*floor(b(n-1)); sequence gives first integer reached.

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A087675 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*floor(b(n-1)); sequence gives first integer reached.

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Maple

Mathematica

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A081854 a(n) = (8*n - 3)*(4*n - 1)*(8*n^2 - 5*n + 1).

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Mathematica

PARI

Formula

A081854 a(n) = (8n - 3)(4n - 1)(8n^2 - 5n + 1).