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 1-5 of 5 results.

A064323 a(n) = a(n-1)+ceiling(a(n-2)/2) with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 75, 103, 141, 193, 264, 361, 493, 674, 921, 1258, 1719, 2348, 3208, 4382, 5986, 8177, 11170, 15259, 20844, 28474, 38896, 53133, 72581, 99148, 135439, 185013, 252733, 345240, 471607, 644227, 880031, 1202145
Offset: 0

Views

Author

Henry Bottomley, Sep 11 2001

Keywords

Comments

a(n)/a(n-1) approaches (1+sqrt(3))/2 = 1.3660254... = A332133 for large n.

Examples

			a(5) = a(4)+ceiling(a(3)/2) = 3+ceiling(2/2) = 4.

Links

Crossrefs

Cf. A064324, A332133.

Programs

  • Magma
    [n le 2 select n-1 else Self(n-1)+Ceiling(Self(n-2)/2): n in [1..45]]; // Bruno Berselli, Apr 20 2012
  • Maple
    a:= proc(n) option remember;
      `if`(n<2, n, a(n-1)+ceil(a(n-2)/2))
    end:
seq(a(n), n=0..48);  # Alois P. Heinz, Jan 26 2023
  • Mathematica
    RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+Ceiling[a[n-2]/2]},a,{n,50}] (* Harvey P. Dale, Nov 06 2013 *)
  • PARI
    for (n=0, 400, if (n>1, a=a1 + ceil(a2/2); a2=a1; a1=a, if (n, a=a1=1, a=a2=0)); write("b064323.txt", n, " ", a) )  \\ Harry J. Smith, Sep 11 2009
  • PARI
    first(n)=if(n<2, return([0,1][1..n+1])); my(v=vector(n+1)); v[2]=1; for(k=3,n+1, v[k]=v[k-1]+(v[k-2]+1)\2); v \\ Charles R Greathouse IV, Jan 26 2023

Formula

a(n) = A064324(n)-1.

A064650 a(n) = floor(a(n-1)/2) + a(n-2) with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 6, 8, 10, 13, 16, 21, 26, 34, 43, 55, 70, 90, 115, 147, 188, 241, 308, 395, 505, 647, 828, 1061, 1358, 1740, 2228, 2854, 3655, 4681, 5995, 7678, 9834, 12595, 16131, 20660, 26461, 33890, 43406, 55593, 71202, 91194, 116799, 149593, 191595
Offset: 0

Views

Author

Henry Bottomley, Oct 04 2001

Keywords

Comments

a(n)/a(n-1) tends to (1+sqrt(17))/4 = 1.2807764...

Links

Crossrefs

Cf. A064324, A064325, A182280, A182281.
Cf. A064651, A000045.

Programs

  • Haskell
    a064650 n = a064650_list !! n
a064650_list = 1 : 2 : zipWith (+)
                       a064650_list (map (flip div 2) $ tail a064650_list)
-- Reinhard Zumkeller, Apr 30 2015
  • Magma
    [n le 2 select n else Floor(Self(n-1)/2)+Self(n-2): n in [1..50]]; // Bruno Berselli, Apr 21 2012
  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 2, a[n] == Floor[a[n - 1]/4] + a[n - 2]}, a, {n, 49}] (* Bruno Berselli, Apr 21 2012 *)
  • PARI
    { for (n=0, 400, if (n>1, a=a1\2 + a2; a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b064650.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 21 2009

Formula

a(n) = A064651(n) + 1.

A182229 a(n) = a(n-1) + floor(a(n-2)/3) with a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 3, 4, 5, 6, 7, 9, 11, 14, 17, 21, 26, 33, 41, 52, 65, 82, 103, 130, 164, 207, 261, 330, 417, 527, 666, 841, 1063, 1343, 1697, 2144, 2709, 3423, 4326, 5467, 6909, 8731, 11034, 13944, 17622, 22270, 28144, 35567, 44948, 56803, 71785, 90719, 114647, 144886, 183101
Offset: 0

Views

Author

Alex Ratushnyak, Apr 19 2012

Keywords

Comments

a(n)/a(n-1) tends to (3+sqrt(21))/6 = 1.263762615825973334... [Bruno Berselli, Apr 23 2012]

Links

Crossrefs

Cf. A064324, A182281, A182230; A176014.

Programs

  • Haskell
    a182229 n = a182229_list !! n
a182229_list = 2 : 3 : zipWith (+)
                       (map (flip div 3) a182229_list) (tail a182229_list)
-- Reinhard Zumkeller, Apr 30 2015
  • Magma
    [n le 2 select n+1 else Self(n-1)+Floor(Self(n-2)/3): n in [1..51]]; // Bruno Berselli, Apr 21 2012
  • Mathematica
    RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == a[n - 1] + Floor[a[n - 2]/3]}, a, {n, 50}] (* Bruno Berselli, Apr 21 2012 *)
  • Python
    prpr = 2
prev = 3
for i in range(2,51):
    current = prev + prpr//3
    print(current, end=',')
    prpr = prev
    prev = current

A182230 a(n) = a(n-1)+floor(a(n-2)/4) with a(0)=3, a(1)=4.

Original entry on oeis.org

3, 4, 4, 5, 6, 7, 8, 9, 11, 13, 15, 18, 21, 25, 30, 36, 43, 52, 62, 75, 90, 108, 130, 157, 189, 228, 275, 332, 400, 483, 583, 703, 848, 1023, 1235, 1490, 1798, 2170, 2619, 3161, 3815, 4605, 5558, 6709, 8098, 9775, 11799, 14242, 17191, 20751, 25048, 30235
Offset: 0

Views

Author

Alex Ratushnyak, Apr 19 2012

Keywords

Comments

a(n)/a(n-1) tends to (1+sqrt(2))/2 = 1.207106781186547524... [Bruno Berselli, Apr 23 2012]

Links

Crossrefs

Cf. A064323, A064324, A182229, A182280; A174968.

Programs

  • Haskell
    a182230 n = a182230_list !! n
a182230_list = 3 : 4 : zipWith (+)
                       (map (flip div 4) a182230_list) (tail a182230_list)
-- Reinhard Zumkeller, Apr 30 2015
  • Magma
    [n le 2 select n+2 else Self(n-1)+Floor(Self(n-2)/4): n in [1..52]]; // Bruno Berselli, Apr 20 2012
  • Maple
    a:= proc(n) a(n):= a(n-1) +floor(a(n-2)/4) end: a(0), a(1):= 3, 4:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 20 2012
  • Mathematica
    RecurrenceTable[{a[0] == 3, a[1] == 4, a[n] == a[n - 1] + Floor[a[n - 2]/4]}, a, {n, 51}] (* Bruno Berselli, Apr 21 2012 *)
  • Python
    prpr = 3
prev = 4
for i in range(2,55):
    current = prev + prpr//4
    print(current, end=',')
    prpr = prev
    prev = current

A064651 a(n) = ceiling(a(n-1)/2) + a(n-2) with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 15, 20, 25, 33, 42, 54, 69, 89, 114, 146, 187, 240, 307, 394, 504, 646, 827, 1060, 1357, 1739, 2227, 2853, 3654, 4680, 5994, 7677, 9833, 12594, 16130, 20659, 26460, 33889, 43405, 55592, 71201, 91193, 116798, 149592
Offset: 0

Views

Author

Henry Bottomley, Oct 04 2001

Keywords

Links

Crossrefs

Cf. A064324, A064325.
Cf. A064650, A000045, A188934.

Programs

  • Haskell
    a064651 n = a064651_list !! n
a064651_list = 0 : 1 : zipWith (+)
   a064651_list (map (flip div 2 . (+ 1)) $ tail a064651_list)
-- Reinhard Zumkeller, Apr 30 2015
  • Mathematica
    RecurrenceTable[{a[0]==0, a[1]==1, a[n]==Ceiling[a[n-1]/2]+a[n-2]}, a, {n,50}] (* Harvey P. Dale, Aug 22 2012 *)
t = {0, 1}; Do[AppendTo[t, Ceiling[t[[-1]]/2] + t[[-2]]], {48}]; t (* T. D. Noe, Aug 22 2012 *)

Formula

a(n) = A064650(n) - 1.
Lim_{n->infinity} a(n)/a(n-1) = (1+sqrt(17))/4 = 1.2807764... = A188934.
Showing 1-5 of 5 results.

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