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 1-4 of 4 results.

A075317 Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1,3),(5,9),(7,13),(11,19),(15,25),(17,29),(21,35),(23,39),(27,45),... This is the sequence of the first member of pairs.

Original entry on oeis.org

1, 5, 7, 11, 15, 17, 21, 23, 27, 31, 33, 37, 41, 43, 47, 49, 53, 57, 59, 63, 65, 69, 73, 75, 79, 83, 85, 89, 91, 95, 99, 101, 105, 109, 111, 115, 117, 121, 125, 127, 131, 133, 137, 141, 143, 147, 151, 153, 157, 159, 163, 167, 169, 173, 175, 179, 183, 185, 189, 193
Offset: 1

Views

Author

Amarnath Murthy, Sep 14 2002

Keywords

Comments

(a(n), A075318(n)) = (2A(n)-1, 2B(n)-1), where A and B are the basic Wythoff sequences A(n)=A000201(n) and B(n)=A001950(n). For a proof cf. Section 2 of the Carlitz et al. paper. - Michel Dekking, Sep 05 2016

Links

Crossrefs

Cf. A075318, A075319, A075320.

Programs

  • Magma
    [2*Floor(n*(1+Sqrt(5))/2)-1: n in [1..60]]; // Vincenzo Librandi, Sep 05 2016
  • Maple
    A075317 := proc(nmax) local r,k,a,pairs ; a := [1] ; pairs := [1,3] ; k := 2 ; r := 5 ; while nops(a) < nmax do while r in pairs do r := r+2 ; od ; a := [op(a),r] ; if r+2*k in pairs then printf("inconsistency",k) ; fi ; pairs := [op(pairs),r,r+2*k] ; k := k+1 ; od ; RETURN(a) ; end: a := A075317(200) ; for n from 1 to nops(a) do printf("%d,",op(n,a)) ; od ; # R. J. Mathar, Nov 12 2006
  • Mathematica
    Table[2 Floor[n (1 + Sqrt[5]) / 2] - 1, {n, 80}] (* Vincenzo Librandi, Sep 05 2016 *)
  • Python
    from math import isqrt
def A075317(n): return (n+isqrt(5*n**2)&-2)-1 # Chai Wah Wu, Aug 16 2022

Formula

a(n) = 2*floor(n*phi)-1, where phi=(1+sqrt(5))/2. - Michel Dekking, Sep 05 2016

Extensions

More terms from R. J. Mathar, Nov 12 2006

A075319 Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1,3),(5,9),(7,13),(11,19),(15,25),(17,29),(21,35),(23,39),(27,45),... This is the sequence of the sum of the members of pairs.

Original entry on oeis.org

4, 14, 20, 30, 40, 46, 56, 62, 72, 82, 88, 98, 108, 114, 124, 130, 140, 150, 156, 166, 172, 182, 192, 198, 208, 218, 224, 234, 240, 250, 260, 266, 276, 286, 292, 302, 308, 318, 328, 334, 344, 350, 360, 370, 376, 386, 396, 402, 412, 418, 428, 438, 444, 454, 460
Offset: 1

Views

Author

Amarnath Murthy, Sep 14 2002

Keywords

Crossrefs

Cf. A075317, A075318, A075320.

Programs

  • Maple
    A075319 := proc(nmax) local r,k,a,pairs ; a := [4] ; pairs := [1,3] ; k := 2 ; r := 5 ; while nops(a) < nmax do while r in pairs do r := r+2 ; od ; if r+2*k in pairs then printf("inconsistency",k) ; fi ; a := [op(a),2*(r+k)] ; pairs := [op(pairs),r,r+2*k] ; k := k+1 ; od ; RETURN(a) ; end: a := A075319(200) : for n from 1 to nops(a) do printf("%d,",op(n,a)) ; od ; # R. J. Mathar, Nov 12 2006
  • Python
    from math import isqrt
def A075319(n): return (n+isqrt(5*n**2)&-2)+n-1<<1 # Chai Wah Wu, Aug 16 2022

Formula

a(n) = A075317(n)+A075318(n). - R. J. Mathar, Nov 12 2006

Extensions

More terms from R. J. Mathar, Nov 12 2006

A075320 Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1, 3), (5, 9), (7, 13), (11, 19), (15, 25), (17, 29), (21, 35), (23, 39), (27, 45), ... This is the sequence of the product of the members of pairs.

Original entry on oeis.org

3, 45, 91, 209, 375, 493, 735, 897, 1215, 1581, 1815, 2257, 2747, 3053, 3619, 3969, 4611, 5301, 5723, 6489, 6955, 7797, 8687, 9225, 10191, 11205, 11815, 12905, 13559, 14725, 15939, 16665, 17955, 19293, 20091, 21505, 22347, 23837, 25375, 26289
Offset: 1

Views

Author

Amarnath Murthy, Sep 14 2002

Keywords

Crossrefs

Cf. A075317, A075318, A075319.

Programs

  • Maple
    A075320 := proc(nmax) local r,k,a,pairs ; a := [3] ; pairs := [1,3] ; k := 2 ; r := 5 ; while nops(a) < nmax do while r in pairs do r := r+2 ; od ; if r+2*k in pairs then printf("inconsistency",k) ; fi ; a := [op(a),r*(r+2*k)] ; pairs := [op(pairs),r,r+2*k] ; k := k+1 ; od ; RETURN(a) ; end: a := A075320(200) : for n from 1 to nops(a) do printf("%d,",op(n,a)) ; od ; # R. J. Mathar, Nov 12 2006
  • Python
    from math import isqrt
def A075320(n): return (m:=(n+isqrt(5*n**2)&-2)-1)*((n<<1)+m) # Chai Wah Wu, Aug 16 2022

Formula

a(n) = A075317(n)*A075318(n). - R. J. Mathar, Nov 12 2006

Extensions

More terms from R. J. Mathar, Nov 12 2006

A287802 Positions of 0 in A287801; complement of A287803.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 11, 12, 13, 14, 17, 18, 19, 20, 23, 24, 26, 27, 28, 29, 32, 33, 35, 36, 37, 38, 41, 42, 43, 44, 47, 48, 50, 51, 52, 53, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 71, 72, 74, 75, 76, 77, 80, 81, 82, 83, 86, 87, 89, 90, 91, 92, 95, 96, 98, 99
Offset: 1

Views

Author

Clark Kimberling, Jun 03 2017

Keywords

Comments

Let d(n) = 3n/2 - a(n). Then d(n) is in {-1/2, 0, 1/2, 1} for n >= 1. Indeed,
d(n) = - 1/2 if n is in A075317; d(n) = 1/2 if n is in A075318;
d(n) = 0 if n is in A283234; d(n) = 1 if n is in A283233.

Links

Crossrefs

Cf. A075317, A075318, A283233, A283234, A287801, A287803.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" -> "100", "1" -> "001"}]
st = ToCharacterCode[w1] - 48    (* A287801 *)
Flatten[Position[st, 0]]  (* A287802 *)
Flatten[Position[st, 1]]  (* A287803 *)
Showing 1-4 of 4 results.

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