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.

A260708 a(2n) = n*(2*n+1), a(2n+7) = a(2n+1) + 12*n + 28, with a(1)=1, a(3)=6, a(5)=16.

Original entry on oeis.org

0, 1, 3, 6, 10, 16, 21, 29, 36, 46, 55, 68, 78, 93, 105, 122, 136, 156, 171, 193, 210, 234, 253, 280, 300, 329, 351, 382, 406, 440, 465, 501, 528, 566, 595, 636, 666, 709, 741, 786, 820, 868, 903, 953, 990, 1042, 1081, 1136, 1176, 1233, 1275, 1334, 1378
Offset: 0

Views

Author

Paul Curtz, Nov 17 2015

Keywords

Comments

Conjecture: this sequence is 0 followed by A264041.
After 3, if a(n) is prime then n == 1 (mod 6).
a(n) is a square for n = 0, 1, 5, 8, 145, 288, 1777, 6533, 9800, 168097, 332928, 2051425, 7539845, ...

Examples

			a(0) = 0*1 = 0,
a(1) = 1,
a(2) = 1*3 = 3,
a(3) = 6,
a(4) = 2*5 = 10,
a(5) = 16,
a(6) = 3*7 = 21,
a(7) = a(1) +12*0 +28 = 29, etc.

Links

Crossrefs

Cf. A000217, A008615, A014105, A260160, A260699, A264041.

Programs

  • Mathematica
    LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 1, 3, 6, 10, 16, 21, 29, 36}, 50] (* Bruno Berselli, Nov 18 2015 *)
  • PARI
    concat(0, Vec(-x*(x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^3*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Nov 18 2015
  • Sage
    [n*(n+1)/2+(1-(-1)^n)*floor(n/6+1/3)/2 for n in (0..60)] # Bruno Berselli, Nov 18 2015

Formula

From Colin Barker, Nov 17 2015: (Start)
G.f.: x*(1 + 2*x + 2*x^2 + 2*x^3 + 3*x^4 + x^5 + x^6) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) for n>8. (End)
a(2*k) = A000217(2*k) by definition; for odd indices:
a(6*k+1) = 18*k^2 + 10*k + 1,
a(6*k+3) = 2*(9*k^2 + 11*k + 3),
a(6*k+5) = 2*(k + 1)*(9*k + 8), that is A178574.
a(n) = A260699(n) + A008615(n).
a(n) = n*(n + 1)/2 + (1 - (-1)^n)*floor(n/6 + 1/3)/2. [Bruno Berselli, Nov 18 2015]

Extensions

Edited by Bruno Berselli, Nov 18 2015

A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, 17, 14, 20, 15, 22, 17, 24, 19, 27, 20, 29, 22, 31, 24, 34, 25, 36, 27, 38, 29, 41, 30, 43, 32, 45, 34, 48, 35, 50, 37, 52, 39, 55, 40, 57, 42, 59, 44, 62, 45, 64, 47, 66, 49, 69, 50, 71, 52, 73, 54, 76, 55, 78
Offset: 0

Views

Author

Paul Curtz, Nov 22 2015

Keywords

Comments

A260708 difference table rows have the same nine-step recurrence:
0, 1, 3, 6, 10, 16, 21, 29, 36, 46, 55, 65, 78, 93, ...
1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, ... = a(n)
1, 1, 1, 2, -1, 3, -1, 3, -1, 4, -3, 5, -3, 5, ... = b(n)
0, 0, 1, -3, 4, -4, 4, -4, 5, -7, 8, -8, 8, -8, ... (see A042965(n)).
(b(2n) + b(2n+1) = A052901(n+2).)

Links

Crossrefs

Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.

Programs

  • Magma
    I:=[1,2,3,4,6,5,8,7];[n le 8 select I[n] else Self(n-2) + Self(n-6) - Self(n-8): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015
  • Mathematica
    RecurrenceTable[{a[n] == a[n-2] + a[n-6] - a[n-8], a[0]=1, a[1]=2, a[2]=3, a[3]=4, a[4]=6, a[5]=5, a[6]=8, a[7]=7}, a, {n,0,100}] (* G. C. Greubel, Nov 23 2015 *)
  • PARI
    Vec((x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 22 2015
  • PARI
    vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ Altug Alkan, Nov 24 2015

Formula

a(2n) = A047282(n). a(2n+1) = A047212(n+1).
a(n) = A260708(n+1) - A260708(n).
a(n+6) = a(n) + period of length 2: repeat 7, 5.
a(2n) + a(2n+1) = 3 + 4*n.
a(n) = n + 1 + (-1)^n*A152467(n+2).
From Colin Barker, Nov 22 2015: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) for n>7.
G.f.: (x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)).
(End)

A265228 Interleave the even numbers with the numbers that are congruent to {1, 3, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 9, 8, 11, 10, 15, 12, 17, 14, 19, 16, 23, 18, 25, 20, 27, 22, 31, 24, 33, 26, 35, 28, 39, 30, 41, 32, 43, 34, 47, 36, 49, 38, 51, 40, 55, 42, 57, 44, 59, 46, 63, 48, 65, 50, 67, 52, 71, 54, 73, 56, 75, 58, 79, 60, 81, 62, 83, 64, 87, 66
Offset: 0

Views

Author

Paul Curtz, Dec 06 2015

Keywords

Comments

b(n) denotes the sequence:
0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, 2, -2, 2, -2, 2, -2, 2, 3, -3, 3, -3, 3, -3, 3, 4, -4, ..., and
c(n) = n + b(n) = n + floor((n+1)/7)*(-1)^((n+1) mod 7) provides:
0, 1, 2, 3, 4, 5, 7, 6, 9, 8, 11, 10, 13, 15, 12, 17, 14, 19, 16, 21, 23, 18, 25, 20, 27, 22, 29, ..., which is a permutation of A001477.
a(n) differs from c(n) because c(n) contains the terms of the form 8*k+5.

Links

Crossrefs

Cf. A001477, A004770, A005843, A047529, A079979, A260160, A260699, A260708.

Programs

  • Mathematica
    lim = 11; Riffle[Range[0, 6 lim, 2], Select[Range[8 lim], MemberQ[{1, 3, 7}, Mod[#, 8]] &]] (* Michael De Vlieger, Dec 06 2015 *)
  • PARI
    concat(0, Vec(x*(1+2*x+2*x^2+2*x^3+4*x^4+2*x^5+x^6)/((1-x)^2 *(1+x)^2*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Dec 06 2015
  • PARI
    vector(100, n, n--; n+(1-(-1)^n)*floor(n/6+1/3)) \\ Altug Alkan, Dec 09 2015

Formula

a(n) = n + 2*A260160(n) = n + (1-(-1)^n)*floor(n/6+1/3). Therefore, for odd n, a(n) = A047529((n+1)/2); otherwise, a(n) = n.
a(n) = a(n-6) - (-1)^n + 7.
a(n) = A260708(n) - A260699(n-1) - A079979(n+3), with A260699(-1) = 0.
From Colin Barker, Dec 06 2015: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) for n > 7.
G.f.: x*(1+2*x+2*x^2+2*x^3+4*x^4+2*x^5+x^6) / ((1-x)^2*(1+x)^2*(1-x+x^2)*(1+x+x^2)). (End)
Showing 1-3 of 3 results.

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