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.

A275175 a(n) = (2 * a(n-3) + a(n-1) * a(n-5)) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 5, 7, 13, 23, 83, 147, 215, 423, 771, 2801, 4971, 7281, 14351, 26181, 95133, 168845, 247317, 487493, 889373, 3231703, 5735737, 8401475, 16560393, 30212491, 109782751, 194846191, 285402811, 562565851, 1026335311, 3729381813, 6619034735, 9695294077, 19110678523
Offset: 0

Views

Author

Seiichi Manyama, Jul 19 2016

Keywords

Comments

Inspired by A048736.

Links

Crossrefs

Cf. A048736, A275173, A275176.

Programs

  • Mathematica
    RecurrenceTable[{a[n] == (2 a[n - 3] + a[n - 1] a[n - 5])/a[n - 6], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 40}] (* Michael De Vlieger, Jul 19 2016 *)
  • PARI
    Vec((1 +x +x^2 +x^3 +x^4 -34*x^5 -32*x^6 -30*x^7 -28*x^8 -22*x^9 +23*x^10 +13*x^11 +7*x^12 +5*x^13 +3*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -34*x^5 +x^10)) + O(x^50)) \\ Colin Barker, Jul 19 2016
  • Ruby
    def A(k, l, n)
  a = Array.new(k * 2, 1)
  ary = [1]
  while ary.size < n + 1
    break if (a[1] * a[-1] + a[k] * l) % a[0] > 0
    a = *a[1..-1], (a[1] * a[-1] + a[k] * l) / a[0]
    ary << a[0]
  end
  ary
end
def A275175(n)
  A(3, 2, n)
end

Formula

G.f.: (1 +x +x^2 +x^3 +x^4 -34*x^5 -32*x^6 -30*x^7 -28*x^8 -22*x^9 +23*x^10 +13*x^11 +7*x^12 +5*x^13 +3*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -34*x^5 +x^10)). - Colin Barker, Jul 19 2016
a(n) = 35*a(n-5) - 35*a(n-10) + a(n-15). - G. C. Greubel, Jul 20 2016

A275176 a(n) = (3 * a(n-3) + a(n-1) * a(n-5)) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 7, 10, 22, 43, 202, 370, 547, 1264, 2521, 11881, 21781, 32221, 74521, 148681, 700744, 1284667, 1900450, 4395442, 8769643, 41331982, 75773530, 112094287, 259256524, 517260241, 2437886161, 4469353561, 6611662441, 15291739441, 30509584561
Offset: 0

Views

Author

Seiichi Manyama, Jul 19 2016

Keywords

Comments

Inspired by A048736.

Links

Crossrefs

Cf. A048736, A275173, A275175.

Programs

  • Mathematica
    RecurrenceTable[{a[n] == (3 a[n - 3] + a[n - 1] a[n - 5])/a[n - 6], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 36}] (* Michael De Vlieger, Jul 19 2016 *)
  • PARI
    Vec((1 +x +x^2 +x^3 +x^4 -59*x^5 -56*x^6 -53*x^7 -50*x^8 -38*x^9 +43*x^10 +22*x^11 +10*x^12 +7*x^13 +4*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -59*x^5 +x^10)) + O(x^50)) \\ Colin Barker, Jul 19 2016
  • Ruby
    def A(k, l, n)
  a = Array.new(k * 2, 1)
  ary = [1]
  while ary.size < n + 1
    break if (a[1] * a[-1] + a[k] * l) % a[0] > 0
    a = *a[1..-1], (a[1] * a[-1] + a[k] * l) / a[0]
    ary << a[0]
  end
  ary
end
def A275176(n)
  A(3, 3, n)
end

Formula

G.f.: (1 +x +x^2 +x^3 +x^4 -59*x^5 -56*x^6 -53*x^7 -50*x^8 -38*x^9 +43*x^10 +22*x^11 +10*x^12 +7*x^13 +4*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -59*x^5 +x^10)). - Colin Barker, Jul 19 2016
a(n) = 60*a(n-5) - 60*a(n-10) + a(n-15).

A275174 a(n) = (a(n-4) + a(n-1) * a(n-7)) / a(n-8), a(0) = a(1) = ... = a(7) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 33, 53, 74, 96, 141, 209, 300, 714, 1151, 1611, 2094, 3083, 4578, 6579, 15665, 25257, 35355, 45959, 67673, 100497, 144431, 343906, 554491, 776186, 1008991, 1485711, 2206346, 3170896, 7550257, 12173533, 17040724
Offset: 0

Views

Author

Seiichi Manyama, Jul 19 2016

Keywords

Comments

Inspired by A048736.

Links

Crossrefs

Cf. A101879, A048736, A275173.

Programs

  • Mathematica
    RecurrenceTable[{a[n] == (a[n - 4] + a[n - 1] a[n - 7])/a[n - 8], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1, a[7] == 1, a[8] == 1}, a, {n, 42}] (* Michael De Vlieger, Jul 19 2016 *)
  • PARI
    Vec((1 +x +x^2 +x^3 +x^4 +x^5 +x^6 -22*x^7 -21*x^8 -20*x^9 -19*x^10 -18*x^11 -16*x^12 -13*x^13 +14*x^14 +10*x^15 +7*x^16 +5*x^17 +4*x^18 +3*x^19 +2*x^20) / ((1 -x)*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)*(1 -22*x^7 +x^14)) + O(x^20)) \\ Colin Barker, Jul 19 2016
  • Ruby
    def A(k, l, n)
  a = Array.new(k * 2, 1)
  ary = [1]
  while ary.size < n + 1
    break if (a[1] * a[-1] + a[k] * l) % a[0] > 0
    a = *a[1..-1], (a[1] * a[-1] + a[k] * l) / a[0]
    ary << a[0]
  end
  ary
end
def A275174(n)
  A(4, 1, n)
end

Formula

G.f.: (1 +x +x^2 +x^3 +x^4 +x^5 +x^6 -22*x^7 -21*x^8 -20*x^9 -19*x^10 -18*x^11 -16*x^12 -13*x^13 +14*x^14 +10*x^15 +7*x^16 +5*x^17 +4*x^18 +3*x^19 +2*x^20) / ((1 -x)*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)*(1 -22*x^7 +x^14)). - Colin Barker, Jul 19 2016
a(n) = 23*a(n-7) - 23*a(n-14) + a(n-21).

A276529 a(n) = (a(n-1) * a(n-5) + 1) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 20, 27, 34, 41, 89, 137, 185, 233, 281, 610, 939, 1268, 1597, 1926, 4181, 6436, 8691, 10946, 13201, 28657, 44113, 59569, 75025, 90481, 196418, 302355, 408292, 514229, 620166, 1346269, 2072372, 2798475, 3524578, 4250681, 9227465
Offset: 0

Views

Author

Seiichi Manyama, Nov 16 2016

Keywords

Comments

Thanks to the linear recurrence signature, we see that this is actually five separate linear recurrence sequences, each with signature (7,-1), interwoven together. - Greg Dresden, Oct 16 2021

Links

Crossrefs

Cf. A275173, A102276, A276530.
5th-sections: A049685, A033891, A033889.

Programs

  • Mathematica
    LinearRecurrence[{0,0,0,0,7,0,0,0,0,-1}, {1,1,1,1,1,1,2,3,4,5}, 50] (* G. C. Greubel, Nov 18 2016 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==1,a[n]==(a[n-1]a[n-5]+ 1)/a[n-6]},a,{n,50}] (* Harvey P. Dale, Oct 08 2020 *)
Flatten[Table[{LucasL[4 n - 2]/3, Fibonacci[4 n - 1], LucasL[4 n + 2]/3 - Fibonacci[4 n], LucasL[4 n - 2]/3 + Fibonacci[4 n], Fibonacci[4 n + 1]}, {n, 0, 10}]] (* Greg Dresden, Oct 16 2021 *)
  • PARI
    Vec((1 +x +x^2 +x^3 +x^4 -6*x^5 -5*x^6 -4*x^7 -3*x^8 -2*x^9)/(1 -7*x^5 +x^10) + O(x^50)) \\ Colin Barker, Nov 16 2016
  • Ruby
    def A(k, m, n)
  a = Array.new(2 * k, 1)
  ary = [1]
  while ary.size < n + 1
    i = a[-1] * a[1] + a[k] ** m
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A276529(n)
  A(3, 0, n)
end

Formula

a(n) + a(n+10) = 7*a(n+5).
a(5-n) = a(n).
G.f.: (1 +x +x^2 +x^3 +x^4 -6*x^5 -5*x^6 -4*x^7 -3*x^8 -2*x^9) / (1 -7*x^5 +x^10). - Colin Barker, Nov 16 2016
From Greg Dresden, Oct 16 2021: (Start)
a(5*n) = L(4*n-2)/3 = A049685(n-1),
a(5*n+1) = F(4*n-1) = A033891(n-1),
a(5*n+2) = L(4*n+2)/3 - F(4*n),
a(5*n+3) = L(4*n-2)/3 + F(4*n),
a(5*n+4) = F(4*n+1) = A033889(n). (End)

A276530 a(n) = (a(n-1) * a(n-5) + a(n-3)^3) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 12, 39, 142, 1077, 21209, 779449, 106636837, 245010524697, 3336696488691229, 1125981890791313205482, 693480182652378523758257457499, 47660918720485535883730945247863294175948, 13387114027268508450553229985503810242341235794343085252
Offset: 0

Views

Author

Seiichi Manyama, Nov 16 2016

Keywords

Links

Crossrefs

Cf. A276529, A275173, A102276.

Programs

  • Mathematica
    RecurrenceTable[{a[n] == (a[n - 1] a[n - 5] + a[n - 3]^3)/a[n - 6], a[0] == a[1] == a[2] == a[3] == a[4] == a[5] == 1}, a, {n, 0, 21}] (* Michael De Vlieger, Nov 16 2016 *)
nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,(f b+d^3)/a}; NestList[nxt,{1,1,1,1,1,1},25][[;;,1]] (* Harvey P. Dale, Apr 21 2023 *)
  • Ruby
    def A(k, m, n)
  a = Array.new(2 * k, 1)
  ary = [1]
  while ary.size < n + 1
    i = a[-1] * a[1] + a[k] ** m
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A276530(n)
  A(3, 3, n)
end
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.