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.

A255983 a(n) = 1 for n <= 5; a(n) = 36*a(n-1) - 450*a(n-2) + 3000*a(n-3) - 11250*a(n-4) + 22500*a(n-5) - 18750*a(n-6) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, -4914, -181854, -4339944, -89153184, -1746815574, -33850986114, -655203251304, -12686085675144, -245683477042884, -4758284508073524, -92156792465163564, -1784855834560787004, -34568319709081645344, -669504074781304567584, -12966661247726595160224
Offset: 0

Views

Author

Alexander Samokrutov, Mar 13 2015

Keywords

Comments

a(n)/a(n-1) tends to 19.367561... = 6 + 6^(1/6) + 6^(2/6) + 6^(3/6) + 6^(4/6) + 6^(5/6), the largest real root of the polynomial x^6 - 36*x^5 + 450*x^4 - 3000*x^3 + 11250*x^2 - 22500*x + 18750.
In general, the polynomial x^6 - k6*x^5 - k5*x^4 - k4*x^3 - k3*x^2 - k2*x -k1 has a root r+b*m^(1/6)+c*m^(2/6)+d*m^(3/6)+g*m^(4/6)+h*m^(5/6), see links for coefficients k1, k2, k3, k4, k5, k6.

Links

Crossrefs

Cf. A247344, A255985.

Programs

  • Magma
    [n le 6 select 1 else 36*Self(n-1)-450*Self(n-2)+3000*Self(n-3)-11250*Self(n-4)+22500*Self(n-5)-18750*Self(n-6): n in [1..30]]; // Vincenzo Librandi, Mar 21 2015
  • Mathematica
    LinearRecurrence[{36, -450, 3000, -11250, 22500, -18750}, {1, 1, 1, 1, 1, 1}, 30] (* Vincenzo Librandi, Mar 21 2015 *)
  • PARI
    Vec(-(13835*x^5-8665*x^4+2585*x^3-415*x^2+35*x-1) / (18750*x^6-22500*x^5+11250*x^4-3000*x^3+450*x^2-36*x+1) + O(x^100)) \\ Colin Barker, Mar 23 2015

Formula

a(n) = 36*a(n-1) - 450*a(n-2) + 3000*a(n-3) - 11250*a(n-4) + 22500*a(n-5) - 18750*a(n-6).
G.f.: -(13835*x^5-8665*x^4+2585*x^3-415*x^2+35*x-1) / (18750*x^6-22500*x^5+11250*x^4-3000*x^3+450*x^2-36*x+1). - Colin Barker, Mar 23 2015

A255985 a(n) = 1 for n <= 6; a(n) = 49*a(n-1) - 882*a(n-2) + 8820*a(n-3) - 52920*a(n-4) + 190512*a(n-5) - 381024*a(n-6) + 326592*a(n-7) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 91147, 4557301, 143008075, 3791855893, 95039848267, 2350059062869, 58037421216523, 1434206075225749, 35454497256469963, 876533685507121621, 21670381641194181259, 535748905642908896533, 13245082208240954261323
Offset: 0

Views

Author

Alexander Samokrutov, Mar 13 2015

Keywords

Comments

a(n)/a(n-1) tends to 24.7225... = 7 + 7^(1/7) + 7^(2/7) + 7^(3/7) + 7^(4/7) + 7^(5/7) + 7^(6/7).
In general, the polynomial x^7 - k7*x^6 - k6*x^5 - k5*x^4 - k4*x^3 - k3*x^2 - k2*x - k1 has a root r + b*m^(1/7) + c*m^(2/7) + d*m^(3/7) + e*m^(4/7) + g*m^(5/7) + h*m^(6/7), see links for coefficients k1, k2, k3, k4, k5, k6, k7.

Links

Crossrefs

Cf. A247344, A255983.

Programs

  • Magma
    [n le 7 select 1 else 49*Self(n-1)-882*Self(n-2)+8820*Self(n-3)-52920*Self(n-4)+190512*Self(n-5) -381024*Self(n-6) +326592*Self(n-7): n in [1..30]]; // Vincenzo Librandi, Mar 21 2015
  • Mathematica
    LinearRecurrence[{49, -882, 8820, -52920, 190512, -381024, 326592}, {1, 1, 1, 1, 1, 1, 1}, 20] (* Vincenzo Librandi, Mar 21 2015 *)
  • PARI
    Vec(-(235446*x^6 -145578*x^5 +44934*x^4 -7986*x^3 +834*x^2 -48*x +1) / (326592*x^7 -381024*x^6 +190512*x^5 -52920*x^4 +8820*x^3 -882*x^2 +49*x -1) + O(x^100)) \\ Colin Barker, Mar 13 2015

Formula

a(n) = 49*a(n-1) -882*a(n-2) +8820*a(n-3) -52920*a(n-4) +190512*a(n-5) -381024*a(n-6) +326592*a(n-7).
G.f.: -(235446*x^6 -145578*x^5 +44934*x^4 -7986*x^3 +834*x^2 -48*x +1) / (326592*x^7 -381024*x^6 +190512*x^5 -52920*x^4 +8820*x^3 -882*x^2 +49*x -1). - Colin Barker, Mar 13 2015

A237988 a(n) = 3*a(n-3) + 3*a(n-6) + a(n-9) for n>8, a(0)=0, a(1)=a(2)=1, a(3)=a(4)=2, a(5)=3, a(6)=7, a(7)=9, a(8)=11.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 7, 9, 11, 27, 34, 43, 104, 131, 165, 400, 504, 635, 1539, 1939, 2443, 5921, 7460, 9399, 22780, 28701, 36161, 87642, 110422, 139123, 337187, 424829, 535251, 1297267, 1634454, 2059283, 4991004, 6288271, 7922725, 19202000, 24193004, 30481275, 73876279, 93078279, 117271283
Offset: 0

Views

Author

Alexander Samokrutov, May 01 2015

Keywords

Comments

These three sequences:
b(3n+3) = b(3n) + b(3n+1) + b(3n+2),
b(3n+4) = 2*b(3n) + b(3n+1) + b(3n+2),
b(3n+5) = 2*b(3n) + 2*b(3n+1) + b(3n+2),
give the polynomial x^3-3*x^2-3*x-1 with root 1 + 2^(1/3) + 2^(2/3). More generally, see link the roots of the equation of the third degree.
Equation: 4*x^3 - 6*x*y*z + 2*y^3 + z^3 = 3, if x = a(3n), y = a(3n+1), z = a(3n+2).

Links

Crossrefs

Cf. A247344, A255983, A255985.

Programs

  • Mathematica
    LinearRecurrence[{0, 0, 3, 0, 0, 3, 0, 0, 1}, {0, 1, 1, 2, 2, 3, 7, 9, 11}, 60] (* Vincenzo Librandi, May 15 2015 *)
CoefficientList[ Series[(x^8 - x^6 + x^4 - 2x^3 - x^2 - x)/(x^9 + 3x^6 + 3x^3 - 1), {x, 0, 44}], x] (* Robert G. Wilson v, Jul 24 2015 *)
  • PARI
    concat(0, Vec(x*(x^7-x^5+x^3-2*x^2-x-1)/(x^9+3*x^6+3*x^3-1) + O(x^100))) \\ Colin Barker, May 01 2015

Formula

G.f.: x*(x^7-x^5+x^3-2*x^2-x-1) / (x^9+3*x^6+3*x^3-1). - Colin Barker, May 01 2015

A238188 a(n) = 4*a(n-4) + 6*a(n-8) + 4*a(n-12) + a(n-16) for n>15, with the sixteen initial values as shown.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 9, 11, 13, 15, 48, 57, 68, 81, 254, 302, 359, 427, 1342, 1596, 1898, 2257, 7093, 8435, 10031, 11929, 37488, 44581, 53016, 63047, 198132, 235620, 280201, 333217, 1047170, 1245302, 1480922, 1761123, 5534517, 6581687, 7826989, 9307911, 29251104, 34785621, 41367308
Offset: 0

Views

Author

Alexander Samokrutov, May 01 2015

Keywords

Comments

These four sequences:
b(4n+4) = b(4n) + b(4n+1) + b(4n+2) + b(4n+3),
b(4n+5) = 2*b(4n) + b(4n+1) + b(4n+2) + b(4n+3),
b(4n+6) = 2*b(4n) + 2*b(4n+1) + b(4n+2) + b(4n+3),
b(4n+7) = 2*b(4n) + 2*b(4n+1) + 2*b(4n+2) + b(4n+3),
give the polynomial: x^4-4*x^3-6*x^2-4*x-1 with root 1 + 2^(1/4) + 2^(2/4) + 2^(3/4). More generally, see link the roots of the equation of the fourth degree.
Equation: 8*x^4-t^4-2*(-z^4+4*t*y*z^2-4*t^2*x*z-2*t^2*y^2)-4*(x^2*(2*z^2+4*t*y)-4*x*y^2*z+y^4) = (-1)^(ceiling(n/2)-floor(n/2)), if x = a(4n), y = a(4n+1), z = a(4n+2), t = a(4n+3).

Links

Crossrefs

Cf. A247344, A255983, A255985.

Programs

  • Mathematica
    LinearRecurrence[{0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1}, {0, 0, 1, 1, 2, 2, 2, 3, 9, 11, 13, 15, 48, 57, 68, 81}, 60] (* Vincenzo Librandi, May 15 2015 *)
  • PARI
    concat([0,0], Vec(x^2*(x^2+x+1)*(x^3-x^2+1)*(x^8-x^6+2*x^4-2*x^2-1) / (x^16+4*x^12+6*x^8+4*x^4-1) + O(x^100))) \\ Colin Barker, May 02 2015

Formula

G.f.: x^2*(x^2+x+1)*(x^3-x^2+1)*(x^8-x^6+2*x^4-2*x^2-1) / (x^16+4*x^12+6*x^8+4*x^4-1). - Colin Barker, May 02 2015

A247584 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 13, 43, 113, 253, 509, 969, 1849, 3719, 8009, 18027, 40897, 91257, 198697, 423777, 894081, 1886011, 4007301, 8594411, 18560081, 40181493, 86872293, 187197193, 402060793, 861827743, 1846685729, 3960390059, 8504658049, 18283290609, 39325827729
Offset: 0

Views

Author

Alexander Samokrutov, Sep 20 2014

Keywords

Comments

a(n)/a(n-1) tends to 2.1486... = 1 + 2^(1/5), the real root of the polynomial x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 3.
If x^5 = 2 and n >= 0, then there are unique integers a, b, c, d, g such that (1 + x)^n = a + b*x + c*x^2 + d*x^3 + g*x^4. The coefficient a is a(n) (from A052102). - Alexander Samokrutov, Jul 11 2015
If x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4) then x, y, z, s, t satisfies Diophantine equation (see link). - Alexander Samokrutov, Jul 11 2015

Links

Crossrefs

The following sequences belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A052101, A052102, A052103, A063727, A083098, A083099, A083100, A084057, A093406, A247344.
Cf. A005531.

Programs

  • Magma
    [n le 5 select 1 else 5*Self(n-1) -10*Self(n-2) +10*Self(n-3) -5*Self(n-4) +3*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jul 11 2015
  • Maple
    m:=50; S:=series( (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 15 2021
  • Mathematica
    LinearRecurrence[{5,-10,10,-5,3}, {1,1,1,1,1}, 50] (* Vincenzo Librandi, Jul 11 2015 *)
  • Maxima
    makelist(sum(2^k*binomial(n,5*k), k, 0, floor(n/5)), n, 0, 50); /* Alexander Samokrutov, Jul 11 2015 */
  • PARI
    Vec((1-x)^4/(1-5*x+10*x^2-10*x^3+5*x^4-3*x^5) + O(x^100)) \\ Colin Barker, Sep 22 2014
  • Sage
    [sum(2^j*binomial(n, 5*j) for j in (0..n//5)) for n in (0..50)] # G. C. Greubel, Apr 15 2021

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5).
a(n) = Sum_{k=0...floor(n/5)} (2^k*binomial(n,5*k)). - Alexander Samokrutov, Jul 11 2015
G.f.: (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5). - Colin Barker, Sep 22 2014
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