A255983 -id:A255983 - cp's OEIS frontend

A247344 a(n) = 1 for n <= 4; a(n) = 25a(n-1) - 200a(n-2) + 800a(n-3) - 1600a(n-4) + 1280*a(n-5) otherwise.

1, 1, 1, 1, 1, 305, 7905, 137105, 2090305, 30673905, 446213025, 6483539025, 94216001025, 1369259983025, 19900452349025, 289229603172625, 4203610924242625, 61094494859232625, 887935798190222625

Offset: 0

Alexander Samokrutov, Sep 14 2014

a(n)/a(n-1) tends to 14.5338... = 5 + 5^(1/5)+5^(2/5)+5^(3/5)+5^(4/5) = 4/(1-5^(-1/5)), the real root of the polynomial x^5 - 25*x^4 + 200*x^3 - 800*x^2 + 1600*x - 1280.

In general, the polynomial x^5 - k5*x^4 - k4*x^3 - k3*x^2 - k2*x - k1 has a root r+b*m^(1/5)+c*m^(2/5)+d*m^(3/5)+g*m^(4/5), see links for coefficients k1, k2, k3, k4, k5.

Alexander Samokrutov, Table of n, a(n) for n = 0..22
Alexander Samokrutov, Coefficients k1, k2, k3, k4, k5
Index entries for linear recurrences with constant coefficients, signature (25,-200,800,-1600,1280).

Cf. A255985, A255983.

[n le 5 select 1 else 25*Self(n-1)-200*Self(n-2)+800*Self(n-3)-1600*Self(n-4)+1280*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Nov 19 2014

CoefficientList[Series[(976 x^4 - 624 x^3 + 176 x^2 - 24 x + 1) / (-1280 x^5 + 1600 x^4 - 800 x^3 + 200 x^2 - 25 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 19 2014 *)

Vec( (976*x^4 - 624*x^3 + 176*x^2 - 24*x + 1)/(-1280*x^5 + 1600*x^4 - 800*x^3 + 200*x^2 - 25*x + 1) + O(x^66) ) \\ Joerg Arndt, Sep 14 2014

a(n) = 25*a(n-1)-200*a(n-2)+800*a(n-3)-1600*a(n-4)+1280*a(n-5).

G.f.: (976*x^4 - 624*x^3 + 176*x^2 - 24*x + 1)/(-1280*x^5 + 1600*x^4 - 800*x^3 + 200*x^2 - 25*x + 1).

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

Alexander Samokrutov, Mar 13 2015

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.

Colin Barker, Table of n, a(n) for n = 0..720
Alexander Samokrutov, Coefficients k1, k2, k3, k4, k5, k6, k7
Index entries for linear recurrences with constant coefficients, signature (49,-882,8820,-52920,190512,-381024,326592).

Cf. A247344, A255983.

[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

LinearRecurrence[{49, -882, 8820, -52920, 190512, -381024, 326592}, {1, 1, 1, 1, 1, 1, 1}, 20] (* Vincenzo Librandi, Mar 21 2015 *)

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

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) = 3a(n-3) + 3a(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

Alexander Samokrutov, May 01 2015

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).

Alexander Samokrutov, Table of n, a(n) for n = 0..83
Alexander Samokrutov, The roots of the equation of the third degree
Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, 3, 0, 0, 1).

Cf. A247344, A255983, A255985.

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 *)

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

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) = 4a(n-4) + 6a(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

Alexander Samokrutov, May 01 2015

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).

Alexander Samokrutov, Table of n, a(n) for n = 0..91
Alexander Samokrutov, The roots of the equation of the fourth degree
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,6,0,0,0,4,0,0,0,1).

Cf. A247344, A255983, A255985.

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 *)

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

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

cp's OEIS Frontend

A247344 a(n) = 1 for n <= 4; a(n) = 25*a(n-1) - 200*a(n-2) + 800*a(n-3) - 1600*a(n-4) + 1280*a(n-5) otherwise.

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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.

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Formula

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.

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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.

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Formula

A247344 a(n) = 1 for n <= 4; a(n) = 25a(n-1) - 200a(n-2) + 800a(n-3) - 1600a(n-4) + 1280*a(n-5) otherwise.

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

A237988 a(n) = 3a(n-3) + 3a(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.

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