Alexander Samokrutov - cp's OEIS frontend

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.

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

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

Alexander Samokrutov, Mar 13 2015

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.

Alexander Samokrutov, Table of n, a(n) for n = 0..25
Alexander Samokrutov, Coefficients k1, k2, k3, k4, k5, k6
Index entries for linear recurrences with constant coefficients, signature (36,-450,3000,-11250,22500,-18750).

Cf. A247344, A255985.

[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

LinearRecurrence[{36, -450, 3000, -11250, 22500, -18750}, {1, 1, 1, 1, 1, 1}, 30] (* Vincenzo Librandi, Mar 21 2015 *)

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

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) = 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

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(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

Alexander Samokrutov, Sep 20 2014

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

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..48 from Alexander Samokrutov)
Alexander Samokrutov, Diophantine equation
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,3).

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.

[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

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

LinearRecurrence[{5,-10,10,-5,3}, {1,1,1,1,1}, 50] (* Vincenzo Librandi, Jul 11 2015 *)

makelist(sum(2^k*binomial(n,5*k), k, 0, floor(n/5)), n, 0, 50); /* Alexander Samokrutov, Jul 11 2015 */

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

[sum(2^j*binomial(n, 5*j) for j in (0..n//5)) for n in (0..50)] # G. C. Greubel, Apr 15 2021

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

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

User: Alexander Samokrutov

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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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.

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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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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.

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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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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.

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

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

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.