A056578 -id:A056578 - cp's OEIS frontend

A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

1, 3, 57, 319, 1065, 2691, 5713, 10767, 18609, 30115, 46281, 68223, 97177, 134499, 181665, 240271, 312033, 398787, 502489, 625215, 769161, 936643, 1130097, 1352079, 1605265, 1892451, 2216553, 2580607, 2987769, 3441315

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A084849, A056109, A056105, A056578, A056579.

Table[5n^4-4n^3+3n^2-2n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,57,319,1065},30] (* Harvey P. Dale, Nov 01 2020 *)

a(n)=5*n^4-4*n^3+3*n^2-2*n+1 \\ Charles R Greathouse IV, Oct 21 2022

From Chai Wah Wu, Nov 13 2018: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-15*x^4 - 54*x^3 - 52*x^2 + 2*x - 1)/(x - 1)^5. (End)

A268644 a(n) = 4n^3 - 3n^2 - 2*n - 1.

Original entry on oeis.org

-1, -2, 15, 74, 199, 414, 743, 1210, 1839, 2654, 3679, 4938, 6455, 8254, 10359, 12794, 15583, 18750, 22319, 26314, 30759, 35678, 41095, 47034, 53519, 60574, 68223, 76490, 85399, 94974, 105239, 116218, 127935, 140414, 153679, 167754, 182663, 198430, 215079, 232634, 251119

Offset: 0

Ilya Gutkovskiy, Feb 09 2016

In general, the ordinary generating function for the values of cubic polynomial p*n^3 + q*n^2 + k*n + m is (m + (p + q + k - 3*m)*x + (4*p - 2*k + 3*m)*x^2 + (p - q + k - m)*x^3)/(1 - x)^4.
Primes in this sequence: 199, 743, 15583, 105239, 435359, 620999, 770239, 853079, 1738423, 3511103, 7580119, 8737039, 10006063, ...
If a(n) is a positive prime then n is congruent to 0 or 4 (mod 6).

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of cubic polynomialK
Eric Weisstein's World of Mathematics, Cubic Polynomial
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408, A056578, A103532, A131464.

[4*n^3-3*n^2-2*n-1: n in [0..40]]; // Vincenzo Librandi, Feb 10 2016

Table[4 n^3 - 3 n^2 - 2 n - 1, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {-1, -2, 15, 74}, 41]
CoefficientList[Series[(-1 + 2 x + 17 x^2 + 6 x^3) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 10 2016 *)

a(n)=4*n^3-3*n^2-2*n-1 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: (-1 + 2*x + 17*x^2 + 6*x^3)/(1 - x)^4.
a(n) = A103532(n - 1) - A005408(n), n>0.
a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4).
Sum_{n>=0} 1/a(n) = -1.407823506818026589265...
E.g.f.: exp(x)*(-1 - x + 9*x^2 + 4*x^3). - Stefano Spezia, Nov 17 2024

A113618 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8*n^7.

Original entry on oeis.org

1, 36, 1793, 24604, 167481, 756836, 2620201, 7526268, 18831569, 42374116, 87654321, 169343516, 309160393, 538155684, 899445401, 1451432956, 2271560481, 3460629668, 5147732449, 7495831836, 10708033241, 15034586596, 20780659593

Offset: 0

Jonathan Vos Post, Jan 14 2006

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 + 8*n^7 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 = (x^9 - 1)/(x-1).

			1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 = 18831569 = 173 * 199 * 547.
1 + 2*26 + 3*26^2 + 4*26^3 + 5*26^4 + 6*26^5 + 7*26^6 + 8*26^7 = 66490537361 is prime, the smallest prime in the sequence.

Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A000012, A005408, A056109, A056578, A056579.

[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7: n in [1..43]] // Vincenzo Librandi, Dec 21 2010

Join[{1},Table[Total[Table[p*n^(p-1),{p,8}]],{n,30}]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,36,1793,24604,167481,756836,2620201,7526268},30] (* Harvey P. Dale, Jul 16 2014 *)

G.f.: (1+28*x+1533*x^2+11212*x^3+18907*x^4+7956*x^5+679*x^6+4*x^7)/(x-1)^8. - R. J. Mathar, Dec 21 2010

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=1, a(1)=36, a(2)=1793, a(3)=24604, a(4)=167481, a(5)=756836, a(6)=2620201, a(7)=7526268. - Harvey P. Dale, Jul 16 2014

A113632 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8 + 10*n^9.

Original entry on oeis.org

1, 55, 9217, 280483, 3378745, 23803711, 118513705, 462945547, 1512003793, 4303999495, 10987654321, 25678050355, 55776799177, 113924725903, 220792014745, 408951042331, 728121033505, 1252121211607, 2087920281313

Offset: 0

Jonathan Vos Post, Jan 14 2006

1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 = (x^11 - 1)/(x-1).

			a(5) = 1 + 2*5 + 3*5^2 + 4*5^3 + 5*5^4 + 6*5^5 + 7*5^6 + 8*5^7 + 9*5^8 + 10*5^9 = 23803711 is prime.
a(30) = 1 + 2*30 + 3*30^2 + 4*30^3 + 5*30^4 + 6*30^5 + 7*30^6 + 8*30^7 + 9*30^8 + 10*30^9 = 202915112960761 is prime.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A056578, A056579.

With[{eq=Total[Range[10](n^Range[0,9])]},Table[eq,{n,0,20}]] (* Harvey P. Dale, Mar 14 2011 *)

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8 + 10*n^9.

G.f.: (1+x*(45+x*(8712+x*(190668+x*(982290+x*(1543254+x*(784080+x*(116268+x*(3477+5*x)))))))))/(x-1)^10. - Harvey P. Dale, Mar 14 2011

cp's OEIS Frontend

A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

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A268644 a(n) = 4*n^3 - 3*n^2 - 2*n - 1.

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A113618 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7.

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A113632 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8 + 10*n^9.

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A268644 a(n) = 4n^3 - 3n^2 - 2*n - 1.

A113618 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8*n^7.

A113632 a(n) = 1 + 2n + 3n^2 + 4n^3 + 5n^4 + 6n^5 + 7n^6 + 8n^7 + 9n^8 + 10*n^9.