A127874 -id:A127874 - cp's OEIS frontend

A127873 a(n) = (n^3)/2 + (3n^2)/2 + 3n + 3.

8, 19, 39, 71, 118, 183, 269, 379, 516, 683, 883, 1119, 1394, 1711, 2073, 2483, 2944, 3459, 4031, 4663, 5358, 6119, 6949, 7851, 8828, 9883, 11019, 12239, 13546, 14943, 16433, 18019, 19704, 21491, 23383, 25383, 27494, 29719, 32061, 34523, 37108

Offset: 1

Artur Jasinski, Feb 04 2007

Generating polynomial is Schur's polynomial of degree 3. Schur's n-degree polynomials are the first n terms of the series expansion of the e^x function. All polynomials are irreducible and belong to the An alternating Galois transitive group if n is divisible by 4 or to the Sn symmetric Galois Group otherwise (proof: Schur, 1930).
Number of terms < 10^k: 0, 1, 4, 11, 26, 57, 124, 270, 583, 1258, 2713, 5847, 12598, 27143, 58479, ... - Muniru A Asiru, Jan 13 2018
For n > 1, a(n-1) is the number of ternary strings of length n that contain at most one 1 and at most two 2s. For example, for n=3, a(2)=19 since from the 27 ternary strings of length 3 we exclude 110 (3 of this type), 112 (3 of this type), 111 and 222. - Enrique Navarrete, Apr 16 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A127874.

A127873 := List([1..10^3],n->(n^3)/2+(3*n^2)/2+3*n+3); # Muniru A Asiru, Jan 13 2018

[(n^3)/2 + (3*n^2)/2 + 3*n + 3: n in [0..50]]; // Vincenzo Librandi, Jan 14 2018

A127873 := [seq((n^3)/2+(3*n^2)/2+3*n+3,n=1..10^3)]; # Muniru A Asiru, Jan 13 2018

Table[3 + 3 x + (3 x^2)/2 + x^3/2, {x, 41}]
Rest@ CoefficientList[ Series[-x (3 x^3 -11 x^2 +13 x - 8)/(x -1)^4, {x, 0, 41}], x] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {8, 19, 39, 71}, 41] (* Robert G. Wilson v, Jan 06 2018 *)

a(n)=n^3/2+3*n*(n+2)/2+3 \\ Charles R Greathouse IV, May 15 2013

def A127873(n): return (n*(n*(n+3)+6)>>1)+3 # Chai Wah Wu, Jul 12 2025

G.f.: x*(8-13*x+11*x^2-3*x^3)/(1-x)^4. - Colin Barker, Apr 17 2012

E.g.f.: (1+x+x^2/2)*(1+x)*exp(x) - 3x - 1. - Enrique Navarrete, Apr 16 2025

A127876 Integers of the form (x^3)/6 + (x^2)/2 + x + 1.

Original entry on oeis.org

1, 13, 61, 172, 373, 691, 1153, 1786, 2617, 3673, 4981, 6568, 8461, 10687, 13273, 16246, 19633, 23461, 27757, 32548, 37861, 43723, 50161, 57202, 64873, 73201, 82213, 91936, 102397, 113623, 125641, 138478, 152161, 166717, 182173, 198556, 215893, 234211

Offset: 1

Artur Jasinski, Feb 04 2007

Generating polynomial is Schur's polynomial of degree 3. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-2)=-coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 27 2010

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A127873, A127874, A127875.

Filtered(List([0..150],x->(x^3)/6+(x^2)/2+x+1),IsInt); # Muniru A Asiru, Apr 30 2018

[(9*n^3-18*n^2+15*n-4)/2: n in [1..30]]; // G. C. Greubel, Apr 29 2018

a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6], AppendTo[a, 1 + x + x^2/2 + x^3/6]], {x, 1, 300}]; a
Select[Table[x^3/6 + x^2/2 + x + 1, {x, 0, 200}], IntegerQ] (* Harvey P. Dale, Jan 06 2011 *)

Vec(x*(1+2*x)*(1+7*x+x^2)/(1-x)^4 + O(x^50)) \\ Colin Barker, May 15 2016

From Colin Barker, May 15 2016: (Start)
a(n) = (9*n^3-18*n^2+15*n-4)/2.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x*(1+2*x)*(1+7*x+x^2) / (1-x)^4.
(End)
E.g.f.: 2 + (9*x^3 + 9*x^2 + 6*x - 4)*exp(x)/2. - G. C. Greubel, Apr 29 2018

a(1) = 1 added by Harvey P. Dale, Jan 06 2011

A127875 Numbers x for which (x^3)/2+(3x^2)/2+3x+3 is prime.

Original entry on oeis.org

2, 4, 7, 8, 10, 11, 20, 23, 26, 28, 31, 34, 44, 50, 56, 62, 71, 74, 76, 79, 82, 83, 88, 91, 103, 104, 110, 112, 118, 122, 131, 134, 139, 140, 142, 148, 152, 163, 170, 175, 176, 179, 199, 202, 206, 226, 227, 235, 238, 239, 242, 244, 266, 271, 274, 278, 296, 299

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of degree 3. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

Cf. A127873, A127874.

a = {}; Do[If[PrimeQ[3 + 3 x + (3 x^2)/2 + x^3/2], AppendTo[a, x]], {x, 1, 300}]; a

A127882 Primes of the form 60*(x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1).

Original entry on oeis.org

163, 977611, 12294697, 37985853397, 49252877161, 137434331779, 830329719061, 1626105882361, 8060524420261, 11467771684597, 13008402510163, 15315610041211, 43633838254429, 71635442712061, 125119099806661

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of 5-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A127873, A127874, A127875, A127876, A127877, A127878, A127879, A127880, A127881, A127883.

Filtered(List([1..2000],x->60*(x^5/120+x^4/24+x^3/6+x^2/2+x+1)),IsPrime); # Muniru A Asiru, Apr 30 2018

select(isprime,[seq(60*(x^5/120+x^4/24+x^3/6+x^2/2+x+1),x=1..2000)]); # Muniru A Asiru, Apr 30 2018

a = {}; Do[If[PrimeQ[60 + 60*x + 30*x^2 + 10*x^3 + (5*x^4)/2 + x^5/2], AppendTo[a, 60 + 60*x + 30*x^2 + 10*x^3 + (5*x^4)/2 + x^5/2]], {x, 1, 1000}]; a

A127877 Integers of the form (x^4)/24 + (x^3)/6 + (x^2)/2 + x + 1 with x > 0.

Original entry on oeis.org

7, 115, 297, 1237, 2171, 5527, 8221, 16441, 22335, 38731, 49697, 78445, 96787, 142927, 171381, 240817, 282551, 382051, 440665, 577861, 657387, 840775, 945677, 1184617, 1319791, 1624507, 1795281, 2176861, 2388995, 2859391, 3119077, 3691105, 4004967, 4692307

Offset: 1

Artur Jasinski, Feb 04 2007

Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A127873, A127874, A127875, A127876, A127878, A127879, A127880, A127881, A127882, A127883.

Filtered(List([0..150],x->(x^4)/24+(x^3)/6+(x^2)/2+x+1),IsInt); # Muniru A Asiru, Apr 30 2018

[(11 +5*(-1)^n +16*(2+(-1)^n)*n +18*(3+(-1)^n)*n^2 +36*(1+(-1)^n)*n^3 +54*n^4)/16: n in [1..30]]; // G. C. Greubel, Apr 29 2018

a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6 + x^4/24], AppendTo[a, 1 + x + x^2/2 + x^3/6 + x^4/24]], {x, 1, 100}]; a
Select[Table[(x^4)/24+(x^3)/6+(x^2)/2+x+1,{x,100}],IntegerQ] (* Harvey P. Dale, Aug 14 2012 *)

Vec(x*(7+108*x+154*x^2+508*x^3+248*x^4+244*x^5+22*x^6+4*x^7+x^8)/((1-x)^5*(1+x)^4) + O(x^50)) \\ Colin Barker, May 15 2016

From Colin Barker, May 15 2016: (Start)
a(n) = (11 +5*(-1)^n +16*(2+(-1)^n)*n +18*(3+(-1)^n)*n^2 +36*(1+(-1)^n)*n^3 +54*n^4)/16.
a(n) = (27*n^4+36*n^3+36*n^2+24*n+8)/8 for n even.
a(n) = (27*n^4+18*n^2+8*n+3)/8 for n odd.
a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9) for n>9.
G.f.: x*(7+108*x+154*x^2+508*x^3+248*x^4+244*x^5+22*x^6+4*x^7+x^8) / ((1-x)^5*(1+x)^4).
(End)

A127879 Primes of the form x^4 + 4x^3 + 12x^2 + 24*x + 24.

Original entry on oeis.org

3760073, 9853769, 117051593, 181145609, 2517933833, 8999750153, 10486376969, 20852229449, 26640445193, 56713997513, 65555973569, 136653695753, 172008443273, 262819256009, 330127243553, 340704528713, 362619554249

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A127873, A127874, A127875, A127876, A127877, A127878, A127880, A127881, A127882, A127883.

Filtered(List([1..2000],x->x^4+4*x^3+12*x^2+24*x+24),IsPrime); # Muniru A Asiru, Apr 30 2018

select(isprime,[seq(x^4+4*x^3+12*x^2+24*x+24,x=1..2000)]); # Muniru A Asiru, Apr 30 2018

a = {}; Do[If[PrimeQ[24 + 24 x + 12 x^2 + 4 x^3 + x^4], AppendTo[a, 24 + 24 x + 12 x^2 + 4 x^3 + x^4]], {x, 1, 1000}]; a
Select[Table[x^4+4x^3+12x^2+24x+24,{x,780}],PrimeQ[#]&] (* Harvey P. Dale, Jan 24 2013 *)

A127880 Numbers x for which x^4 + 4x^3 + 12x^2 + 24x + 24 is prime.

Original entry on oeis.org

43, 55, 103, 115, 223, 307, 319, 379, 403, 487, 505, 607, 643, 715, 757, 763, 775, 799, 883, 925, 979, 1063, 1069, 1135, 1147, 1165, 1189, 1279, 1309, 1369, 1543, 1567, 1585, 1627, 1693, 1729, 1783, 1813, 1819, 1855, 1903, 1939, 1945, 2083, 2149, 2155

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of 4-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A127873, A127874, A127875, A127876, A127877, A127878, A127879, A127881, A127882, A127883.

Filtered([1..3000],x->IsPrime(x^4+4*x^3+12*x^2+24*x+24)); # Muniru A Asiru, Apr 30 2018

select(x->isprime(x^4+4*x^3+12*x^2+24*x+24),[$1..3000]); # Muniru A Asiru, Apr 30 2018

a = {}; Do[If[PrimeQ[24 + 24 x + 12 x^2 + 4 x^3 + x^4], AppendTo[a, x]], {x, 1, 1000}]; a

isok(x) = isprime(x^4 + 4*x^3 + 12*x^2 + 24*x + 24); \\ Michel Marcus, Apr 30 2018

A127881 Integers of the form x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1 with x > 0.

Original entry on oeis.org

241231, 7057861, 21166951, 52066891, 216295321, 654480151, 1619368381, 2411089396, 3486017011, 6776093041, 12182173471, 20592045301, 26260194241, 33113005531, 51096161161, 76160729191, 110218336621, 131302849486

Offset: 1

Artur Jasinski, Feb 04 2007

nonn

Generating polynomial is Schur's polynomial of 5-degree. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A127873, A127874, A127875, A127876, A127877, A127878, A127879, A127880, A127882, A127883, A127884.

a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120], AppendTo[a, 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120]], {x, 1, 1000}]; a
Select[Table[ x^5/120+x^4/24+x^3/6+x^2/2+x+1,{x,450}],IntegerQ] (* Harvey P. Dale, Jan 20 2019 *)

for(x=1,500,y=x^5+5*x^4+20*x^3+60*x^2+120*x+120;if(y%120==0,print1(y/120, ", "))) \\ Michael B. Porter, Jan 29 2010

isA127881(n)={local(r);r=0;fordiv(120*n-120,x,if(x^5/120+x^4/24+x^3/6+x^2/2+x+1==n,r=1));r} \\ Michael B. Porter, Jan 29 2010

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A127873 a(n) = (n^3)/2 + (3*n^2)/2 + 3*n + 3.

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A127876 Integers of the form (x^3)/6 + (x^2)/2 + x + 1.

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A127875 Numbers x for which (x^3)/2+(3x^2)/2+3x+3 is prime.

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A127882 Primes of the form 60*(x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1).

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A127877 Integers of the form (x^4)/24 + (x^3)/6 + (x^2)/2 + x + 1 with x > 0.

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A127879 Primes of the form x^4 + 4*x^3 + 12*x^2 + 24*x + 24.

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A127880 Numbers x for which x^4 + 4x^3 + 12x^2 + 24x + 24 is prime.

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A127873 a(n) = (n^3)/2 + (3n^2)/2 + 3n + 3.

A127879 Primes of the form x^4 + 4x^3 + 12x^2 + 24*x + 24.