A269442 -id:A269442 - cp's OEIS frontend

A269527 a(n) = n^20 + n^15 + n^10 + n^5 + 1.

1, 5, 1082401, 3501192601, 1100586419201, 95397958987501, 3656628685521505, 79797014141614001, 1152956690052710401, 12157871353675866901, 100001000010000100001, 672754172206667010505, 3833775399530967321601, 19005014960911672392701, 83668410993669614565601

Offset: 0

Ilya Gutkovskiy, Feb 29 2016

a(n) = Phi_25(n) where Phi_k(x) is the k-th cyclotomic polynomial.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Cf. similar sequences of the type Phi_k(n) listed in A269442.

List([0..20], n-> Sum([0..4], j-> n^(5*j))); # G. C. Greubel, Apr 24 2019

[n^20+n^15+n^10+n^5+1: n in [0..20]]; // Vincenzo Librandi, Feb 29 2016

a:= n-> add(n^(5*j), j=0..4):
seq(a(n), n=0..18);  # Alois P. Heinz, Apr 24 2019

Table[Cyclotomic[25, n], {n, 0, 12}]
Join[{1},Table[Total[n^(5*Range[0,4])],{n,20}]] (* Harvey P. Dale, Sep 01 2016 *)

a(n) = polcyclo(25, n); \\ Michel Marcus, Feb 29 2016

[sum(n^(5*j) for j in (0..4)) for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 - 16*x + 1082506*x^2 + 3478461900*x^3 + 1027288678125*x^4 + 73019455046736*x^5 + 1879744586779368*x^6 + 21598535807594352*x^7 + 124747954270666290*x^8 + 388588494914595280*x^9 + 679562623971548876*x^10 + 679561811581541512*x^11 + 388588026599578578*x^12 + 124748409900243120*x^13 + 21598656801538440*x^14 + 1879672759361136*x^15 + 72997577685261*x^16 + 1025730080064*x^17 + 3451141690*x^18 + 1016780*x^19 + x^20)/(1 - x)^21.

Sum_{n>=0} 1/a(n) = 1.200000924158557891...

A270204 a(n) = n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1.

Original entry on oeis.org

1, 1, 3277, 478297, 15790321, 234750601, 2117950381, 13564461457, 67662254017, 278985273841, 990099009901, 3112703553961, 8854610100337, 23161037562937, 56406126018061, 129172239050401, 280379743338241, 580613195032417, 1153271900252557, 2207200789455481

Offset: 0

Ilya Gutkovskiy, Mar 13 2016

a(n) = Phi_28(n) where Phi_k(x) is the k-th cyclotomic polynomial.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. similar sequences of the type Phi_k(n) listed in A269442.

List([0..20], n-> Sum([0..6], j-> (-n^2)^j)); # G. C. Greubel, Apr 24 2019

[(&+[(-n^2)^j: j in [0..6]]): n in [0..20]]; // G. C. Greubel, Apr 24 2019

a:= n-> add((-n^2)^j, j=0..6):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 24 2019

Table[n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1, {n, 0, 17}]
Table[Cyclotomic[28, n], {n, 0, 17}]

a(n) = polcyclo(28, n); \\ Altug Alkan, Mar 13 2016

[sum((-n^2)^j for j in (0..6)) for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 - 12*x + 3342*x^2 + 435488*x^3 + 9828495*x^4 + 65845800*x^5 + 163388148*x^6 + 163386432*x^7 + 65847087*x^8 + 9827780*x^9 + 435774*x^10 + 3264*x^11 + x^12)/(1 - x)^13.

Sum_{n>=0} 1/a(n) = 2.000307316...

A326618 a(n) = n^18 + n^9 + 1.

Original entry on oeis.org

1, 3, 262657, 387440173, 68719738881, 3814699218751, 101559966746113, 1628413638264057, 18014398643699713, 150094635684419611, 1000000001000000001, 5559917315850179173, 26623333286045024257, 112455406962561892503, 426878854231297789441, 1477891880073843750001

Offset: 0

Richard N. Smith, Jul 15 2019

nonn

a(n) = Phi_27(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Richard N. Smith, Table of n, a(n) for n = 0..2000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument

Sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A269442 (k=17), A060891 (k=18), A269446 (k=19), A060892 (k=20), A269483 (k=21), A269486 (k=22), A060893 (k=24), A269527 (k=25), A266229 (k=26), this sequence (k=27), A270204 (k=28), A060894 (k=30), A060895 (k=32), A060896 (k=36).

Cf. A153440 (indices of prime terms).

[n^18+n^9+1: n in [0..17]]; // Vincenzo Librandi, Jul 15 2019

Table[n^18 + n^9 + 1, {n, 0, 17}] (* Vincenzo Librandi, Jul 15 2019 *)
Table[Cyclotomic[27, n], {n, 0, 17}]

a(n) = polcyclo(27, n); \\ Michel Marcus, Jul 20 2019

cp's OEIS Frontend

A269527 a(n) = n^20 + n^15 + n^10 + n^5 + 1.

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A270204 a(n) = n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A326618 a(n) = n^18 + n^9 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI