A269442 -id:A269442 - cp's OEIS frontend

A060885 a(n) = Sum_{j=0..10} n^j.

1, 11, 2047, 88573, 1398101, 12207031, 72559411, 329554457, 1227133513, 3922632451, 11111111111, 28531167061, 67546215517, 149346699503, 311505013051, 617839704241, 1172812402961, 2141993519227, 3780494710543, 6471681049901, 10778947368421, 17513875027111, 27824681019587

Offset: 0

N. J. A. Sloane, May 05 2001

a(n) = Phi_11(n), where Phi_k is the k-th cyclotomic polynomial.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are:

A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), this sequence (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..10]]): n in [0..20]]; // G. C. Greubel, Apr 15 2019

A060885 := proc(n)
        numtheory[cyclotomic](11,n) ;
end proc:
seq(A060885(n),n=0..20) ; # R. J. Mathar, Feb 07 2014

Join[{1},Table[Total[n^Range[0,10]],{n,20}]] (* Harvey P. Dale, Jun 19 2011 *)

a(n) = n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 \\ Harry J. Smith, Jul 14 2009

a(n) = polcyclo(11, n); \\ Michel Marcus, Apr 06 2016

[sum(n^j for j in (0..10)) for n in (0..20)] # G. C. Greubel, Apr 15 2019

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

G.f.: (1+x^2*(1981+x*(66496+x*(534898+x*(1364848+x*(1233970+ x*(389104+x*(36829+x*(672+x)))))))))/(1-x)^11. - Harvey P. Dale, Jun 19 2011

A102909 a(n) = Sum_{j=0..8} n^j.

Original entry on oeis.org

1, 9, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111, 235794769, 469070941, 883708281, 1589311291, 2745954241, 4581298449, 7411742281, 11668193551, 17927094321, 26947368421, 39714002329, 57489010371, 81870575521, 114861197401

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Mar 01 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A001016, A060890, A059839.

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), this sequence (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..8]]): n in [0..30]]; // G. C. Greubel, Feb 13 2018

1 + Sum[Range[0, 30]^j, {j, 1, 8}] (* G. C. Greubel, Feb 13 2018 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,9,511,9841,87381,488281,2015539,6725601,19173961},30] (* Harvey P. Dale, Feb 01 2025 *)

a(n)=n^8+n^7+n^6+n^5+n^4+n^3+n^2+n+1 \\ Charles R Greathouse IV, Oct 07 2015

[sum(n^j for j in (0..8)) for n in (0..30)] # G. C. Greubel, Apr 14 2019

a(n) = (n^2+n+1) * (n^6+n^3+1) and so is never prime. - Jonathan Vos Post, Dec 21 2012
G.f.: (x^8 + 162*x^7 + 3418*x^6 + 14212*x^5 + 16578*x^4 + 5482*x^3 + 466*x^2 + 1)/(1-x)^9. - Colin Barker, Nov 05 2012, edited by M. F. Hasler, Dec 31 2012
a(n) = (n^9-1)/(n-1) with a(1) = 9. - L. Edson Jeffery and M. F. Hasler, Dec 30 2012
E.g.f.: exp(x)*(1 + 8*x + 247*x^2 + 1389*x^3 + 2127*x^4 + 1206*x^5 + 288*x^6 + 29*x^7 + x^8). - Stefano Spezia, Oct 03 2024

Offset corrected by N. J. A. Sloane, Dec 30 2012

A269446 a(n) = n(n^6 + n^3 + 1)(n^6 - n^3 + 1)(n^2 + n + 1)(n^2 - n + 1)*(n + 1) + 1.

Original entry on oeis.org

1, 19, 524287, 581130733, 91625968981, 4768371582031, 121871948002099, 1899815864228857, 20587884010836553, 168856464709124011, 1111111111111111111, 6115909044841454629, 29043636306420266077, 121826690864620509223, 459715689149916492091, 1583455585752214704241

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

a(n) = Phi_19(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 (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

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

List([0..20], n-> n*(n^6+n^3+1)*(n^6-n^3+1)*(n^2+n+1)*(n^2-n+1)*(n+1)+1); # G. C. Greubel, Apr 24 2019

[n*(n^6+n^3+1)*(n^6-n^3+1)*(n^2+n+1)*(n^2-n+1)*(n+1)+1: n in [0..20]]; // G. C. Greubel, Apr 24 2019

Mathematica
```
Table[Cyclotomic[19, n], {n, 0, 15}]
```

a(n)=n*(n^6+n^3+1)*(n^6-n^3+1)*(n^2+n+1)*(n^2-n+1)*(n+1)+1 \\ Charles R Greathouse IV, Jul 26 2016

[n*(n^6+n^3+1)*(n^6-n^3+1)*(n^2+n+1)*(n^2-n+1)*(n+1)+1 for n in (0..20)] # G. C. Greubel, Apr 24 2019

Sum_{n>=0} 1/a(n) = 1.0526334880315548541801483535546024...

A104376 a(n) = Sum_{j=0..13} n^j.

Original entry on oeis.org

1, 14, 16383, 2391484, 89478485, 1525878906, 15672832819, 113037178808, 628292358729, 2859599056870, 11111111111111, 37974983358324, 116719860413533, 328114698808274, 854769755812155, 2085209001813616, 4803839602528529, 10523614159962558, 22047845151887119

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 16 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), this sequence (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..13]]): n in [0..20]]; // Vincenzo Librandi, May 01 2011

Table[1+Sum[n^j, {j,1,13}], {n,0,20}] (* G. C. Greubel, Apr 14 2019 *)
LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,14,16383,2391484,89478485,1525878906,15672832819,113037178808,628292358729,2859599056870,11111111111111,37974983358324,116719860413533,328114698808274},20] (* Harvey P. Dale, Sep 04 2023 *)

a(n)=sum(j=0,13, n^j) \\ Charles R Greathouse IV, Oct 07 2015

[sum(n^j for j in (0..13)) for n in (0..20)] # G. C. Greubel, Apr 14 2019

a(n) = n^13 + n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n^1 + 1.

G.f.: (5461*x^12 + 1119288*x^11 + 37443654*x^10 + 372458048*x^9 + 1409085783*x^8 + 2263446576*x^7 + 1598944452*x^6 + 484853760*x^5 + 57484467*x^4 + 2163032*x^3 + 16278*x^2 + 1)/(1-x)^14. - Colin Barker, Nov 04 2012

Name changed by G. C. Greubel, Apr 14 2019

A104682 a(n) = Sum_{j=0..14} n^j.

Original entry on oeis.org

1, 15, 32767, 7174453, 357913941, 7629394531, 94036996915, 791260251657, 5026338869833, 25736391511831, 111111111111111, 417724816941565, 1400638324962397, 4265491084507563, 11966776581370171, 31278135027204241, 76861433640456465, 178901440719363487

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 22 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), this sequence (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..14]]): n in [0..20]]; // Vincenzo Librandi, May 01 2011

With[{f=Total[n^Range[0,14]]},Table[f,{n,0,20}]] (* Harvey P. Dale, Jun 11 2011 *)

a(n) = sum(j=0, 14, n^j) \\ Charles R Greathouse IV, Oct 07 2015

[sum(n^j for j in (0..14)) for n in (0..20)] # G. C. Greubel, Apr 15 2019

a(n) = n^14 + n^13 + n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n^1 + 1.
a(n) = (n^2 + n + 1) * (n^4 + n^3 + n^2 + n + 1) * (n^8 - n^7 + n^5 - n^4 + n^3 - n + 1). - Jonathan Vos Post, Apr 23 2005
G.f.: (x^14 +10908*x^13 +3423487*x^12 +162086420*x^11 +2236727781*x^10 +11806635128*x^9 +27116815299*x^8 +28635678216*x^7 +13957353555*x^6 +2999111468*x^5 +253732221*x^4 +6684068*x^3 +32647*x^2 +1)/(1-x)^15. - Colin Barker, Nov 04 2012

More terms from Harvey P. Dale, Jun 11 2011

Name changed by G. C. Greubel, Apr 15 2019

A105067 a(n) = Sum_{j=0..11} n^j.

Original entry on oeis.org

1, 12, 4095, 265720, 5592405, 61035156, 435356467, 2306881200, 9817068105, 35303692060, 111111111111, 313842837672, 810554586205, 1941507093540, 4361070182715, 9267595563616, 18764998447377, 36413889826860

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 05 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), this sequence (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1: n in [0..20]]; // Vincenzo Librandi, May 01 2011

1+Sum[Range[0,20]^j, {j,1,11}] (* G. C. Greubel, Apr 13 2019 *)

a(n)=polcyclo(11,n)+n^11 \\ Charles R Greathouse IV, Sep 03 2011

[sum(n^j for j in (0..11)) for n in (0..20)] # G. C. Greubel, Apr 13 2019

Factorization of the polynomial into irreducible components over integers: n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 = +- (n + 1) * (n^2 - n + 1) * (n^2 + 1) * (n^2 + n + 1) * (n^4 - n^2 + 1). - Jonathan Vos Post, Apr 06 2005

G.f.: (1365*x^10 + 116480*x^9 + 1851213*x^8 + 8893248*x^7 + 15593370*x^6 + 10568064*x^5 + 2671890*x^4 + 217152*x^3 + 4017*x^2 + 1)/(x - 1)^12. - Colin Barker, Oct 29 2012

Signature changed by Georg Fischer, Apr 13 2019

A105312 a(n) = Sum_{j=0..15} n^j.

Original entry on oeis.org

1, 16, 65535, 21523360, 1431655765, 38146972656, 564221981491, 5538821761600, 40210710958665, 231627523606480, 1111111111111111, 4594972986357216, 16807659899548765, 55451384098598320, 167534872139182395, 469172025408063616, 1229782938247303441

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Apr 30 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are: A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), A060885 (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), this sequence (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..15]]): n in [0..20]]; // Vincenzo Librandi, May 01 2011 (modified by G. C. Greubel, Apr 14 2019)

a:= n-> add(n^k, k=0..15):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 04 2012

Prepend[Table[Total[n^Range[0,15]],{n,20}],1]  (* Harvey P. Dale, Jan 19 2011 *)

vector(20, n, n--; sum(j=0,15, n^j)) \\ G. C. Greubel, Apr 14 2019

[sum(n^j for j in (0..15)) for n in (0..20)] # G. C. Greubel, Apr 14 2019

a(n) = n^15 + n^14 + n^13 + n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n^1 + 1.

G.f.: (21845*x^14 + 10412160*x^13 + 689427979*x^12 + 12966588160*x^11 + 93207091581*x^10 + 296077418240*x^9 + 446019954555*x^8 + 326065923072*x^7 + 113735241015*x^6 + 17786608768*x^5 + 1095139065*x^4 + 20476160*x^3 + 65399*x^2 +1 )/(x-1)^16. - Colin Barker, Nov 04 2012

More terms from Harvey P. Dale, Jan 19 2011

Name changed by G. C. Greubel, Apr 14 2019

A266229 a(n) = Sum_{j=0..12} (-n)^j.

Original entry on oeis.org

1, 1, 2731, 398581, 13421773, 203450521, 1865813431, 12111126301, 61083979321, 254186582833, 909090909091, 2876892678661, 8230246567621, 21633936185161, 52914318216943, 121637191772461, 264917625139441, 550254335161441

Offset: 0

Ilya Gutkovskiy, Mar 13 2016

a(n) = Phi_26(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..12], j-> (-n)^j)); # G. C. Greubel, Apr 24 2019

[(&+[(-n)^j: j in [0..12]]): n in [0..20]]; // G. C. Greubel, Apr 24 2019

Table[n^12-n^11+n^10-n^9+n^8-n^7+n^6-n^5+n^4-n^3+n^2-n+1, {n, 0, 17}]
Table[Cyclotomic[26, n], {n, 0, 17}]

a(n) = polcyclo(26, n); \\ Michel Marcus, Mar 13 2016

[sum((-n)^j for j in (0..12)) for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 - 12*x + 2796*x^2 + 362870*x^3 + 8453667*x^4 + 59275152*x^5 + 155813880*x^6 + 167535876*x^7 + 74215935*x^8 + 12641708*x^9 + 691692*x^10 + 8022*x^11 + 13*x^12)/(1 - x)^13.

Sum_{n>=0} 1/a(n) = 2.0003687552...

Name changed by G. C. Greubel, Apr 24 2019

A269483 a(n) = n^12 - n^11 + n^9 - n^8 + n^6 - n^4 + n^3 - n + 1.

Original entry on oeis.org

1, 1, 2359, 368089, 12783421, 196890121, 1822428931, 11898664849, 60247241209, 251393376241, 900900990991, 2855262053161, 8177824843189, 21515718297529, 52663539957211, 121132473843361, 263947231891441, 548461977100129

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

a(n) = Phi_21(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-> n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1); # G. C. Greubel, Apr 24 2019

[n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1: n in [0..20]]; // Vincenzo Librandi, Feb 28 2016

Table[Cyclotomic[21, n], {n, 0, 17}]
CoefficientList[Series[(1 -12x +2424x^2 +337214x^3 +8182695x^4 +58741344 x^5 +156377856x^6 +168607380x^7 +73943271x^8 +12191420x^9 + 612600 x^10 +5406x^11 +x^12)/(1-x)^13, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 28 2016 *)

a(n) = polcyclo(21, n); \\ Michel Marcus, Feb 29 2016

A269483_list, m = [], [479001600, -2674425600, 6386688000, -8501915520, 6889478400, -3482100720, 1080164160, -194177280, 17948256, -666714, 5418, 0, 1]
for _ in range(10**2):
    A269483_list.append(m[-1])
    for i in range(12):
        m[i+1] += m[i] # Chai Wah Wu, Feb 28 2016

[n^12-n^11+n^9-n^8+n^6-n^4+n^3-n+1 for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 - 12*x + 2424*x^2 + 337214*x^3 + 8182695*x^4 + 58741344*x^5 + 156377856*x^6 + 168607380*x^7 + 73943271*x^8 + 12191420*x^9 + 612600*x^10 + 5406*x^11 + x^12)/(1-x)^13.

Sum_{n>=0} 1/a(n) = 2.00042670913...

A269486 a(n) = Sum_{j=0..10} (-n)^j.

Original entry on oeis.org

1, 1, 683, 44287, 838861, 8138021, 51828151, 247165843, 954437177, 3138105961, 9090909091, 23775972551, 57154490053, 128011456717, 269971011311, 540609741211, 1034834473201, 1903994239313, 3382547898907, 5824512944911

Offset: 0

Ilya Gutkovskiy, Feb 28 2016

a(n) = Phi_22(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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

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

List([0..20], n-> Sum([0..10], j-> (-n)^j)); # G. C. Greubel, Apr 24 2019

[n^10-n^9+n^8-n^7+n^6-n^5+n^4-n^3+n^2-n+1: n in [0..30]]; // Vincenzo Librandi, Feb 29 2016

Table[Cyclotomic[22, n], {n, 0, 19}]
CoefficientList[Series[(1 -10x +727x^2 +36664x^3 +389434x^4 +1233508x^5 + 1365310 x^6 +534568x^7 +66661x^8 +1926x^9 +11x^10)/(1-x)^11, {x,0,33}], x] (* Vincenzo Librandi, Feb 29 2016 *)

a(n) = polcyclo(22, n); \\ Michel Marcus, Feb 28 2016

[sum((-n)^j for j in (0..10)) for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 - 10*x + 727*x^2 + 36664*x^3 + 389434*x^4 + 1233508*x^5 + 1365310*x^6 + 534568*x^7 + 66661*x^8 + 1926*x^9 + 11*x^10)/(1-x)^11. - Vincenzo Librandi, Feb 29 2016
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11).
Sum_{n>=0} 1/a(n) = 2.0014880486975...

Name changed by G. C. Greubel, Apr 24 2019

cp's OEIS Frontend

A060885 a(n) = Sum_{j=0..10} n^j.

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A102909 a(n) = Sum_{j=0..8} n^j.

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A269446 a(n) = n*(n^6 + n^3 + 1)*(n^6 - n^3 + 1)*(n^2 + n + 1)*(n^2 - n + 1)*(n + 1) + 1.

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A104682 a(n) = Sum_{j=0..14} n^j.

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A105067 a(n) = Sum_{j=0..11} n^j.

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A269446 a(n) = n(n^6 + n^3 + 1)(n^6 - n^3 + 1)(n^2 + n + 1)(n^2 - n + 1)*(n + 1) + 1.