A059839 -id:A059839 - cp's OEIS frontend

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

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

A059830 a(n) = n^6 + n^4 + n^2 + 1.

Original entry on oeis.org

1, 4, 85, 820, 4369, 16276, 47989, 120100, 266305, 538084, 1010101, 1786324, 3006865, 4855540, 7568149, 11441476, 16843009, 24221380, 34117525, 47176564, 64160401, 85961044, 113614645, 148316260, 191435329, 244531876, 309373429, 387952660, 482505745, 595531444

Offset: 0

N. J. A. Sloane, Feb 25 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A059839.

Cf. A002522, A002523, A024006, A067998.

Table[(n^2+1)*(n^4+1),{n,0,40}] (* Alexander Adamchuk, Apr 13 2006 *)

a(n) = { my(f=n^2); (f + 1)*(f^2 + 1) } \\ Harry J. Smith, Jun 29 2009

a(n) = (n^2+1)*(n^4+1) = A002522(n)*A002523(n) = A002522(n)*A002522(n^2). a(n) = (n^8-1)/(n^2-1) = -A024006(n)/A067998(n+1), n>1. - Alexander Adamchuk, Apr 13 2006

G.f.: -(4*x^6+57*x^5+309*x^4+274*x^3+78*x^2-3*x+1)/(x-1)^7. - Colin Barker, Nov 05 2012

A059848 As a square table by antidiagonals, the n-digit number which in base k starts 1010101...

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 0, 0, 1, 3, 5, 2, 1, 0, 1, 4, 10, 10, 3, 0, 0, 1, 5, 17, 30, 21, 3, 1, 0, 1, 6, 26, 68, 91, 42, 4, 0, 0, 1, 7, 37, 130, 273, 273, 85, 4, 1, 0, 1, 8, 50, 222, 651, 1092, 820, 170, 5, 0, 0, 1, 9, 65, 350, 1333, 3255, 4369, 2460, 341, 5, 1, 0, 1, 10

Offset: 0

Henry Bottomley, Feb 26 2001

			T(5,3)=10101 base 3=81+9+1=91; T(4,6)=1010 base 6=216+6=222. Table starts {0,0,0,0,...}, {1,1,1,1,...}, {0,1,2,3,...}, {1,2,5,10,...}, ...

Rows include A000004, A000012, A001477, A002522, A034262, A059826, A059830, A059839. Columns include A000035, A008619, A000975, A033113, A033114, A033115, A033116, A033117, A033118, A033119, A056830.

T(n, k)=floor[k^(n+1)/(k^2-1)] =T(n-2, k)+k^(n-1) =k*T(n-1, k)-((-1)^n-1)/2

A241855 Array t(n,k) of sum of successive even powers of primes, where t(n,k) = sum_(j=0..k-1) prime(n)^(2j), with n>=1 and k>=0, read by ascending antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 5, 0, 1, 10, 21, 0, 1, 26, 91, 85, 0, 1, 50, 651, 820, 341, 0, 1, 122, 2451, 16276, 7381, 1365, 0, 1, 170, 14763, 120100, 406901, 66430, 5461, 0, 1, 290, 28731, 1786324, 5884901, 10172526, 597871, 21845, 0, 1, 362, 83811, 4855540, 216145205, 288360150, 254313151, 5380840, 87381

Offset: 1

Jean-François Alcover, Apr 30 2014

Conjecture: any term, except 0 and 1, is never a square.
Row n=1 is A002450,
row n=2 is A002452,
row n=3 is A218728,
row n=4 is A218753,
rows n>=5 are not in the OEIS,
column k=2 is A066872,
columns k>=3 are not in the OEIS.

			Array begins:
0,  1,   5,    21,      85,       341,        1365, ...
0,  1,  10,    91,     820,      7381,       66430, ...
0,  1,  26,   651,   16276,    406901,    10172526, ...
0,  1,  50,  2451,  120100,   5884901,   288360150, ...
0,  1, 122, 14763, 1786324, 216145205, 26153569806, ...
etc.

Cf. A002450, A002452, A066872, A059826, A059830, A059839, A218728, A218753.

t[n_, k_] := ((Prime[n]^2)^k-1)/(Prime[n]^2-1); Table[t[n-k+1, k], {n, 0, 10}, {k, 0, n}] // Flatten

t(n,k) = ((prime(n)^2)^k-1)/(prime(n)^2-1).

cp's OEIS Frontend

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

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A059830 a(n) = n^6 + n^4 + n^2 + 1.

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A059848 As a square table by antidiagonals, the n-digit number which in base k starts 1010101...

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A241855 Array t(n,k) of sum of successive even powers of primes, where t(n,k) = sum_(j=0..k-1) prime(n)^(2j), with n>=1 and k>=0, read by ascending antidiagonals.

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