A024004 -id:A024004 - cp's OEIS frontend

A339529 Decimal expansion of Sum_{k>=1} (zeta(6*k)-1).

0, 1, 7, 5, 9, 3, 0, 2, 6, 3, 8, 5, 3, 2, 1, 5, 7, 6, 2, 1, 3, 7, 5, 6, 6, 0, 4, 4, 9, 7, 9, 5, 5, 2, 0, 0, 2, 0, 6, 5, 0, 4, 3, 4, 9, 4, 8, 0, 7, 4, 7, 0, 9, 4, 9, 7, 3, 2, 4, 6, 9, 3, 0, 1, 6, 4, 3, 3, 2, 4, 3, 7, 4, 9, 8, 3, 2, 0, 9, 0, 1, 8, 6, 7, 6, 0, 6, 4, 4, 1, 8, 8, 0, 4, 7, 5, 9, 9, 5, 2, 1, 2, 0, 4, 3

Offset: 0

Vaclav Kotesovec, Dec 08 2020

			0.017593026385321576213756604497955200206504349480747094973246930164332437...

Cf. A024004, A256919, A339530.

Join[{0}, RealDigits[11/12 - Pi*Tanh[Sqrt[3]*Pi/2]/(2*Sqrt[3]), 10, 100][[1]]]

Equals Sum_{k>=2} 1/(k^6 - 1).

Equals 11/12 - Pi*tanh(sqrt(3)*Pi/2)/(2*sqrt(3)).

A024006 a(n) = 1 - n^8.

Original entry on oeis.org

1, 0, -255, -6560, -65535, -390624, -1679615, -5764800, -16777215, -43046720, -99999999, -214358880, -429981695, -815730720, -1475789055, -2562890624, -4294967295, -6975757440, -11019960575, -16983563040, -25599999999, -37822859360, -54875873535

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..420
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A024004.

[1-n^8: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[1-n^8,{n,0,40}] (* and *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 0, -255, -6560, -65535, -390624, -1679615, -5764800, -16777215}, 40] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

a(n)=1-n^8 \\ Charles R Greathouse IV, Oct 07 2015

Sum_{n>=2} -1/a(n) = 15/16 - Pi*(coth(Pi)/8) + Pi * (sin(sqrt(2)*Pi) + sinh(sqrt(2)*Pi)) / (4*sqrt(2) * (cos(sqrt(2)*Pi) - cosh(sqrt(2)*Pi))) = A339530 = 0.0040926982992862873... . - Vaclav Kotesovec, Feb 14 2015

A123866 a(n) = n^6 - 1.

Original entry on oeis.org

0, 63, 728, 4095, 15624, 46655, 117648, 262143, 531440, 999999, 1771560, 2985983, 4826808, 7529535, 11390624, 16777215, 24137568, 34012223, 47045880, 63999999, 85766120, 113379903, 148035888, 191102975, 244140624, 308915775, 387420488

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 7 = 0 iff n mod 7 > 0: a(A008589(n))=6; a(A047304(n)) = 0; a(n) mod 7 = 6*(1-A082784(n)).

a(n) = A005563(n-1)*A059826(n) = A068601(n)*A001093(n). - Reinhard Zumkeller, Feb 02 2007

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

Cf. A024004, A001014, A005563, A123865, A123867, A123868.

List([1..30], n-> n^6 -1); # G. C. Greubel, Aug 08 2019

a123866 = (subtract 1) . (^ 6)  -- Reinhard Zumkeller, Mar 11 2014

[n^6 - 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A123866:=n->n^6 - 1; seq(A123866(n), n=1..40); # Wesley Ivan Hurt, Feb 26 2014

Table[n^6-1,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,63,728,4095, 15624,46655, 117648}, 30] (* Harvey P. Dale, Nov 18 2012 *)

A123866(n):=n^6 -1 $ makelist(A123866(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n)=n^6-1 \\ Charles R Greathouse IV, Oct 07 2015

[n^6 -1 for n in (1..30)] # G. C. Greubel, Aug 08 2019

G.f.: x^2*(63 + 287*x + 322*x^2 + 42*x^3 + 7*x^4 - x^5)/(1-x)^7. - Colin Barker, May 08 2012
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(1)=0, a(2)=63, a(3)=728, a(4)=4095, a(5)=15624, a(6)=46655, a(7)=117648. - Harvey P. Dale, Nov 18 2012
Sum_{n>=2} 1/a(n) = 11/12 - Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/6. - Vaclav Kotesovec, Feb 14 2015
E.g.f.: 1 + (-1 + x + 31*x^2 + 90*x^3 + 65*x^4 + 15*x^5 + x^6)*exp(x). - G. C. Greubel, Aug 08 2019
Product_{n>=2} (1 + 1/a(n)) = 6*Pi^2*sech(sqrt(3)*Pi/2)^2. - Amiram Eldar, Jan 20 2021

A002604 a(n) = n^6 + 1.

Original entry on oeis.org

1, 2, 65, 730, 4097, 15626, 46657, 117650, 262145, 531442, 1000001, 1771562, 2985985, 4826810, 7529537, 11390626, 16777217, 24137570, 34012225, 47045882, 64000001, 85766122, 113379905, 148035890

Offset: 0

N. J. A. Sloane

Because of Fermat's little theorem, a(n) is never divisible by 7. - Altug Alkan, Apr 08 2016

Vincenzo Librandi, 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).

Equals A001014 + 1. Cf. A024004, A002522.

[n^6 + 1: n in [0..50]]; // Vincenzo Librandi, May 02 2011

Table[n^6+1,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,2,65,730,4097,15626,46657},30] (* Harvey P. Dale, Jul 28 2021 *)

PARI
```
a(n)=n^6+1
```

G.f. (-1 + 5*x - 72*x^2 - 282*x^3 - 317*x^4 - 51*x^5 - 2*x^6) / (x - 1)^7. - R. J. Mathar, Aug 06 2012
Sum_{n>=0} 1/a(n) = 1/2 + Pi * (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi))) / 6 = 1.5171007340332164261529... . - Vaclav Kotesovec, Feb 14 2015
Sum_{n>=0} (-1)^n/a(n) = 1/2 + Pi/(6*sinh(Pi)) + Pi * (sqrt(3)*cosh(Pi/2) * sin((sqrt(3)*Pi)/2) + cos((sqrt(3)*Pi)/2) * sinh(Pi/2)) / (3*(cosh(Pi) - cos(sqrt(3)*Pi))) = 0.514210347292695053493... . - Vaclav Kotesovec, Feb 14 2015

A024007 a(n) = 1 - n^9.

Original entry on oeis.org

1, 0, -511, -19682, -262143, -1953124, -10077695, -40353606, -134217727, -387420488, -999999999, -2357947690, -5159780351, -10604499372, -20661046783, -38443359374, -68719476735, -118587876496, -198359290367

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..388

Cf. A024004, A024006.

[1-n^9: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[1 - n^9, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

A108495 a(n) = (n^7 - n)/6.

Original entry on oeis.org

0, 0, 21, 364, 2730, 13020, 46655, 137256, 349524, 797160, 1666665, 3247860, 5971966, 10458084, 17568915, 28476560, 44739240, 68389776, 102036669, 148978620, 213333330, 300181420, 415726311, 567470904, 764411900, 1017252600

Offset: 0

Henry Bottomley, Jun 06 2005

Also integer sequences for (n^2-n)/1 (A002378 offset), (n^3-n)/2 (A027480 offset), (n^43-n)/42 (A108496) and (n^1807-n)/1806.

			a(2) = (2^7 - 2)/6 = 126/6 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Zagier, Problems posed at the St Andrews Colloquium, 1996.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A014117, A108497, A108499.

[(n^7-n)/6: n in [0..40]]; // Vincenzo Librandi, May 02 2011

Table[(n^7-n)/6,{n,0,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,0,21,364,2730,13020,46655,137256},30] (* Harvey P. Dale, Apr 16 2014 *)

[(n**7-n)//6 for n in range(41)] # David Radcliffe, Jun 06 2025

a(n) = (n-1)*A059721(n) = -A024004(n)*n/6.

G.f.: 7*x^2*(3 + 28*x + 58*x^2 + 28*x^3 + 3*x^4)/(1-x)^8. [Colin Barker, May 08 2012]

A258837 a(n) = 1 - n^2.

Original entry on oeis.org

1, 0, -3, -8, -15, -24, -35, -48, -63, -80, -99, -120, -143, -168, -195, -224, -255, -288, -323, -360, -399, -440, -483, -528, -575, -624, -675, -728, -783, -840, -899, -960, -1023, -1088, -1155, -1224, -1295, -1368, -1443, -1520, -1599, -1680, -1763, -1848

Offset: 0

Vincenzo Librandi, Jun 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067998, A131386, A007531.

Sequences of the type 1-n^k: A024000 (k=1), this sequence (k=2), A024001 (k=3), A024002 (k=4), A024003 (k=5), A024004 (k=6), A024005 (k=7), A024006 (k=8), A024007 (k=9), A024008 (k=10), A024009 (k=11), A024010 (k=12).

Magma
```
[1-n^2: n in [0..50]];
```

I:=[1,0,-3]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Table[1 - n^2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 0, -3}, 50]

my(x='x+O('x^50)); Vec((1-3*x)/(1-x)^3) \\ G. C. Greubel, May 11 2017

G.f.: (1-3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = -A067998(n+1). - Joerg Arndt, Jun 13 2015
a(n) = (-1)^n*A131386(n+1). - Bruno Berselli, Jun 15 2015
E.g.f.: (1 - x - x^2)*exp(x). - G. C. Greubel, May 11 2017
Sum_{n>=2} 1/a(n) = -3/4. - Amiram Eldar, Feb 17 2023

cp's OEIS Frontend

A339529 Decimal expansion of Sum_{k>=1} (zeta(6*k)-1).

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A024006 a(n) = 1 - n^8.

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A123866 a(n) = n^6 - 1.

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A002604 a(n) = n^6 + 1.

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A024007 a(n) = 1 - n^9.

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A108495 a(n) = (n^7 - n)/6.

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A258837 a(n) = 1 - n^2.

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