A001016 Eighth powers: a(n) = n^8. (Formerly M5426 N2357)

0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 25600000000, 37822859361, 54875873536, 78310985281, 110075314176

Offset

0,3

Comments

Besides the first term, this sequence is the denominator of ((Pi)^8)/9450 = 1+1/256+1/6561+1/65536+1/390625+1/1679616+... - Mohammad K. Azarian, Nov 01 2011

For n > 0, a(n) is the largest number k such that k + n^4 divides k^2 + n^4. - Derek Orr, Oct 01 2014

Fourth powers of squares and squares of 4th powers. Squares composed with themselves twice. - Wesley Ivan Hurt, Apr 01 2016

References

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

Multiplicative with a(p^e) = p^(8e). - David W. Wilson, Aug 01 2001

Totally multiplicative sequence with a(p) = p^8 for primes p. - Jaroslav Krizek, Nov 01 2009

G.f.: -x*(1+x)*(x^6+246*x^5+4047*x^4+11572*x^3+4047*x^2+246*x+1)/(x-1)^9. - R. J. Mathar, Jan 07 2011

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 40320. - Ant King, Sep 24 2013

From Wesley Ivan Hurt, Apr 01 2016: (Start)

a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9) for n>8.

a(n) = A000290(n)^4 = A000290(A000290(A000290(n))).

a(n) = A000583(n)^2. (End)

From Amiram Eldar, Oct 08 2020: (Start)

Sum_{n>=1} 1/a(n) = zeta(8) = Pi^8/9450 (A013666).

Sum_{n>=1} (-1)^(n+1)/a(n) = 127*zeta(8)/128 = 127*Pi^8/1209600. (End)

E.g.f.: exp(x)*x*(1 + 127*x + 966*x^2 + 1701*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7). - Stefano Spezia, Jul 29 2022

Maple

A001016:=n->n^8: seq(A001016(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2016

Mathematica

Table[n^8, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)

Prog

(Maxima) A001016(n):=n^8$
makelist(A001016(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(PARI) a(n)=n^8 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [n^8 : n in [0..50]]; // Wesley Ivan Hurt, Apr 01 2016

Crossrefs

Cf. A000290 (squares), A000583 (fourth powers), A013666.

Cf. A000542 (partial sums), A022524 (first differences).

Sequence in context: A016900 A017680 A210840 * A352054 A351606 A343288

Adjacent sequences: A001013 A001014 A001015 * A001017 A001018 A001019

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Sep 19 2000

Status

approved