|
%I M5426 N2357 #79 May 05 2024 20:02:28
%S 0,1,256,6561,65536,390625,1679616,5764801,16777216,43046721,
%T 100000000,214358881,429981696,815730721,1475789056,2562890625,
%U 4294967296,6975757441,11019960576,16983563041,25600000000,37822859361,54875873536,78310985281,110075314176
%N Eighth powers: a(n) = n^8.
%C 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
%C 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
%C Fourth powers of squares and squares of 4th powers. Squares composed with themselves twice. - _Wesley Ivan Hurt_, Apr 01 2016
%D Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001016/b001016.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F Multiplicative with a(p^e) = p^(8e). - _David W. Wilson_, Aug 01 2001
%F Totally multiplicative sequence with a(p) = p^8 for primes p. - _Jaroslav Krizek_, Nov 01 2009
%F 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
%F 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
%F From _Wesley Ivan Hurt_, Apr 01 2016: (Start)
%F 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.
%F a(n) = A000290(n)^4 = A000290(A000290(A000290(n))).
%F a(n) = A000583(n)^2. (End)
%F From _Amiram Eldar_, Oct 08 2020: (Start)
%F Sum_{n>=1} 1/a(n) = zeta(8) = Pi^8/9450 (A013666).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 127*zeta(8)/128 = 127*Pi^8/1209600. (End)
%F 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
%p A001016:=n->n^8: seq(A001016(n), n=0..50); # _Wesley Ivan Hurt_, Apr 01 2016
%t Table[n^8, {n, 0, 20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)
%o (Maxima) A001016(n):=n^8$
%o makelist(A001016(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (PARI) a(n)=n^8 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [n^8 : n in [0..50]]; // _Wesley Ivan Hurt_, Apr 01 2016
%Y Cf. A000290 (squares), A000583 (fourth powers), A013666.
%Y Cf. A000542 (partial sums), A022524 (first differences).
%K nonn,easy,mult,changed
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Sep 19 2000
| 