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a(n) = n^5 - 1.
7

%I #37 Sep 08 2022 08:46:12

%S 0,31,242,1023,3124,7775,16806,32767,59048,99999,161050,248831,371292,

%T 537823,759374,1048575,1419856,1889567,2476098,3199999,4084100,

%U 5153631,6436342,7962623,9765624,11881375,14348906,17210367,20511148,24299999,28629150,33554431

%N a(n) = n^5 - 1.

%H Muniru A Asiru, <a href="/A258807/b258807.txt">Table of n, a(n) for n = 1..5000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: x^2*(31 + 56*x + 36*x^2 - 4*x^3 + x^4)/(1 - x)^6.

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

%F a(n) = -A024003(n). - _Bruno Berselli_, Jun 11 2015

%F Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(5*n) - 1) = 0.0379539032... - _Amiram Eldar_, Nov 06 2020

%p seq(n^5-1,n=1..35); # _Muniru A Asiru_, Oct 28 2018

%t Table[n^5 - 1, {n, 1, 50}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 31, 242, 1023, 3124, 7775}, 50]

%o (Magma) [n^5-1: n in [1..50]]; /* or */ I:=[0,31,242,1023, 3124,7775]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+ 6*Self(n-5)-Self(n-6): n in [1..50]];

%o (Sage) [n^5-1 for n in (1..50)] # _Bruno Berselli_, Jun 11 2015

%o (PARI) a(n)=n^5-1 \\ _Charles R Greathouse IV_, Jun 11 2015

%o (GAP) List([1..35],n->n^5-1); # _Muniru A Asiru_, Oct 28 2018

%o (Python) for n in range(1, 50): print(n**5 - 1, end=', ') # _Stefano Spezia_, Oct 28 2018

%Y Subsequence of A181124.

%Y Cf. A024003, A024049, A300656.

%Y Sequences of the type n^k-1: A132411 (k=2), A068601 (k=3), A123865 (k=4), this sequence (k=5), A123866 (k=6), A258808 (k=7), A258809 (k=8), A258810 (k=9), A123867 (k=10), A258812 (k=11), A123868 (k=12).

%K nonn,easy

%O 1,2

%A _Vincenzo Librandi_, Jun 11 2015