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%I #52 Dec 17 2022 12:43:36
%S 0,5,85,1365,21845,349525,5592405,89478485,1431655765,22906492245,
%T 366503875925,5864062014805,93824992236885,1501199875790165,
%U 24019198012642645,384307168202282325,6148914691236517205,98382635059784275285,1574122160956548404565
%N a(n) = (16^n-1)/3.
%C Numbers of A002450 that are multiples of 5. Also sequence found by reading the line from 0, in the direction 0, 5,..., in the square spiral whose edges are the Jacobsthal numbers A001045 and whose vertices are the numbers A000975. This is a semi-diagonal in the spiral.
%C In binary, these numbers are 101...01 (see A031982). - _Alonso del Arte_, May 20 2017
%C 0 together with Jacobsthal numbers ending with the decimal digit 5. - _Jianing Song_, Aug 30 2022
%H Vincenzo Librandi, <a href="/A195156/b195156.txt">Table of n, a(n) for n = 0..800</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (17,-16).
%F From _Bruno Berselli_, Sep 19 2011: (Start)
%F G.f.: 5*x/((1-x)*(1-16*x)).
%F a(n) = A002450(2n) = (16^n-1)/3.
%F a(n) = 5*A131865(n-1) = a(n-1) + 5*A001025(n-1) = 16*a(n-1) + 5 for n > 0. (End)
%F From _Jianing Song_, Aug 30 2022: (Start)
%F a(n) = A001045(4*n).
%F a(n+1) - a(n) = 10*A013777(n-1) = 80*A001025(n-1) for n >= 1. (End)
%F E.g.f.: exp(x)*(exp(15*x) - 1)/3. - _Stefano Spezia_, Dec 17 2022
%p A195156:=n->(16^n-1)/3; seq(A195156(k), k=0..50); # _Wesley Ivan Hurt_, Oct 24 2013
%t Table[(16^n - 1)/3, {n, 0, 63}] (* _Wesley Ivan Hurt_, Oct 24 2013 *)
%o (Magma) [(16^n-1)/3:n in [0..20]]; // _Vincenzo Librandi_, Sep 20 2011
%o (PARI) for(n=0,50, print1((16^n - 1)/3, ", ")) \\ _G. C. Greubel_, Oct 11 2017
%Y Bisection of A002450.
%Y Cf. A000975, A001025, A013777, A031982, A131865.
%Y First quadrisection of Jacobsthal numbers A001045; the other quadrisections are A139792 (second), A144864 (third), and A141060 (fourth).
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Sep 10 2011
%E New sequence name suggested by _Charles R Greathouse IV_ using Berselli's formula. - Sep 19 2011
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