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%I #36 Sep 29 2019 15:09:44
%S 1,17,33,49,65,97,129,161,193,241,289,337,385,449,513,577,641,721,801,
%T 881,961,1057,1153,1249,1345,1457,1569,1681,1793,1921,2049,2177,2305,
%U 2449,2593,2737,2881,3041,3201,3361,3521,3697,3873,4049,4225,4417,4609,4801
%N 8*A200975(n)-7 where A200975 are the numbers on the diagonals in Ulam's spiral.
%C All elements are odd.
%C The pair (a(n), a(n+1)) is separated by A002265(n-1) elements in A158057.
%H Todd Silvestri, <a href="/A249356/b249356.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F a(n) = 2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2).
%F G.f.: - x*(x^5-x^4+15*x+1)/((x-1)^3*(x^3+x^2+x+1)).
%F a(n) = 2*a(n-1) - a(n-2) + 16 if n == 2 mod 4, a(n) = 2*a(n-1) - a(n-2) otherwise. - _Robert Israel_, Nov 04 2014
%F a(n) = 2*n*(n+2) - (1-(-1)^n)*(1-2*i^(n+1)) + 1, where i=sqrt(-1). - _Bruno Berselli_, Nov 18 2014
%p seq(2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2), n=1..100); # _Robert Israel_, Nov 04 2014
%t a[n_Integer/;n>0]:=2 n (n+2)+(-1)^n-4 Mod[n^2 (3 n+2),4,-1]
%t CoefficientList[Series[-(x^5 - x^4 + 15 x + 1) / ((x - 1)^3 (x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 16 2014 *)
%t Table[2 n (n + 2) - (1 - (-1)^n) (1 - 2 I^(n + 1)) + 1, {n, 1, 50}] (* _Bruno Berselli_, Nov 18 2014 *)
%t LinearRecurrence[{2,-1,0,1,-2,1},{1,17,33,49,65,97},50] (* _Harvey P. Dale_, Sep 29 2019 *)
%o (PARI) a(n) = 2*n*(n+2)+(-1)^n-4*round(sin((Pi*n)/2)) \\ _Charles R Greathouse IV_, Nov 17 2014
%K nonn,easy
%O 1,2
%A _Todd Silvestri_, Oct 27 2014
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