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%I #37 Sep 08 2022 08:45:42
%S 1,17,33,49,65,81,97,113,129,145,161,177,193,209,225,241,257,273,289,
%T 305,321,337,353,369,385,401,417,433,449,465,481,497,513,529,545,561,
%U 577,593,609,625,641,657,673,689,705,721,737,753,769,785,801,817,833,849
%N First differences of A051870: 16*n + 1.
%C The identity (16*n+1)^2 - (16*n^2+2*n)*(4)^2 = 1 can be written as a(n+1)^2 - A158056(n)*(4)^2 = 1. - _Vincenzo Librandi_, Feb 09 2012
%C This sequence gives the 18-gonal (or octadecagonal) gnomonic numbers. Name suggested by _Todd Silvestri_, Nov 22 2014
%C All elements are odd and contains subsequence A249356. - _Todd Silvestri_, Nov 22 2014
%H Vincenzo Librandi, <a href="/A158057/b158057.txt">Table of n, a(n) for n = 0..1000</a>
%H OEIS Wiki, <a href="http://oeis.org/wiki/Gnomonic_numbers">Gnomonic numbers</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 16*n + 1.
%F a(n) = 2*a(n-1) - a(n-2), a(0) = 1, a(1) = 17.
%F G.f.: (1+15*x)/(1-x)^2. - _Vincenzo Librandi_, Nov 23 2014
%F E.g.f.: (1 + 18*x)*exp(x). - _G. C. Greubel_, Sep 18 2019
%p seq(16*n+1, n=0..60); # _G. C. Greubel_, Sep 18 2019
%t LinearRecurrence[{2,-1}, {1,17}, 60]
%t Table[16*n+1,{n,0,60}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 10 2010 *)
%t a[n_Integer/;n>=0]:=16 n+1 (* _Todd Silvestri_, Nov 22 2014 *)
%t CoefficientList[Series[(1+15x)/(1-x)^2, {x,0,60}], x] (* _Vincenzo Librandi_, Nov 23 2014 *)
%o (Magma) I:=[1, 17]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
%o (PARI) a(n)=n<<4+1 \\ _Charles R Greathouse IV_, Dec 23 2011
%o (Sage) [16*n+1 for n in (0..60)] # _G. C. Greubel_, Sep 18 2019
%o (GAP) List([0..60], n-> 16*n+1); # _G. C. Greubel_, Sep 18 2019
%Y Cf. A051870, A158056.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Mar 12 2009
%E Name clarified and offset changed by _Todd Silvestri_, Nov 22 2014
%E Edited by _Vincenzo Librandi_ Nov 23 2014
%E Edited: Offset changed to 0 according to the
%E Todd Silvestri proposal. Name changed. - _Wolfdieter Lang_, Nov 29 2014
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