cp's OEIS Frontend

A108195 a(n) = n^2 + 5*n - 1.

%I A108195 #60 May 12 2025 06:44:35
%S A108195 5,13,23,35,49,65,83,103,125,149,175,203,233,265,299,335,373,413,455,
%T A108195 499,545,593,643,695,749,805,863,923,985,1049,1115,1183,1253,1325,
%U A108195 1399,1475,1553,1633,1715,1799,1885,1973,2063,2155,2249,2345,2443,2543,2645,2749
%N A108195 a(n) = n^2 + 5*n - 1.
%C A108195 a(n-2) = n*(n + 1) - 7, n >= 0, with a(-2) = -7, a(-1) = -5 and a(0) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 29 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - _Wolfdieter Lang_, Aug 16 2013
%C A108195 Numbers m such that 4*m + 29 is an odd square, starting with 7^2 = A016754(3). - _Bruce J. Nicholson_, Jul 11 2017
%H A108195 Vincenzo Librandi, <a href="/A108195/b108195.txt">Table of n, a(n) for n = 1..1000</a>
%H A108195 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaullistCross.html">Gaullist Cross</a>.
%H A108195 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GreekCross.html">Greek Cross</a>.
%H A108195 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A108195 For n > 1: a(n) = A176271(n+2,n-1). - _Reinhard Zumkeller_, Apr 13 2010
%F A108195 a(n) = 2*n + a(n-1) + 4, with n > 1, a(1)=5. - _Vincenzo Librandi_, Nov 13 2010
%F A108195 G.f.: x*(5 - 2*x - x^2)/(1 - x)^3. - _Vincenzo Librandi_, Jun 11 2014
%F A108195 From _Elmo R. Oliveira_, Nov 01 2024: (Start)
%F A108195 E.g.f.: exp(x)*(x^2 + 6*x - 1) + 1.
%F A108195 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%F A108195 Sum_{n>=1} 1/a(n) = 47/35 + tan(sqrt(29)*Pi/2)*Pi/sqrt(29). - _Amiram Eldar_, May 12 2025
%e A108195 The Cross of Lorraine having n=2 crossbeams consists of a(2)=13 squares
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%p A108195 with (combinat):seq(fibonacci(3, n)+n-8, n=3..51); # _Zerinvary Lajos_, Jun 07 2008
%t A108195 CoefficientList[Series[(5 + 3 x - x^2 - 5 x)/(1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 11 2014 *)
%t A108195 Array[#^2 + 5 # - 1 &, 49] (* _Michael De Vlieger_, Jul 12 2017 *)
%o A108195 (Magma) [n^2+5*n-1: n in [1..40]]; // _Vincenzo Librandi_, Jun 11 2014
%o A108195 (PARI) a(n)=n^2+5*n-1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A108195 Cf. A002522, A016754, A176271.
%K A108195 nonn,easy
%O A108195 1,1
%A A108195 _Reinhard Zumkeller_, Jun 15 2005