Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #52 Nov 02 2024 03:43:27
%S 5,13,23,35,49,65,83,103,125,149,175,203,233,265,299,335,373,413,455,
%T 499,545,593,643,695,749,805,863,923,985,1049,1115,1183,1253,1325,
%U 1399,1475,1553,1633,1715,1799,1885,1973,2063,2155,2249,2345,2443,2543,2645,2749
%N a(n) = n^2 + 5*n - 1.
%C 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 Numbers m such that 4*m + 29 is an odd square, starting with 7^2 = A016754(3). - _Bruce J. Nicholson_, Jul 11 2017
%H Vincenzo Librandi, <a href="/A108195/b108195.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreekCross.html">Greek Cross</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GaullistCross.html">Gaullist Cross</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F For n > 1: a(n) = A176271(n+2,n-1). - _Reinhard Zumkeller_, Apr 13 2010
%F a(n) = 2*n + a(n-1) + 4, with n > 1, a(1)=5. - _Vincenzo Librandi_, Nov 13 2010
%F G.f.: x*(5 - 2*x - x^2)/(1 - x)^3. - _Vincenzo Librandi_, Jun 11 2014
%F From _Elmo R. Oliveira_, Nov 01 2024: (Start)
%F E.g.f.: exp(x)*(x^2 + 6*x - 1) + 1.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%e ....... +---+ ......... The Cross of Lorraine
%e ....... | + | ......... having n=2 crossbeams
%e ... +---+---+---+ ..... consists of a(2)=13 squares
%e ... | + | + | + |
%e ... +---+---+---+
%e ....... | + |
%e +---+---+---+---+---+
%e | + | + | + | + | + |
%e +---+---+---+---+---+
%e ....... | + |
%e ....... +---+
%e ....... | + |
%e ....... +---+
%e ....... | + |
%e ....... +---+
%p with (combinat):seq(fibonacci(3, n)+n-8, n=3..51); # _Zerinvary Lajos_, Jun 07 2008
%t CoefficientList[Series[(5 + 3 x - x^2 - 5 x)/(1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 11 2014 *)
%t Array[#^2 + 5 # - 1 &, 49] (* _Michael De Vlieger_, Jul 12 2017 *)
%o (Magma) [n^2+5*n-1: n in [1..40]]; // _Vincenzo Librandi_, Jun 11 2014
%o (PARI) a(n)=n^2+5*n-1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A002522, A016754, A176271.
%K nonn,easy
%O 1,1
%A _Reinhard Zumkeller_, Jun 15 2005