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a(n) = n^2 + 5*n - 1.
6

%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