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

5, 13, 23, 35, 49, 65, 83, 103, 125, 149, 175, 203, 233, 265, 299, 335, 373, 413, 455, 499, 545, 593, 643, 695, 749, 805, 863, 923, 985, 1049, 1115, 1183, 1253, 1325, 1399, 1475, 1553, 1633, 1715, 1799, 1885, 1973, 2063, 2155, 2249, 2345, 2443, 2543, 2645, 2749

Offset

1,1

Comments

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

Numbers m such that 4*m + 29 is an odd square, starting with 7^2 = A016754(3). - Bruce J. Nicholson, Jul 11 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Greek Cross.

Eric Weisstein's World of Mathematics, Gaullist Cross.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

For n > 1: a(n) = A176271(n+2,n-1). - Reinhard Zumkeller, Apr 13 2010

a(n) = 2*n + a(n-1) + 4, with n > 1, a(1)=5. - Vincenzo Librandi, Nov 13 2010

G.f.: x*(5 - 2*x - x^2)/(1 - x)^3. - Vincenzo Librandi, Jun 11 2014

From Elmo R. Oliveira, Nov 01 2024: (Start)

E.g.f.: exp(x)*(x^2 + 6*x - 1) + 1.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Example

....... +---+ ......... The Cross of Lorraine
....... | + | ......... having n=2 crossbeams
... +---+---+---+ ..... consists of a(2)=13 squares
... | + | + | + |
... +---+---+---+
....... | + |
+---+---+---+---+---+
| + | + | + | + | + |
+---+---+---+---+---+
....... | + |
....... +---+
....... | + |
....... +---+
....... | + |
....... +---+

Maple

with (combinat):seq(fibonacci(3, n)+n-8, n=3..51); # Zerinvary Lajos, Jun 07 2008

Mathematica

CoefficientList[Series[(5 + 3 x - x^2 - 5 x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 11 2014 *)
Array[#^2 + 5 # - 1 &, 49] (* Michael De Vlieger, Jul 12 2017 *)

Prog

(Magma) [n^2+5*n-1: n in [1..40]]; // Vincenzo Librandi, Jun 11 2014
(PARI) a(n)=n^2+5*n-1 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A002522, A016754, A176271.

Sequence in context: A129806 A125830 A049882 * A074798 A031336 A099958

Adjacent sequences: A108192 A108193 A108194 * A108196 A108197 A108198

Keyword

nonn,easy

Author

Reinhard Zumkeller, Jun 15 2005

Status

approved