A177059 a(n) = 25*n^2 + 25*n + 6.

6, 56, 156, 306, 506, 756, 1056, 1406, 1806, 2256, 2756, 3306, 3906, 4556, 5256, 6006, 6806, 7656, 8556, 9506, 10506, 11556, 12656, 13806, 15006, 16256, 17556, 18906, 20306, 21756, 23256, 24806, 26406, 28056, 29756, 31506, 33306, 35156, 37056, 39006, 41006, 43056

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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

Formula

a(n) = (5*n + 2)*(5*n + 3).

a(n) = 50*n + a(n-1) with a(0)=6.

a(n) = 25*A002061(n+1) - 19. - Reinhard Zumkeller, Jun 16 2010

G.f.: (6 + 38*x + 6*x^2)/(1-x)^3. - Vincenzo Librandi, Feb 03 2012

From Amiram Eldar, Jan 23 2022: (Start)

Sum_{n>=0} 1/a(n) = sqrt(1 - 2/sqrt(5))*Pi/5.

Sum_{n>=0} (-1)^n/a(n) = 2*log(phi)/sqrt(5) - 2*log(2)/5, where phi is the golden ratio (A001622).

Product_{n>=0} (1 - 1/a(n)) = 2*sqrt(2/(5+sqrt(5))) * cos(Pi/(2*sqrt(5))).

Product_{n>=0} (1 + 1/a(n)) = sqrt(2 - 2/sqrt(5)) * cosh(sqrt(3)*Pi/10).

Product_{n>=0} (1 - 2/a(n)) = 1/phi. (End)

From Elmo R. Oliveira, Oct 24 2024: (Start)

E.g.f.: exp(x)*(6 + 25*x*(2 + x)).

a(n) = A016873(n)*A016885(n) = 2*A061793(n).

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

Mathematica

LinearRecurrence[{3, -3, 1}, {6, 56, 156}, 50] (* Vincenzo Librandi, Feb 03 2012 *)
Table[25n^2+25n+6, {n, 0, 40}] (* Harvey P. Dale, Mar 30 2019 *)

Prog

(PARI) a(n)=25*n^2+25*n+6 \\ Charles R Greathouse IV, Dec 28 2011
(Magma) I:=[6, 56, 156]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 03 2012

Crossrefs

Cf. A001545, A001622, A002061, A177060, A177065, A177071, A177072, A177073.

Cf. A016873, A016885, A061793.

Sequence in context: A045526 A164579 A137034 * A345473 A053127 A068495

Adjacent sequences: A177056 A177057 A177058 * A177060 A177061 A177062

Keyword

nonn,easy

Author

Vincenzo Librandi, May 31 2010

Status

approved