A028563 a(n) = n*(n+7).

0, 8, 18, 30, 44, 60, 78, 98, 120, 144, 170, 198, 228, 260, 294, 330, 368, 408, 450, 494, 540, 588, 638, 690, 744, 800, 858, 918, 980, 1044, 1110, 1178, 1248, 1320, 1394, 1470, 1548, 1628, 1710, 1794, 1880, 1968, 2058, 2150, 2244, 2340, 2438, 2538, 2640, 2744, 2850, 2958, 3068, 3180, 3294

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

a(m), for m >= 1, are the only positive integer values of t for which the Binet-de Moivre formula of the recurrence b(n) = 7*b(n-1) + t*b(n-2) has a square root whose radicand is a square. In particular, sqrt(7^2+4*t) is a positive integer since 7^2 + 4*t = 7^2 + 4*a(m) = (2*m + 7)^2. Thus the characteristic roots are r1 = 7 + m and r2 = -m. - Felix P. Muga II, Mar 28 2014 (edited - Wolfdieter Lang, Apr 17 2014)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
• Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002522, A055999.

Programs

Magma

[n*(n+7): n in [0..60]]; // Vincenzo Librandi, Oct 18 2013

Maple

A028563:=n->n*(n + 7); seq(A028563(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Mathematica

CoefficientList[Series[2 x (4 - 3 x)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Oct 18 2013 *)
LinearRecurrence[{3,-3,1},{0,8,18},60] (* Harvey P. Dale, Oct 07 2015 *)

PARI

a(n)=n*(n+7) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 2*A055999(n). - Zerinvary Lajos, Feb 12 2007

a(n) = 2*n + a(n-1) + 6. - Vincenzo Librandi, Aug 05 2010

Sum_{n>=1} 1/a(n) = 363/980 = 0.37040816... - R. J. Mathar, Mar 22 2011

G.f.: 2*x*(4-3*x)/(1-x)^3. - Colin Barker, Feb 17 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/7 - 319/2940. - Amiram Eldar, Jan 15 2021

From Amiram Eldar, Feb 12 2024: (Start)

Product_{n>=1} (1 - 1/a(n)) = 720*cos(sqrt(53)*Pi/2)/(143*Pi).

Product_{n>=1} (1 + 1/a(n)) = -112*cos(3*sqrt(5)*Pi/2)/(11*Pi). (End)

From Elmo R. Oliveira, Dec 12 2024: (Start)

E.g.f.: exp(x)*x*(8 + x).

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