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A028563 n(n+7). 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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)

REFERENCES

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Patrick De Geest, Palindromic Quasipronics of the form n(n+x)

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

FORMULA

Equals 2*A055999. - 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

MAPLE

with(combinat): seq(fibonacci(3, n+3)+n-10, n=0..50); # Zerinvary Lajos, Jun 07 2008

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

MATHEMATICA

lst={}; Do[AppendTo[lst, n*(n+7)], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 06 2008 *)

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 *)

PROG

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

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

CROSSREFS

Cf. A002522.

Sequence in context: A278546 A092163 A100592 * A171523 A120091 A211526

Adjacent sequences:  A028560 A028561 A028562 * A028564 A028565 A028566

KEYWORD

nonn,easy,changed

AUTHOR

Patrick De Geest

STATUS

approved

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Last modified June 28 13:24 EDT 2017. Contains 288838 sequences.