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A082111 A row of number array A082110. 9
1, 7, 15, 25, 37, 51, 67, 85, 105, 127, 151, 177, 205, 235, 267, 301, 337, 375, 415, 457, 501, 547, 595, 645, 697, 751, 807, 865, 925, 987, 1051, 1117, 1185, 1255, 1327, 1401, 1477, 1555, 1635, 1717, 1801, 1887, 1975, 2065, 2157, 2251, 2347, 2445, 2545, 2647 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Gary W. Adamson, Jul 29 2009: (Start)

Let (a,b) = roots to x^2 -5x + 1 = 0 = 4.79128... and 0.208712...

Then a(n) = (n + a) * (n + b). Example: a(5) = 51 = (5 + 4.79128...) * (5 + 0.208712...) (End)

For n>0: a(n) = A176271(n+2,n). - Reinhard Zumkeller, Apr 13 2010

a(n-2) = n*(n+1) - 5, n>= 0, with a(-2) = -5 and a(-1) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 21 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 15 2013

Numbers m>0 such that 4m+21 is a square. - Bruce J. Nicholson, Jul 19 2017

Numbers represented as 151 in number base B. If 'digits' from B upwards are allowed then 151(2)=15, 151(3)=25, 151(4)=37, 151(5)=51 also. - Ron Knott, Nov 14 2017

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

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

FORMULA

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

a(n) = 2*n + a(n-1) + 4 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010

a(0)=1, a(1)=7, a(2)=15, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Apr 22 2012

Sum_{n>=0} 1/a(n) = 8/15 + Pi*tan(sqrt(21)*Pi/2)/sqrt(21) = 1.424563592286456286... . - Vaclav Kotesovec, Apr 10 2016

From G. C. Greubel, Jul 19 2017: (Start)

G.f.: (1 + 4*x - 3*x^2)/(1 - x)^3.

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

MATHEMATICA

Table[n^2 + 5*n + 1, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

LinearRecurrence[{3, -3, 1}, {1, 7, 15}, 80] (* Harvey P. Dale, Apr 22 2012 *)

PROG

(PARI) a(n)=n^2+5*n+1 \\ Charles R Greathouse IV, Jun 17 2017

CROSSREFS

Cf. A002522, A028387, A028872.

Sequence in context: A056119 A284758 A211430 * A236582 A268662 A260558

Adjacent sequences:  A082108 A082109 A082110 * A082112 A082113 A082114

KEYWORD

easy,nonn,changed

AUTHOR

Paul Barry, Apr 04 2003

STATUS

approved

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Last modified November 21 13:53 EST 2017. Contains 295001 sequences.