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%I #88 Sep 06 2022 22:25:45
%S 1,7,15,25,37,51,67,85,105,127,151,177,205,235,267,301,337,375,415,
%T 457,501,547,595,645,697,751,807,865,925,987,1051,1117,1185,1255,1327,
%U 1401,1477,1555,1635,1717,1801,1887,1975,2065,2157,2251,2347,2445,2545,2647
%N a(n) = n^2 + 5*n + 1.
%C From _Gary W. Adamson_, Jul 29 2009: (Start)
%C Let (a,b) = roots to x^2 - 5*x + 1 = 0 = 4.79128... and 0.208712...
%C Then a(n) = (n + a) * (n + b). Example: a(5) = 51 = (5 + 4.79128...) * (5 + 0.208712...) (End)
%C For n > 0: a(n) = A176271(n+2,n). - _Reinhard Zumkeller_, Apr 13 2010
%C 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
%C Numbers m > 0 such that 4m+21 is a square. - _Bruce J. Nicholson_, Jul 19 2017
%C 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
%C If A and B are sequences satisfying the recurrence t(n) = 5*t(n-1) - t(n-2) with initial values A(0) = 1, A(1) = n+5 and B(0) = -1, B(1) = n, then a(n) = A(i)^2 - A(i-1)*A(i+1) = B(j)^2 - B(j-1)*B(j+1) for i, j > 0. - _Klaus Purath_, Oct 18 2020
%C The prime terms in this sequence are listed in A089376. The prime factors are given in A038893. With the exception of 3 and 7, each prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -5 (mod p). - _Klaus Purath_, Nov 24 2020
%H G. C. Greubel, <a href="/A082111/b082111.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2*n + a(n-1) + 4 (with a(0)=1). - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=7, a(2)=15. - _Harvey P. Dale_, Apr 22 2012
%F Sum_{n>=0} 1/a(n) = 8/15 + Pi*tan(sqrt(21)*Pi/2)/sqrt(21) = 1.424563592286456286... . - _Vaclav Kotesovec_, Apr 10 2016
%F From _G. C. Greubel_, Jul 19 2017: (Start)
%F G.f.: (1 + 4*x - 3*x^2)/(1 - x)^3.
%F E.g.f.: (x^2 + 6*x + 1)*exp(x). (End)
%F a(n) = A014209(n+1) - 2 = A338041(2*n+1). - _Hugo Pfoertner_, Oct 08 2020
%F a(n) = A249547(n+1) - A024206(n-4), n >= 5. - _Klaus Purath_, Nov 24 2020
%t Table[n^2 + 5*n + 1,{n,0,80}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 19 2011 *)
%t LinearRecurrence[{3,-3,1},{1,7,15},80] (* _Harvey P. Dale_, Apr 22 2012 *)
%o (PARI) a(n)=n^2+5*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y First row of A082110.
%Y Cf. A002522, A014209, A028387, A028872, A338041.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 04 2003
%E New title (using given formula) from _Hugo Pfoertner_, Oct 08 2020
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