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a(n) = (5*n+1)*(5*n+4).
2

%I #48 Oct 24 2024 15:42:26

%S 4,54,154,304,504,754,1054,1404,1804,2254,2754,3304,3904,4554,5254,

%T 6004,6804,7654,8554,9504,10504,11554,12654,13804,15004,16254,17554,

%U 18904,20304,21754,23254,24804,26404,28054,29754,31504,33304,35154,37054,39004,41004,43054

%N a(n) = (5*n+1)*(5*n+4).

%H T. D. Noe, <a href="/A001545/b001545.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 50*A000217(n) + 4.

%F a(n) = 50*n + a(n-1) with a(0)=4. - _Vincenzo Librandi_, Jan 20 2011

%F From _Amiram Eldar_, Jan 23 2022: (Start)

%F Sum_{n>=0} 1/a(n) = sqrt(1 + 2/sqrt(5))*Pi/15 = 0.2882687....

%F Sum_{n>=0} (-1)^n/a(n) = 2*log(phi)/(3*sqrt(5)) + 2*log(2)/15, where phi is the golden ratio (A001622).

%F Product_{n>=0} (1 - 1/a(n)) = sqrt(2 + 2/sqrt(5)) * cos(sqrt(13)*Pi/10).

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

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

%F G.f.: 2*(2+21*x+2*x^2)/(1-x)^3. - _R. J. Mathar_, May 30 2022

%F Sum_{n>=0} 1/a(n) = (Psi(4/5) - Psi(1/5))/15. See A200135, A200138. - _R. J. Mathar_, May 30 2022

%F From _Elmo R. Oliveira_, Oct 23 2024: (Start)

%F E.g.f.: exp(x)*(4 + 25*x*(2 + x)).

%F a(n) = A016861(n)*A016897(n).

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

%t Table[(5n+1)(5n+4),{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{4,54,154},60] (* _Harvey P. Dale_, Mar 17 2019 *)

%o (PARI) a(n)=(5*n+1)*(5*n+4) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000217, A001622, A016861, A016897, A177059, A200135, A200138.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_