|
%I #29 Sep 08 2022 08:45:32
%S 5,16,29,44,61,80,101,124,149,176,205,236,269,304,341,380,421,464,509,
%T 556,605,656,709,764,821,880,941,1004,1069,1136,1205,1276,1349,1424,
%U 1501,1580,1661,1744,1829,1916,2005,2096,2189,2284,2381,2480,2581,2684
%N a(n) = n^2 + 10*n + 5: coefficients of the irrational part of (1 + sqrt(n))^5.
%C (1+sqrt(n))^5 = (5*n^2 + 10*n + 1) + (n^2 + 10*n + 5)*sqrt(n). For coefficients of the rational part see A134593.
%H G. C. Greubel, <a href="/A134594/b134594.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) = ((1+sqrt(n))^5 - (5*n^2 + 10*n + 1))/sqrt(n), for n > 0. [corrected by _Jon E. Schoenfield_, Nov 23 2018]
%F G.f.: (1+x)*(5-4*x)/(1-x)^3. - _R. J. Mathar_, Nov 14 2007
%F a(n) = 2*n + a(n-1) + 9 (with a(0)=5). - _Vincenzo Librandi_, Nov 23 2010
%F E.g.f.: (5 +11*x +x^2)*exp(x). - _G. C. Greubel_, Nov 23 2018
%t Table[(n^2 + 10n + 5), {n, 0, 50}]
%t LinearRecurrence[{3,-3,1}, {5,16,29}, 50] (* _G. C. Greubel_, Nov 23 2018 *)
%o (PARI) a(n)=n^2+10*n+5 \\ _Charles R Greathouse IV_, Jun 17 2017
%o (Magma) [n^2 +10*n +5: n in [0..50]]; // _G. C. Greubel_, Nov 23 2018
%o (Sage) [n^2 +10*n +5 for n in range(50)] # _G. C. Greubel_, Nov 23 2018
%o (GAP) List([0..50],n->n^2+10*n+5); # _Muniru A Asiru_, Nov 24 2018
%Y Cf. A134593.
%K nonn,easy
%O 0,1
%A _Artur Jasinski_, Nov 04 2007
|
