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A226199 a(n) = 7^n + n.

%I A226199 #37 Jun 16 2025 16:12:31
%S A226199 1,8,51,346,2405,16812,117655,823550,5764809,40353616,282475259,
%T A226199 1977326754,13841287213,96889010420,678223072863,4747561509958,
%U A226199 33232930569617,232630513987224,1628413597910467,11398895185373162,79792266297612021,558545864083284028,3909821048582988071
%N A226199 a(n) = 7^n + n.
%C A226199 Smallest prime of this form is a(34) = 54116956037952111668959660883.
%C A226199 In general, the g.f. of a sequence of numbers of the form k^n + n is (1-x-(k-1)*x^2)/((1-k*x)*(x-1)^2) with main linear recurrence (k+2)*a(n-1) - (2*k+1)*a(n-2) + k*a(n-3). - _Bruno Berselli_, Jun 16 2013
%H A226199 Vincenzo Librandi, <a href="/A226199/b226199.txt">Table of n, a(n) for n = 0..300</a>
%H A226199 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-15,7).
%F A226199 G.f.: (1-x-6*x^2)/((1-7*x)*(1-x)^2).
%F A226199 a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
%F A226199 E.g.f.: exp(x)*(exp(6*x) + x). - _Elmo R. Oliveira_, Mar 05 2025
%t A226199 Table[7^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - x - 6 x^2) / ((1 - 7 x) (1 - x)^2), {x, 0, 20}], x]
%t A226199 LinearRecurrence[{9,-15,7},{1,8,51},30] (* _Harvey P. Dale_, Jun 16 2025 *)
%o A226199 (Magma) [7^n+n: n in [0..20]];
%o A226199 (Magma) I:=[1, 8, 51]; [n le 3 select I[n] else 9*Self(n-1)-15*Self(n-2)+7*Self(n-3): n in [1..30]];
%o A226199 (PARI) a(n)=7^n+n \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A226199 Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), this sequence (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).
%Y A226199 Cf. A199483 (first differences), A370657.
%K A226199 nonn,easy
%O A226199 0,2
%A A226199 _Vincenzo Librandi_, Jun 16 2013