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a(n) = (39^n - 1)/38.
3

%I #22 Aug 29 2024 14:31:26

%S 0,1,40,1561,60880,2374321,92598520,3611342281,140842348960,

%T 5492851609441,214221212768200,8354627297959801,325830464620432240,

%U 12707388120196857361,495588136687677437080,19327937330819420046121,753789555901957381798720,29397792680176337890150081

%N a(n) = (39^n - 1)/38.

%C Partial sums of powers of 39 (A009983).

%H Vincenzo Librandi, <a href="/A218742/b218742.txt">Table of n, a(n) for n = 0..600</a>

%H <a href="/index/Par#partial">Index entries related to partial sums</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (40,-39).

%F a(n) = floor(39^n/38).

%F From _Vincenzo Librandi_, Nov 07 2012: (Start)

%F G.f.: x/((1-x)*(1-39*x)).

%F a(n) = 40*a(n-1) - 39*a(n-2). (End)

%F E.g.f.: exp(20*x)*sinh(19*x)/19. - _Elmo R. Oliveira_, Aug 29 2024

%t LinearRecurrence[{40, -39}, {0, 1}, 30] (* _Vincenzo Librandi_, Nov 07 2012 *)

%t (39^Range[0,20]-1)/38 (* _Harvey P. Dale_, Mar 05 2023 *)

%o (PARI) a(n)=39^n\38

%o (Magma) [n le 2 select n-1 else 40*Self(n-1) - 39*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 07 2012

%o (Maxima) A218742(n):=(39^n-1)/38$

%o makelist(A218742(n),n,0,30); /* _Martin Ettl_, Nov 07 2012 */

%Y Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

%Y Cf. A009983.

%K nonn,easy

%O 0,3

%A _M. F. Hasler_, Nov 04 2012