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a(n) = (42^n - 1)/41.
4

%I #20 Aug 29 2024 16:00:28

%S 0,1,43,1807,75895,3187591,133878823,5622910567,236162243815,

%T 9918814240231,416590198089703,17496788319767527,734865109430236135,

%U 30864334596069917671,1296302053034936542183,54444686227467334771687,2286676821553628060410855,96040426505252378537255911

%N a(n) = (42^n - 1)/41.

%C Partial sums of powers of 42 (A009986).

%H Vincenzo Librandi, <a href="/A218745/b218745.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/Q#q-numbers">Index entries related to q-numbers</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (43,-42).

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

%F G.f.: x/((1-x)*(1-42*x)).

%F a(n) = 43*a(n-1) - 42*a(n-2).

%F a(n) = floor(42^n/41). (End)

%F E.g.f.: exp(x)*(exp(41*x) - 1)/41. - _Elmo R. Oliveira_, Aug 29 2024

%t LinearRecurrence[{43, -42}, {0, 1}, 30] (* _Vincenzo Librandi_, Nov 07 2012 *)

%t (42^Range[0,20]-1)/41 (* _Harvey P. Dale_, May 08 2024 *)

%o (PARI) A218745(n)=42^n\41

%o (Magma) [n le 2 select n-1 else 43*Self(n-1) - 42*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 07 2012

%o (Maxima) A218745(n):=(42^n-1)/41$

%o makelist(A218745(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. A009986.

%K nonn,easy

%O 0,3

%A _M. F. Hasler_, Nov 04 2012