cp's OEIS Frontend

A057651 a(n) = (3*5^n - 1)/2.

%I A057651 #35 Mar 29 2025 16:53:54
%S A057651 1,7,37,187,937,4687,23437,117187,585937,2929687,14648437,73242187,
%T A057651 366210937,1831054687,9155273437,45776367187,228881835937,
%U A057651 1144409179687,5722045898437,28610229492187,143051147460937,715255737304687,3576278686523437,17881393432617187,89406967163085937
%N A057651 a(n) = (3*5^n - 1)/2.
%C A057651 Sum of n-th row of triangle of powers of 5: 1; 1 5 1; 1 5 25 5 1 ; 1 5 25 125 25 5 1; ... - _Philippe Deléham_, Feb 23 2014
%H A057651 Vincenzo Librandi, <a href="/A057651/b057651.txt">Table of n, a(n) for n = 0..1000</a>
%H A057651 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-5).
%F A057651 G.f.: (1+x)/(1 - 6*x + 5*x^2).
%F A057651 a(0)=1, a(n) = 5*a(n-1) + 2; a(n) = a(n-1) + 6*(5^(n-1)). - _Amarnath Murthy_, May 27 2001
%F A057651 a(n) = 6*a(n-1) - 5*a(n-2), n > 1. - _Vincenzo Librandi_, Oct 30 2011
%F A057651 a(n) = Sum_{k=0..n} A112468(n,k)*6^k. - _Philippe Deléham_, Feb 23 2014
%F A057651 From _Elmo R. Oliveira_, Mar 29 2025: (Start)
%F A057651 E.g.f.: exp(x)*(3*exp(4*x) - 1)/2.
%F A057651 a(n) = A097162(2*n) = A198762(n)/2. (End)
%e A057651 a(0) = 1;
%e A057651 a(1) = 1 + 5 + 1 = 7;
%e A057651 a(2) = 1 + 5 + 25 + 5 + 1 = 37;
%e A057651 a(3) = 1 + 5 + 25 + 125 + 25 + 5 + 1 = 187; etc. - _Philippe Deléham_, Feb 23 2014
%e A057651 G.f. = 1 + 7*x + 37*x^2 + 187*x^3 + 937*x^4 + 4687*x^5 + 23437*x^6 + ...
%p A057651 G.f=(1+x)/(1-5*x)/(1-x): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=0..30); # _Zerinvary Lajos_, Jan 11 2009
%t A057651 Table[(3*5^n-1)/2,{n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2012 *)
%o A057651 (Magma) [(3*5^n-1)/2: n in [0..30]]; // _Vincenzo Librandi_, Oct 30 2011
%o A057651 (PARI) a(n)=3*5^n\2 \\ _Charles R Greathouse IV_, Dec 22 2011
%Y A057651 Cf. A024049, A081655, A097162, A198762.
%Y A057651 Cf. A020989, A061801, A112468, A112739.
%K A057651 nonn,easy
%O A057651 0,2
%A A057651 _N. J. A. Sloane_, Oct 13 2000