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

1, 7, 37, 187, 937, 4687, 23437, 117187, 585937, 2929687, 14648437, 73242187, 366210937, 1831054687, 9155273437, 45776367187, 228881835937, 1144409179687, 5722045898437, 28610229492187, 143051147460937, 715255737304687, 3576278686523437, 17881393432617187, 89406967163085937

Offset: 0

Author

N. J. A. Sloane, Oct 13 2000

Keywords

• nonn
• easy

Comments

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

Examples

			a(0) = 1;
a(1) = 1 + 5 + 1 = 7;
a(2) = 1 + 5 + 25 + 5 + 1 = 37;
a(3) = 1 + 5 + 25 + 125 + 25 + 5 + 1 = 187; etc. - _Philippe Deléham_, Feb 23 2014
G.f. = 1 + 7*x + 37*x^2 + 187*x^3 + 937*x^4 + 4687*x^5 + 23437*x^6 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-5).

Crossrefs

Cf. A024049, A081655, A097162, A198762.

Cf. A020989, A061801, A112468, A112739.

Programs

Magma

[(3*5^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 30 2011

Maple

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

Mathematica

Table[(3*5^n-1)/2,{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

PARI

a(n)=3*5^n\2 \\ Charles R Greathouse IV, Dec 22 2011

Formula

G.f.: (1+x)/(1 - 6*x + 5*x^2).

a(0)=1, a(n) = 5*a(n-1) + 2; a(n) = a(n-1) + 6*(5^(n-1)). - Amarnath Murthy, May 27 2001

a(n) = 6*a(n-1) - 5*a(n-2), n > 1. - Vincenzo Librandi, Oct 30 2011

a(n) = Sum_{k=0..n} A112468(n,k)*6^k. - Philippe Deléham, Feb 23 2014

From Elmo R. Oliveira, Mar 29 2025: (Start)

E.g.f.: exp(x)*(3*exp(4*x) - 1)/2.

a(n) = A097162(2*n) = A198762(n)/2. (End)