OFFSET
1,2
COMMENTS
From Gary Detlefs, Aug 31 2021 (Start)
This is the x=5 member of the x-family of sequences with member a(x, n) = x^n*Sum_{k=1..n} S(x, k), with S(x, k) = Sum_{j=1..k} 1/x^j.
S(x, k) = (x^k - 1)/((x-1)*x^k) = (1/x^k)*Sum_{j=0..k-1} x^j, and a(x,n) = ((n*(x-1) - 1)*x^n + 1)/(x-1)^2.
The sequence {x^k*S(x, k)} with recurrence signature (x+1, -x) leads to sequence {a(x, n)} with recurrence signature (2*x+1, -x*(x+2), x^2). (End) [Rewritten by Wolfdieter Lang, Nov 30 2021]
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11,-35,25).
FORMULA
From Vincenzo Librandi, Oct 23 2012: (Start)
a(n) = 10*a(n-1) - 25*a(n-2) + 1; a(1)=1, a(2)=11.
G.f.: x/((1-x)*(1-5*x)^2). (End)
a(n) = 11*a(n-1) - 35*a(n-2) + 25*a(n-3); a(1)=1, a(2)=11, a(3)=86. - Harvey P. Dale, May 06 2013
a(n) = 5^n*Sum_{k=1..n} (Sum_{j=1..k} 1/x^j) = ((4*n - 1)*5^n + 1)/4^2. See the general comment above, and the first formula. -Gary Detlefs, Aug 31 2021 [Edited by Wolfdieter Lang, Nov 30 2021]
MATHEMATICA
CoefficientList[Series[1/((1 - x)(1 - 5 x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 23 2012 *)
LinearRecurrence[{11, -35, 25}, {1, 11, 86}, 20] (* Harvey P. Dale, May 06 2013 *)
PROG
(Magma) I:=[1, 11]; [n le 2 select I[n] else 10*Self(n-1)-25*Self(n-2)+ 1: n in [1..30]]; // Vincenzo Librandi, Oct 23 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved