A014901 a(1)=1, a(n) = 18*a(n-1) + n.
1, 20, 363, 6538, 117689, 2118408, 38131351, 686364326, 12354557877, 222382041796, 4002876752339, 72051781542114, 1296932067758065, 23344777219645184, 420205989953613327, 7563707819165039902, 136146740744970718253, 2450641333409472928572, 44111544001370512714315
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- Index entries for linear recurrences with constant coefficients, signature (20,-37,18).
Programs
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Magma
I:=[1, 20, 363]; [n le 3 select I[n] else 20*Self(n-1) - 37*Self(n-2) + 18*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
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Mathematica
LinearRecurrence[{20, -37, 18}, {1, 20, 363}, 20] (* Vincenzo Librandi, Oct 20 2012 *) nxt[{n_,a_}]:={n+1,18a+n+1}; NestList[nxt,{1,1},20][[All,2]] (* Harvey P. Dale, Oct 08 2017 *) -
Maxima
a[1]:1$ a[2]:20$ a[3]:363$ a[n]:=20*a[n-1]-37*a[n-2]+18*a[n-3]$ A014901(n):=a[n]$ makelist(A014901(n),n,1,30); /* Martin Ettl, Nov 06 2012 */
Formula
a(1)=1, a(2)=20, a(3)=363, a(n) = 20*a(n-1) - 37*a(n-2) + 18*a(n-3). - Vincenzo Librandi, Oct 20 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-18*x)*(1-x)^2).
E.g.f.: exp(x)*(18*exp(17*x) - 17*x - 18)/289.
a(n) = (18^(n+1) - 17*n - 18)/289. (End)
a(-1-n) = A014935(n)/18^n for all n in Z. - Michael Somos, Mar 29 2025