A153037 a(n) = 2*n^2 + 16*n + 23.
23, 41, 63, 89, 119, 153, 191, 233, 279, 329, 383, 441, 503, 569, 639, 713, 791, 873, 959, 1049, 1143, 1241, 1343, 1449, 1559, 1673, 1791, 1913, 2039, 2169, 2303, 2441, 2583, 2729, 2879, 3033, 3191, 3353, 3519, 3689, 3863, 4041, 4223, 4409, 4599, 4793, 4991, 5193
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
-
Magma
I:=[23, 41, 63]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 22 2012
-
Mathematica
Table[2*n^2 + 16*n + 23, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *) LinearRecurrence[{3, -3, 1}, {23, 41, 63}, 50] (* Vincenzo Librandi, Feb 22 2012 *) CoefficientList[Series[(23 - 28*x +9*x^2)/(1 -x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 04 2013 *) -
PARI
a(n)=2*n^2+16*n+23 \\ Charles R Greathouse IV, Jan 11 2012
Formula
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012
G.f.: (23 - 28*x + 9*x^2)/(1-x)^3. - Vincenzo Librandi, Jan 04 2013
From Amiram Eldar, Feb 25 2023: (Start)
Sum_{n>=0} 1/a(n) = 137/126 - cot(3*Pi/sqrt(2))*Pi/(6*sqrt(2)).
Sum_{n>=0} (-1)^n/a(n) = 43/42 - cosec(3*Pi/sqrt(2))*Pi/(6*sqrt(2)). (End)
E.g.f.: exp(x)*(23 + 18*x + 2*x^2). - Elmo R. Oliveira, Feb 08 2025
Extensions
Erroneously duplicated terms removed by Vincenzo Librandi, Feb 22 2012
Comments