A206399 a(0) = 1; for n > 0, a(n) = 41*n^2 + 2.

1, 43, 166, 371, 658, 1027, 1478, 2011, 2626, 3323, 4102, 4963, 5906, 6931, 8038, 9227, 10498, 11851, 13286, 14803, 16402, 18083, 19846, 21691, 23618, 25627, 27718, 29891, 32146, 34483, 36902, 39403, 41986, 44651, 47398, 50227, 53138, 56131, 59206, 62363, 65602

Offset

0,2

Comments

Apart from the first term, numbers of the form (r^2 + 2*s^2)*n^2 + 2 = (r*n)^2 + (s*n - 1)^2 + (s*n + 1)^2: in this case is r = 3, s = 4. After 1, all terms are in A000408.

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

O.g.f.: (1 + x)*(1 + 39*x + x^2)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020

E.g.f.: exp(x)*(41*x^2 + 41*x + 2) - 1. - Elmo R. Oliveira, Nov 29 2024

Mathematica

Join[{1}, 41 Range[39]^2 + 2]
CoefficientList[Series[(1 + x) (1 + 39 x + x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Prog

(Magma) [n eq 0 select 1 else 41*n^2+2: n in [0..39]];
(Magma) I:=[1, 43, 166, 371]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..41]]; // Vincenzo Librandi, Aug 18 2013
(PARI) a(n)=if(n, 41*n^2+2, 1) \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Sequences of the same type: A005893, A005897, A005899, A005901, A005903, A005905, A005914, A005918, A005919, A008527, A010000-A010023.

Cf. A000408.

Sequence in context: A235392 A078852 A078956 * A123040 A142016 A140640

Adjacent sequences: A206396 A206397 A206398 * A206400 A206401 A206402

Keyword

nonn,easy

Author

Bruno Berselli, Feb 07 2012

Status

approved