A005919 -id:A005919 - cp's OEIS frontend

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

Bruno Berselli, Feb 07 2012

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.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Cf. A000408.

[n eq 0 select 1 else 41*n^2+2: n in [0..39]];

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

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 *)

a(n)=if(n,41*n^2+2,1) \\ Charles R Greathouse IV, Sep 24 2015

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

A010018 a(0) = 1, a(n) = 28*n^2 + 2 for n>0.

Original entry on oeis.org

1, 30, 114, 254, 450, 702, 1010, 1374, 1794, 2270, 2802, 3390, 4034, 4734, 5490, 6302, 7170, 8094, 9074, 10110, 11202, 12350, 13554, 14814, 16130, 17502, 18930, 20414, 21954, 23550, 25202, 26910, 28674, 30494, 32370, 34302, 36290, 38334, 40434, 42590, 44802

Offset: 0

N. J. A. Sloane

First bisection of A005919. - Bruno Berselli, Feb 07 2012
a(n) = the second level of difference between the sum of the terms in the n+1 X n+1 matrices and those in the n X n matrices starting at n=1 for an array constructed by using the terms in A016813 as the antidiagonals; the first few antidiagonals are 1; 5,9; 13,17,21; 25,29,33,37. - J. M. Bergot, Jul 05 2013
[More formally: (sum[m(n+1),j {j=0..n+1}]+sum[m(i,n+1) {i=0..n}]) - (sum[m(n,j) {j=0...n}] + sum[m(i,n) {i=0..n-1}])=a(n)]
[The first five rows begin: 1,9,21,37,57; 5,17,33,53,77; 13,29,49,73,101;25,45,69,97,129; 41,65,93,125,161]

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

Join[{1}, 28 Range[40]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
LinearRecurrence[{3, -3, 1}, {1, 30, 114, 254}, 40] (* Robert G. Wilson v, Jul 06 2013 *)

G.f.: (1+x)*(1+26*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 07 2012
E.g.f.: (x*(x+1)*28+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(14)/56*Pi*coth(Pi/sqrt 14) = 1.05615979263340... - R. J. Mathar, May 07 2024
a(n) = 2*A158482(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A195314(n)+A195314(n+1). - R. J. Mathar, May 07 2024

cp's OEIS Frontend

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

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