A017534 - OEIS

%I #29 Feb 27 2023 08:37:56

%S 1,169,625,1369,2401,3721,5329,7225,9409,11881,14641,17689,21025,

%T 24649,28561,32761,37249,42025,47089,52441,58081,64009,70225,76729,

%U 83521,90601,97969,105625,113569,121801

%N a(n) = (12*n + 1)^2.

%H Vincenzo Librandi, <a href="/A017534/b017534.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (1 + 166*x + 121*x^2 )/(1-x)^3. - _R. J. Mathar_, Mar 10 2011

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jul 07 2012

%F E.g.f.: (1 + 168*x + 144*x^2)*exp(x). - _G. C. Greubel_, Dec 24 2022

%t CoefficientList[Series[(1+166*x+121*x^2)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 07 2012 *)

%t LinearRecurrence[{3,-3,1},{1,169,625},30] (* _Harvey P. Dale_, Feb 27 2023 *)

%o (Magma) I:=[1, 169, 625]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Jul 07 2012

%o (PARI) a(n)=(12*n+1)^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%o (SageMath) [(12*n+1)^2 for n in range(51)] # _G. C. Greubel_, Dec 24 2022

%Y Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), this sequence (m=12), A134934 (m=14).

%Y Cf. A082043.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_