login
a(n) = 12*n^2 + 22*n + 11.
5

%I #32 Sep 08 2022 08:45:40

%S 11,45,103,185,291,421,575,753,955,1181,1431,1705,2003,2325,2671,3041,

%T 3435,3853,4295,4761,5251,5765,6303,6865,7451,8061,8695,9353,10035,

%U 10741,11471,12225,13003,13805,14631,15481,16355,17253,18175,19121

%N a(n) = 12*n^2 + 22*n + 11.

%C Sequence found by reading the line from 11, in the direction 11, 45,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - _Omar E. Pol_, Jul 18 2012

%H Vincenzo Librandi, <a href="/A154106/b154106.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 +x)*(11 +x)/(1-x)^3.

%F a(n) = 2*n*A016969(n+1) + 11.

%F a(0) = 11; for n > 0, a(n) = a(n-1) + 24*n + 10.

%F a(n) = 2 + A185918(n+1). - _Omar E. Pol_, Jul 18 2012

%F E.g.f.: (11 + 34*x + 12*x^2)*exp(x). - _G. C. Greubel_, Sep 02 2016

%e a(3) = 12*3^2 + 22*3 + 11 = 185 = 2*3*29 + 11 = 2*3*A016969(4) + 11.

%e a(4) = a(3) +24*4 +10 = 185 +96 +10 = 291.

%t Table[12n^2+22n+11,{n,0,50}] (* _Harvey P. Dale_, Mar 16 2011 *)

%t LinearRecurrence[{3,-3,1},{11,45,103}, 25] (* _G. C. Greubel_, Sep 02 2016 *)

%o (Magma) [ 12*n^2+22*n+11: n in [0..39] ];

%o (PARI) a(n)=12*n^2+22*n+11 \\ _Charles R Greathouse IV_, Oct 16 2015

%Y Cf. A016969 (6n+5), A153286, A185918, A194454.

%K nonn,easy

%O 0,1

%A _Klaus Brockhaus_, Jan 04 2009