login
Centered 44-gonal numbers.
7

%I #27 Nov 15 2024 13:10:53

%S 1,45,133,265,441,661,925,1233,1585,1981,2421,2905,3433,4005,4621,

%T 5281,5985,6733,7525,8361,9241,10165,11133,12145,13201,14301,15445,

%U 16633,17865,19141,20461,21825,23233,24685,26181,27721,29305,30933,32605,34321,36081,37885,39733

%N Centered 44-gonal numbers.

%C Sequence found by reading the line from 1, in the direction 1, 45, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195323 in the same spiral.

%H Vincenzo Librandi, <a href="/A195318/b195318.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = 22*n^2 - 22*n + 1.

%F Sum_{n>=1} 1/a(n) = Pi*tan(3*Pi/(2*sqrt(11)))/(6*sqrt(11)). - _Amiram Eldar_, Feb 11 2022

%F G.f.: -x*(1+42*x+x^2)/(x-1)^3. - _R. J. Mathar_, May 07 2024

%F From _Elmo R. Oliveira_, Nov 15 2024: (Start)

%F E.g.f.: exp(x)*(22*x^2 + 1) - 1.

%F a(n) = 2*A069173(n) - 1.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

%t Table[22n^2-22n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,45,133},50] (* _Harvey P. Dale_, Mar 16 2019 *)

%o (Magma) [22*n^2 - 22*n + 1: n in [1..50]]; // _Vincenzo Librandi_, Sep 21 2011

%o (PARI) a(n)=22*n^2-22*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Bisection of A195149.

%Y Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195316, A195317.

%Y Cf. A069173, A195313, A195323.

%K nonn,easy

%O 1,2

%A _Omar E. Pol_, Sep 16 2011