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Concentric 22-gonal numbers.
15

%I #35 Jan 17 2023 09:24:00

%S 0,1,22,45,88,133,198,265,352,441,550,661,792,925,1078,1233,1408,1585,

%T 1782,1981,2200,2421,2662,2905,3168,3433,3718,4005,4312,4621,4950,

%U 5281,5632,5985,6358,6733,7128,7525,7942,8361,8800,9241,9702,10165,10648,11133

%N Concentric 22-gonal numbers.

%C Sequence found by reading the line from 0, in the direction 0, 22,..., and the same line from 1, in the direction 1, 45,..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Main axis, perpendicular to A152740 in the same spiral.

%H Vincenzo Librandi, <a href="/A195149/b195149.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F G.f.: -x*(1+20*x+x^2) / ( (1+x)*(x-1)^3 ). - _R. J. Mathar_, Sep 18 2011

%F a(n) = (22*n^2+9*(-1)^n-9)/4; a(n) = -a(n-1)+11*n^2-11*n+1. - _Vincenzo Librandi_, Sep 27 2011

%F Sum_{n>=1} 1/a(n) = Pi^2/132 + tan(3*Pi/(2*sqrt(11)))*Pi/(6*sqrt(11)). - _Amiram Eldar_, Jan 17 2023

%p A195149:=n->(22*n^2+9*(-1)^n-9)/4: seq(A195149(n), n=0..50); # _Wesley Ivan Hurt_, Jul 07 2014

%t Table[(22*n^2 + 9*(-1)^n - 9)/4, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jul 07 2014 *)

%o (Magma) [(22*n^2+9*(-1)^n-9)/4: n in [0..50]]; // _Vincenzo Librandi_, Sep 27 2011

%o (PARI) a(n)=(22*n^2+9*(-1)^n-9)/4 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y A195323 and A195318 interleaved.

%Y Cf. A032528, A077221, A195142, A195143, A195145, A195146, A195147, A195148.

%Y Cf. A032527, A195049, A195058. Column 22 of A195040. - _Omar E. Pol_, Sep 29 2011

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Sep 17 2011