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Concentric 9-gonal numbers.
9

%I #34 Jan 16 2023 08:18:59

%S 0,1,9,19,36,55,81,109,144,181,225,271,324,379,441,505,576,649,729,

%T 811,900,991,1089,1189,1296,1405,1521,1639,1764,1891,2025,2161,2304,

%U 2449,2601,2755,2916,3079,3249,3421,3600,3781,3969,4159,4356,4555,4761,4969,5184,5401,5625

%N Concentric 9-gonal numbers.

%C Also concentric enneagonal numbers or concentric nonagonal numbers.

%C A016766 and A069131 interleaved.

%C Partial sums of A056020. - _Reinhard Zumkeller_, Jan 07 2012

%H Vincenzo Librandi, <a href="/A195042/b195042.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 a(n) = (9*n^2 + 5/2*((-1)^n - 1))/4.

%F From _R. J. Mathar_, Sep 28 2011: (Start)

%F G.f.: -x*(1+7*x+x^2) / ( (1+x)*(x-1)^3 ).

%F a(n) + a(n+1) = A060544(n+1). (End)

%F Sum_{n>=1} 1/a(n) = Pi^2/54 + tan(sqrt(5)*Pi/6)*Pi/(3*sqrt(5)). - _Amiram Eldar_, Jan 16 2023

%t LinearRecurrence[{2,0,-2,1},{0,1,9,19},60] (* _Harvey P. Dale_, Nov 24 2019 *)

%o (Magma) [(9*n^2+5/2*((-1)^n-1))/4: n in [0..50]]; // _Vincenzo Librandi_, Sep 29 2011

%o (Haskell)

%o a195042 n = a195042_list !! n

%o a195042_list = scanl (+) 0 a056020_list

%o -- _Reinhard Zumkeller_, Jan 07 2012

%o (PARI) a(n)=(9*n^2+5/2*((-1)^n-1))/4 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Column 9 of A195040.

%Y Cf. A016766, A069131, A077221, A195041, A195142, A195043.

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Sep 27 2011