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%I #59 Jan 16 2023 08:52:52
%S 0,1,10,21,40,61,90,121,160,201,250,301,360,421,490,561,640,721,810,
%T 901,1000,1101,1210,1321,1440,1561,1690,1821,1960,2101,2250,2401,2560,
%U 2721,2890,3061,3240,3421,3610,3801,4000,4201,4410,4621,4840,5061,5290
%N Concentric 10-gonal numbers.
%C Also concentric decagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 10, ..., and the same line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Main axis, perpendicular to A028895 in the same spiral.
%H Vincenzo Librandi, <a href="/A195142/b195142.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+8*x+x^2) / ( (1+x)*(x-1)^3 ). - _R. J. Mathar_, Sep 18 2011
%F a(n) = -a(n-1) + 5*n^2 - 5*n + 1, a(0)=0. - _Vincenzo Librandi_, Sep 27 2011
%F From _Bruno Berselli_, Sep 27 2011: (Start)
%F a(n) = a(-n) = (10*n^2 + 3*(-1)^n - 3)/4.
%F a(n) = a(n-2) + 10*(n-1). (End)
%F a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=10, a(3)=21. - _Harvey P. Dale_, Sep 29 2011
%F Sum_{n>=1} 1/a(n) = Pi^2/60 + tan(sqrt(3/5)*Pi/2)*Pi/(2*sqrt(15)). - _Amiram Eldar_, Jan 16 2023
%t RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+10(n-1)},a[n],{n,50}] (* or *) LinearRecurrence[{2,0,-2,1},{0,1,10,21},50] (* _Harvey P. Dale_, Sep 29 2011 *)
%o (Magma) [(10*n^2+3*(-1)^n-3)/4: n in [0..50]]; // _Vincenzo Librandi_, Sep 27 2011
%o (Haskell)
%o a195142 n = a195142_list !! n
%o a195142_list = scanl (+) 0 a090771_list
%o -- _Reinhard Zumkeller_, Jan 07 2012
%Y A033583 and A069133 interleaved.
%Y Cf. A028895, A032527, A032528, A077221, A085787, A195042, A195143, A195145, A195146, A195147, A195148, A195149.
%Y Cf. A090771 (first differences).
%Y Column 10 of A195040. - _Omar E. Pol_, Sep 28 2011
%K nonn,easy
%O 0,3
%A _Omar E. Pol_, Sep 17 2011
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