A195045 - OEIS

%I #39 Jan 16 2023 08:19:39

%S 0,1,13,27,52,79,117,157,208,261,325,391,468,547,637,729,832,937,1053,

%T 1171,1300,1431,1573,1717,1872,2029,2197,2367,2548,2731,2925,3121,

%U 3328,3537,3757,3979,4212,4447,4693,4941,5200,5461,5733,6007,6292,6579,6877,7177,7488,7801,8125

%N Concentric 13-gonal numbers.

%C Also concentric tridecagonal numbers or concentric triskaidecagonal numbers.

%C Partial sums of A175886. - _Reinhard Zumkeller_, Jan 07 2012

%H Vincenzo Librandi, <a href="/A195045/b195045.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) = 13*n^2/4+9*((-1)^n-1)/8.

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

%F G.f.: -x*(1+11*x+x^2) / ( (1+x)*(x-1)^3 ).

%F a(n)+a(n+1) = A069126(n+1). (End)

%F a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3. - _Wesley Ivan Hurt_, Nov 22 2015

%F Sum_{n>=1} 1/a(n) = Pi^2/78 + tan(3*Pi/(2*sqrt(13)))*Pi/(3*sqrt(13)). - _Amiram Eldar_, Jan 16 2023

%p A195045:=n->13*n^2/4+9*((-1)^n-1)/8: seq(A195045(n), n=0..70); # _Wesley Ivan Hurt_, Nov 22 2015

%t Table[13 n^2/4 + 9 ((-1)^n - 1)/8, {n, 0, 50}] (* _Wesley Ivan Hurt_, Nov 22 2015 *)

%o (Magma) [13*n^2/4+9*((-1)^n-1)/8: n in [0..50]]; // _Vincenzo Librandi_, Sep 29 2011

%o (Haskell)

%o a195045 n = a195045_list !! n

%o a195045_list = scanl (+) 0 a175886_list

%o -- _Reinhard Zumkeller_, Jan 07 2012

%o (PARI) a(n)=13*n^2/4+9*((-1)^n-1)/8 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (PARI) concat(0, Vec(-x*(1+11*x+x^2)/((1+x)*(x-1)^3) + O(x^50))) \\ _Altug Alkan_, Nov 22 2015

%Y Column 13 of A195040.

%Y Cf. A032527, A032528, A069126, A175886, A195043, A195046, A195143, A195145.

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Sep 27 2011