A316725 - OEIS

%I #43 Mar 01 2022 05:33:13

%S 0,1,24,27,73,78,147,154,246,255,370,381,519,532,693,708,892,909,1116,

%T 1135,1365,1386,1639,1662,1938,1963,2262,2289,2611,2640,2985,3016,

%U 3384,3417,3808,3843,4257,4294,4731,4770,5230,5271,5754,5797,6303,6348,6877,6924,7476,7525,8100,8151,8749,8802

%N Generalized 27-gonal (or icosiheptagonal) numbers: m*(25*m - 23)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

%C Note that in the sequences of generalized k-gonal numbers always a(3) = k. In this case k = 27.

%C Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, with k >= 5.

%C A general formula for the generalized k-gonal numbers is given by m*((k-2)*m-k+4)/2, with m = 0, +1, -1, +2, -2, +3, -3, ..., k >= 5.

%C Partial sums of A317323. - _Omar E. Pol_, Jul 28 2018

%H Colin Barker, <a href="/A316725/b316725.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F From _Colin Barker_, Jul 11 2018: (Start)

%F G.f.: x*(1 + 23*x + x^2) / ((1 - x)^3*(1 + x)^2).

%F a(n) = n*(25*n + 46)/8 for n even.

%F a(n) = (25*n - 21)*(n + 1)/8 for n odd.

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

%F (End)

%F Sum_{n>=1} 1/a(n) =  2*(25 + 23*Pi*cot(2*Pi/25))/529. - _Amiram Eldar_, Mar 01 2022

%p a:= n-> (m-> m*(25*m-23)/2)(-ceil(n/2)*(-1)^n):

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Jul 11 2018

%t CoefficientList[Series[-x (x^2 + 23x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 53}], x] (* or *)

%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 24, 27, 73, 78, 147}, 53] (* _Robert G. Wilson v_, Jul 28 2018; corrected by _Georg Fischer_, Apr 03 2019 *)

%t nn=30; Sort[Table[n (25 n - 23) / 2, {n, -nn, nn}]] (* _Vincenzo Librandi_, Jul 29 2018 *)

%o (PARI) concat(0, Vec(x*(1 + 23*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ _Colin Barker_, Jul 11 2018

%o (GAP) a:=[0,1,24,27,73];;  for n in [6..60] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # _Muniru A Asiru_, Jul 16 2018

%Y Cf. A255186; A317323.

%Y Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), this sequence (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Jul 11 2018