A316724 - OEIS

%I #38 Sep 25 2024 10:28:46

%S 0,1,23,26,70,75,141,148,236,245,355,366,498,511,665,680,856,873,1071,

%T 1090,1310,1331,1573,1596,1860,1885,2171,2198,2506,2535,2865,2896,

%U 3248,3281,3655,3690,4086,4123,4541,4580,5020,5061,5523,5566,6050,6095,6601,6648,7176,7225,7775

%N Generalized 26-gonal (or icosihexagonal) numbers: m*(12*m - 11) with m = 0, +1, -1, +2, -2, +3, -3, ...

%C 48*a(n) + 121 is a square. - _Bruno Berselli_, Jul 11 2018

%C Partial sums of A317322. - _Omar E. Pol_, Jul 28 2018

%H Colin Barker, <a href="/A316724/b316724.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 _Bruno Berselli_, Jul 11 2018: (Start)

%F O.g.f.: x*(1 + 22*x + x^2)/((1 + x)^2*(1 - x)^3).

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

%F a(n) = (12*n*(n + 1) + 5*(2*n + 1)*(-1)^n - 5)/4. Therefore:

%F a(n) = n*(6*n + 11)/2 for n even; otherwise, a(n) = (n + 1)*(6*n - 5)/2.

%F (2*n - 1)*a(n) + (2*n + 1)*a(n-1) - n*(12*n^2 - 11) = 0. (End)

%F From _Amiram Eldar_, Mar 01 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 12/121 + (sqrt(3)+2)*Pi/11.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(3)*log(sqrt(3)+2) + 6*log(2) + 3*log(3))/11 - 12/121. (End)

%F E.g.f.: (1/4)*(5*(1 - 2*x)*exp(-x) + (-5 + 24*x + 12*x^2)*exp(x)). - _G. C. Greubel_, Sep 24 2024

%t Table[(12 n (n + 1) + 5 (2 n + 1) (-1)^n - 5)/4, {n, 0, 60}] (* _Bruno Berselli_, Jul 11 2018 *)

%t CoefficientList[ Series[-x (x^2 + 22x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 60}], x] (* or *)

%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 23, 26, 70}, 60] (* _Robert G. Wilson v_, Jul 28 2018 *)

%t nn=30; Sort[Table[n (12 n - 11), {n, -nn, nn}]] (* _Vincenzo Librandi_, Jul 29 2018 *)

%o (PARI) concat(0, Vec(x*(1 + 22*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^60))) \\ _Colin Barker_, Jul 12 2018

%o (Magma)

%o [(12*n*(n+1) + 5*(-1)^n*(2*n+1) -5)/4: n in [0..60]]; // _G. C. Greubel_, Sep 24 2024

%o (SageMath)

%o [(12*n*(n+1) + 5*(-1)^n*(2*n+1) -5)//4 for n in range(61)] # _G. C. Greubel_, Sep 24 2024

%Y Cf. A255185, A317322.

%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), this sequence (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Jul 11 2018