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Generalized 30-gonal (or triacontagonal) numbers: m*(14*m - 13) with m = 0, +1, -1, +2, -2, +3, -3, ...
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%I #30 Mar 01 2022 05:33:09

%S 0,1,27,30,82,87,165,172,276,285,415,426,582,595,777,792,1000,1017,

%T 1251,1270,1530,1551,1837,1860,2172,2197,2535,2562,2926,2955,3345,

%U 3376,3792,3825,4267,4302,4770,4807,5301,5340,5860,5901,6447,6490,7062,7107,7705,7752,8376,8425,9075,9126,9802,9855

%N Generalized 30-gonal (or triacontagonal) numbers: m*(14*m - 13) 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 = 30.

%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 Every sequence of generalized k-gonal numbers can be represented as vertices of a rectangular spiral constructed with line segments on the square grid, with k >= 5.

%C 56*a(n) + 169 is a square. - _Vincenzo Librandi_, Jul 12 2018

%C Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, with k >= 5. - _Omar E. Pol_, Jul 27 2018

%C Also partial sums of A317326. - _Omar E. Pol_, Jul 28 2018

%H Colin Barker, <a href="/A316729/b316729.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 G.f.: x*(1 + 26*x + x^2)/((1 + x)^2*(1 - x)^3). - _Vincenzo Librandi_, Jul 12 2018

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

%F a(n) = (28*n*(n + 1) + 12*(2*n + 1)*(-1)^n - 12)/8.

%F a(n) = n*(7*n + 13)/2, if n is even, or (n + 1)*(7*n - 6)/2 otherwise.

%F Sum_{n>=1} 1/a(n) = 14/169 + Pi*cot(Pi/14)/13. (End)

%t CoefficientList[Series[x (1 + 26 x + x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 55}], x] (* _Vincenzo Librandi_, Jul 12 2018 *)

%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 27, 30, 82}, 47] (* _Robert G. Wilson v_, Jul 28 2018 *)

%o (PARI) concat(0, Vec(x*(1 + 26*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ _Colin Barker_, Jul 16 2018

%Y Cf. A254474, A317326.

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

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Jul 11 2018

%E Duplicated term (1551) deleted by _Colin Barker_, Jul 16 2018