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%I A316729 #30 Mar 01 2022 05:33:09
%S A316729 0,1,27,30,82,87,165,172,276,285,415,426,582,595,777,792,1000,1017,
%T A316729 1251,1270,1530,1551,1837,1860,2172,2197,2535,2562,2926,2955,3345,
%U A316729 3376,3792,3825,4267,4302,4770,4807,5301,5340,5860,5901,6447,6490,7062,7107,7705,7752,8376,8425,9075,9126,9802,9855
%N A316729 Generalized 30-gonal (or triacontagonal) numbers: m*(14*m - 13) with m = 0, +1, -1, +2, -2, +3, -3, ...
%C A316729 Note that in the sequences of generalized k-gonal numbers always a(3) = k. In this case k = 30.
%C A316729 Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, with k >= 5.
%C A316729 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 A316729 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 A316729 56*a(n) + 169 is a square. - _Vincenzo Librandi_, Jul 12 2018
%C A316729 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 A316729 Also partial sums of A317326. - _Omar E. Pol_, Jul 28 2018
%H A316729 Colin Barker, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EzMTY3MjkvYjMxNjcyOS50eHQ">Table of n, a(n) for n = 0..1000</a>
%H A316729 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYyNvcmRlcl8wNQ">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A316729 G.f.: x*(1 + 26*x + x^2)/((1 + x)^2*(1 - x)^3). - _Vincenzo Librandi_, Jul 12 2018
%F A316729 From _Amiram Eldar_, Mar 01 2022: (Start)
%F A316729 a(n) = (28*n*(n + 1) + 12*(2*n + 1)*(-1)^n - 12)/8.
%F A316729 a(n) = n*(7*n + 13)/2, if n is even, or (n + 1)*(7*n - 6)/2 otherwise.
%F A316729 Sum_{n>=1} 1/a(n) = 14/169 + Pi*cot(Pi/14)/13. (End)
%t A316729 CoefficientList[Series[x (1 + 26 x + x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 55}], x] (* _Vincenzo Librandi_, Jul 12 2018 *)
%t A316729 LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 27, 30, 82}, 47] (* _Robert G. Wilson v_, Jul 28 2018 *)
%o A316729 (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 A316729 Cf. A254474, A317326.
%Y A316729 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 A316729 nonn,easy
%O A316729 0,3
%A A316729 _Omar E. Pol_, Jul 11 2018
%E A316729 Duplicated term (1551) deleted by _Colin Barker_, Jul 16 2018

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