login
Generalized 14-gonal numbers: m*(6*m-5), m = 0,+1,-1,+2,-2,+3,-3,...
44

%I #55 Feb 28 2022 04:13:17

%S 0,1,11,14,34,39,69,76,116,125,175,186,246,259,329,344,424,441,531,

%T 550,650,671,781,804,924,949,1079,1106,1246,1275,1425,1456,1616,1649,

%U 1819,1854,2034,2071,2261,2300,2500,2541,2751,2794,3014,3059,3289

%N Generalized 14-gonal numbers: m*(6*m-5), m = 0,+1,-1,+2,-2,+3,-3,...

%C Also generalized tetradecagonal numbers or generalized tetrakaidecagonal numbers.

%C Also A211014 and positive terms of A051866 interleaved. - _Omar E. Pol_, Aug 04 2012

%C Exponents in expansion of Product_{n >= 1} (1 + x^(12*n-11))*(1 + x^(12*n-1))*(1 - x^(12*n)) = 1 + x + x^11 + x^14 + x^34 + .... - _Peter Bala_, Dec 10 2020

%H Vincenzo Librandi, <a href="/A195818/b195818.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = (3*n*(n+1) + (2*n+1)*(-1)^n - 1)/2. - _Vincenzo Librandi_, Sep 30 2011

%F G.f.: -x*(x^2+10*x+1) / ((x-1)^3*(x+1)^2). - _Colin Barker_, Sep 15 2013

%F Sum_{n>=1} 1/a(n) = 6/25 + sqrt(3)*Pi/5. - _Vaclav Kotesovec_, Oct 05 2016

%F E.g.f.: (x*(3*x + 4)*cosh(x) + (3*x^2 + 8*x - 2)*sinh(x))/2. - _Stefano Spezia_, Jun 08 2021

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (5*log(432)-6)/25. - _Amiram Eldar_, Feb 28 2022

%p a:= n-> (m-> m*(6*m-5))(ceil(-(n+1)/2)*(-1)^n):

%p seq(a(n), n=0..46); # _Alois P. Heinz_, Jun 08 2021

%t LinearRecurrence[{1,2,-2,-1,1},{0,1,11,14,34},50] (* _Harvey P. Dale_, Mar 13 2018 *)

%o (Magma) [(3*n*(n+1)+(2*n+1)*(-1)^n-1)/2: n in [0..60]]; // _Vincenzo Librandi_, Sep 30 2011

%o (Magma) A195818:=func<n | n*(6*n-5)>; [0] cat [A195818(n*m): m in [1,-1], n in [1..25]];

%o (PARI) Vec(-x*(x^2+10*x+1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ _Colin Barker_, Sep 15 2013

%Y Partial sums of A195817.

%Y Column 10 of A195152.

%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), this sequence (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), A316729 (k=30).

%Y Cf. A051866, A211014.

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Sep 29 2011