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%I #66 Mar 31 2022 10:17:32
%S 1,12,34,67,111,166,232,309,397,496,606,727,859,1002,1156,1321,1497,
%T 1684,1882,2091,2311,2542,2784,3037,3301,3576,3862,4159,4467,4786,
%U 5116,5457,5809,6172,6546,6931,7327,7734,8152,8581,9021,9472,9934,10407,10891
%N a(n) = (11*n^2 - 11*n + 2)/2.
%C Centered hendecagonal (11-gonal) numbers. - _Omar E. Pol_, Oct 03 2011
%C Numbers of the form (2*m+1)^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - _Bruno Berselli_, Feb 11 2015
%H T. D. Noe, <a href="/A069125/b069125.txt">Table of n, a(n) for n = 1..1000</a>
%H X. Acloque, <a href="http://www.members.fortunecity.fr/polynexus/index.html">Polynexus Numbers and other mathematical wonders</a> [broken link]
%H Leo Tavares, <a href="/A069125/a069125.jpg">Illustration: Clipped Stars</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 1 + Sum_{j=0..n-1} (11*j). - Xavier Acloque, Oct 22 2003
%F Binomial transform of [1, 11, 11, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 11, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 29 2007
%F a(n) = 11*n + a(n-1) - 11 with n>1, a(1)=1. - _Vincenzo Librandi_, Aug 08 2010
%F G.f. -x*(1+9*x+x^2) / (x-1)^3. - _R. J. Mathar_, Jun 05 2011
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=12, a(2)=34. - _Harvey P. Dale_, Jun 25 2011
%F a(n) = A152740(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F From _Amiram Eldar_, Jun 21 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(3/11)*Pi/2)/sqrt(33).
%F Sum_{n>=1} a(n)/n! = 13*e/2 - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 13/(2*e) - 1. (End)
%F a(n) = A003154(n) - A000217(n-1). - _Leo Tavares_, Mar 29 2022
%e a(5)=111 because 111 = (11*5^2 - 11*5 + 2)/2 = (275 - 55 + 2)/2 = 222/2.
%t FoldList[#1 + #2 &, 1, 11 Range@ 45] (* _Robert G. Wilson v_ *)
%t Table[(11n^2-11n+2)/2,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{1,12,34},60] (* _Harvey P. Dale_, Jun 25 2011 *)
%o (PARI) a(n)=(11*n^2-11*n+2)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A001263, A001844, A003215, A005448, A005891, A069099.
%Y Cf. A003154, A000217.
%K nonn,easy,nice
%O 1,2
%A _Terrel Trotter, Jr._, Apr 07 2002
%E More terms from _Harvey P. Dale_, Jun 25 2011
%E Name rewritten by _Bruno Berselli_, Feb 11 2015
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