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%I #107 Dec 26 2022 09:47:20
%S 0,1,11,30,58,95,141,196,260,333,415,506,606,715,833,960,1096,1241,
%T 1395,1558,1730,1911,2101,2300,2508,2725,2951,3186,3430,3683,3945,
%U 4216,4496,4785,5083,5390,5706,6031,6365,6708,7060,7421,7791,8170
%N 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.
%C From _Floor van Lamoen_, Jul 21 2001: (Start)
%C Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the line from 0 in the direction 0,1,...
%C The spiral begins:
%C 15
%C / \
%C 16 14
%C / \
%C 17 3 13
%C / / \ \
%C 18 4 2 12
%C / \ \
%C 5 0---1 11
%C / \
%C 6---7---8---9--10
%C . (End)
%C (1), (4+7), (7+10+13), (10+13+16+19), ... - _Jon Perry_, Sep 10 2004
%C This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - _T. D. Noe_, Apr 13 2011
%C Sequence found by reading the line from 0, in the direction 0, 11, ... and the parallel line from 1, in the direction 1, 30, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - _Omar E. Pol_, Jul 18 2012
%C Starting with offset 1, the sequence is the binomial transform of (1, 10, 9, 0, 0, 0, ...). - _Gary W. Adamson_, Aug 01 2015
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
%H T. D. Noe, <a href="/A051682/b051682.txt">Table of n, a(n) for n = 0..1000</a>
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to 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) = n*(9*n-7)/2.
%F G.f.: x*(1+8*x)/(1-x)^3.
%F Row sums of triangle A131432. - _Gary W. Adamson_, Jul 10 2007
%F a(n) = 9*n + a(n-1) - 8 (with a(0)=0). - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=1, a(2)=11. - _Harvey P. Dale_, May 07 2012
%F a(n) = A218470(9n). - _Philippe Deléham_, Mar 27 2013
%F a(9*a(n)+37*n+1) = a(9*a(n)+37*n) + a(9*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F a(n+y) - a(n-y-1) = (a(n+x) - a(n-x-1))*(2*y+1)/(2*x+1), 0 <= x < n, y <= x, a(0)=0. - _Gionata Neri_, May 03 2015
%F a(n) = A000217(n-1) + A000217(3*n-2) - A000217(n-2). - _Charlie Marion_, Dec 21 2019
%F Product_{n>=2} (1 - 1/a(n)) = 9/11. - _Amiram Eldar_, Jan 21 2021
%F E.g.f.: exp(x)*x*(2 + 9*x)/2. - _Stefano Spezia_, Dec 25 2022
%t Table[n (9n-7)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,1,11},51] (* _Harvey P. Dale_, May 07 2012 *)
%o (PARI) a(n)=(9*n-7)*n/2 \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Magma) [n*(9*n-7)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Aug 01 2015
%Y First differences of A007586.
%Y Cf. A093644 ((9, 1) Pascal, column m=2). Partial sums of A017173.
%Y Cf. A000217, A004188, A131432, A188892, A195160, A218470.
%K nonn,easy
%O 0,3
%A _Barry E. Williams_
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