cp's OEIS Frontend

A051682 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.

%I A051682 #107 Dec 26 2022 09:47:20
%S A051682 0,1,11,30,58,95,141,196,260,333,415,506,606,715,833,960,1096,1241,
%T A051682 1395,1558,1730,1911,2101,2300,2508,2725,2951,3186,3430,3683,3945,
%U A051682 4216,4496,4785,5083,5390,5706,6031,6365,6708,7060,7421,7791,8170
%N A051682 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.
%C A051682 From _Floor van Lamoen_, Jul 21 2001: (Start)
%C A051682 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 A051682 The spiral begins:
%C A051682          15
%C A051682          / \
%C A051682        16  14
%C A051682        /     \
%C A051682      17   3  13
%C A051682      /   / \   \
%C A051682    18   4   2  12
%C A051682        /     \   \
%C A051682       5   0---1  11
%C A051682      /             \
%C A051682     6---7---8---9--10
%C A051682 . (End)
%C A051682 (1), (4+7), (7+10+13), (10+13+16+19), ... - _Jon Perry_, Sep 10 2004
%C A051682 This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - _T. D. Noe_, Apr 13 2011
%C A051682 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 A051682 Starting with offset 1, the sequence is the binomial transform of (1, 10, 9, 0, 0, 0, ...). - _Gary W. Adamson_, Aug 01 2015
%D A051682 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
%D A051682 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D A051682 Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
%H A051682 T. D. Noe, <a href="/A051682/b051682.txt">Table of n, a(n) for n = 0..1000</a>
%H A051682 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 A051682 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H A051682 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A051682 a(n) = n*(9*n-7)/2.
%F A051682 G.f.: x*(1+8*x)/(1-x)^3.
%F A051682 Row sums of triangle A131432. - _Gary W. Adamson_, Jul 10 2007
%F A051682 a(n) = 9*n + a(n-1) - 8 (with a(0)=0). - _Vincenzo Librandi_, Aug 06 2010
%F A051682 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 A051682 a(n) = A218470(9n). - _Philippe Deléham_, Mar 27 2013
%F A051682 a(9*a(n)+37*n+1) = a(9*a(n)+37*n) + a(9*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F A051682 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 A051682 a(n) = A000217(n-1) + A000217(3*n-2) - A000217(n-2). - _Charlie Marion_, Dec 21 2019
%F A051682 Product_{n>=2} (1 - 1/a(n)) = 9/11. - _Amiram Eldar_, Jan 21 2021
%F A051682 E.g.f.: exp(x)*x*(2 + 9*x)/2. - _Stefano Spezia_, Dec 25 2022
%t A051682 Table[n (9n-7)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,1,11},51] (* _Harvey P. Dale_, May 07 2012 *)
%o A051682 (PARI) a(n)=(9*n-7)*n/2 \\ _Charles R Greathouse IV_, Jun 16 2011
%o A051682 (Magma) [n*(9*n-7)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Aug 01 2015
%Y A051682 First differences of A007586.
%Y A051682 Cf. A093644 ((9, 1) Pascal, column m=2). Partial sums of A017173.
%Y A051682 Cf. A000217, A004188, A131432, A188892, A195160, A218470.
%K A051682 nonn,easy
%O A051682 0,3
%A A051682 _Barry E. Williams_