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A051682
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11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.
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63
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0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, 3683, 3945, 4216, 4496, 4785, 5083, 5390, 5706, 6031, 6365, 6708, 7060, 7421, 7791, 8170
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OFFSET
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0,3
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COMMENTS
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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,... - Floor van Lamoen, Jul 21 2001. The spiral begins:
......15
....16..14
..17..3...13
18..4...2...12
..5...0...1...11
6...7...8...9...10
(1), (4+7), (7+10+13), (10+13+16+19), ... - Jon Perry, Sep 10 2004
This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - T. D. Noe, Apr 13 2011
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
Starting with offset 1, the sequence is the binomial transform of (1, 10, 9, 0, 0, 0, ...). - Gary W. Adamson, Aug 01 2015
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pps. 10-20, 79-94.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = n*(9*n-7)/2.
G.f.: x*(1+8*x)/(1-x)^3.
Row sums of triangle A131432. - Gary W. Adamson, Jul 10 2007
a(n) = 9*n + a(n-1) - 8 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
a(0)=0, a(1)=1, a(2)=11, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 07 2012
a(n) = A218470(9n). - Philippe Deléham, Mar 27 2013
a(9*a(n)+37*n+1) = a(9*a(n)+37*n) + a(9*n+1). - Vladimir Shevelev, Jan 24 2014
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
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MAPLE
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a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+9 od: seq(a[n], n=0..43); # Zerinvary Lajos, Feb 18 2008
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MATHEMATICA
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s=0; lst={s}; Do[s+=n++ +1; AppendTo[lst, s], {n, 0, 6!, 9}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Table[n (9n-7)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 11}, 51] (* Harvey P. Dale, May 07 2012 *)
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PROG
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(PARI) a(n)=(9*n-7)*n/2 \\ Charles R Greathouse IV, Jun 16 2011
(MAGMA) [n*(9*n-7)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 01 2015
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CROSSREFS
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First differences of A007586.
Cf. A093644 ((9, 1) Pascal, column m=2). Partial sums of A017173.
Cf. A004188, A131432, A195160, A218470.
Sequence in context: A162734 A163060 A247433 * A109943 A137411 A002755
Adjacent sequences: A051679 A051680 A051681 * A051683 A051684 A051685
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KEYWORD
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nonn,easy
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AUTHOR
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Barry E. Williams
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STATUS
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approved
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