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A051682 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2. 69
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Floor van Lamoen, Jul 21 2001: (Start)
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,...
The spiral begins:
15
/ \
16 14
/ \
17 3 13
/ / \ \
18 4 2 12
/ \ \
5 0---1 11
/ \
6---7---8---9--10
. (End)
(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
REFERENCES
Albert 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, pp. 10-20, 79-94.
LINKS
FORMULA
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(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
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
a(n) = A000217(n-1) + A000217(3*n-2) - A000217(n-2). - Charlie Marion, Dec 21 2019
Product_{n>=2} (1 - 1/a(n)) = 9/11. - Amiram Eldar, Jan 21 2021
E.g.f.: exp(x)*x*(2 + 9*x)/2. - Stefano Spezia, Dec 25 2022
MATHEMATICA
Table[n (9n-7)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 11}, 51] (* Harvey P. Dale, May 07 2012 *)
PROG
(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
CROSSREFS
First differences of A007586.
Cf. A093644 ((9, 1) Pascal, column m=2). Partial sums of A017173.
Sequence in context: A162734 A163060 A247433 * A109943 A303856 A137411
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified June 21 01:13 EDT 2024. Contains 373535 sequences. (Running on oeis4.)