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

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

Offset: 0

Author

Barry E. Williams

Keywords

• nonn
• easy

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

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

First differences of A007586.

Cf. A093644 ((9, 1) Pascal, column m=2). Partial sums of A017173.

Cf. A000217, A004188, A131432, A188892, A195160, A218470.

Programs

Magma

[n*(9*n-7)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 01 2015

Mathematica

Table[n (9n-7)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,1,11},51] (* Harvey P. Dale, May 07 2012 *)

PARI

a(n)=(9*n-7)*n/2 \\ Charles R Greathouse IV, Jun 16 2011

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