A069099 Centered heptagonal numbers.

1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246

Offset: 1

Author

Terrel Trotter, Jr., Apr 05 2002

Keywords

• nonn
• easy
• nice

Comments

Equals the triangular numbers convolved with [ 1, 5, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009

Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and abs(x + y) < n, counting twice pairs of equal numbers. - Reinhard Zumkeller, Jan 23 2012; corrected and extended by Mauro Fiorentini, Jan 01 2018

The number of pairs without repetitions is a(n) - 2n + 3 for n > 1. For example, there are 19 such pairs for n = 3: (-2, 0), (-2, 1), (-2, 2), (-1, -1), (-1, 0), (-1, 1), (-1, 2), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (1, -2), (1, -1), (1, 0), (1, 1), (2, -2), (2, -1), (2, 0). - Mauro Fiorentini, Jan 01 2018

Examples

			a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
From _Bruno Berselli_, Oct 27 2017: (Start)
1   =         -(0) + (1).
8   =       -(0+1) + (2+3+4).
22  =     -(0+1+2) + (3+4+5+6+7).
43  =   -(0+1+2+3) + (4+5+6+7+8+9+10).
71  = -(0+1+2+3+4) + (5+6+7+8+9+10+11+12+13). (End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Nicolas Bělohoubek and Viktor Javor, Centered heptagonal numbers appearing in aperiodic heptagonal tiling
• Leo Tavares, Illustration: Crystal Numbers
• Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
• Index entries for sequences related to centered polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000566 (heptagonal numbers).

Cf. A001263, A057655, A001106.

Cf. A003215, A000217.

Programs

Haskell

a069099 n = length
   [(x,y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n - 1]
-- Reinhard Zumkeller, Jan 23 2012

Mathematica

FoldList[#1 + #2 &, 1, 7 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,8,22},50] (* Harvey P. Dale, Jun 04 2011 *)

PARI

a(n)=(7*n^2-7*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = (7*n^2 - 7*n + 2)/2.

a(n) = 1 + Sum_{k=1..n} 7*k. - Xavier Acloque, Oct 26 2003

Binomial transform of [1, 7, 7, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 7, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007

a(n) = 7*n + a(n-1) - 7 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

G.f.: x*(1+5*x+x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=8, a(2)=22. - Harvey P. Dale, Jun 04 2011

a(n) = A024966(n-1) + 1. - Omar E. Pol, Oct 03 2011

a(n) = 2*a(n-1) - a(n-2) + 7. - Ant King, Jun 17 2012

From Ant King, Jun 17 2012: (Start)

Sum_{n>=1} 1/a(n) = 2*Pi/sqrt(7)*tanh(Pi/(2*sqrt(7))) = 1.264723171685652...

a(n) == 1 (mod 7) for all n.

The sequence of digital roots of the a(n) is period 9: repeat [1, 8, 4, 7, 8, 7, 4, 8, 1] (the period is a palindrome).

The sequence of a(n) mod 10 is period 20: repeat [1, 8, 2, 3, 1, 6, 8, 7, 3, 6, 6, 3, 7, 8, 6, 1, 3, 2, 8, 1] (the period is a palindrome).

(End)

E.g.f.: -1 + (2 + 7*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jun 30 2016

a(n) = A101321(7,n-1). - R. J. Mathar, Jul 28 2016

From Amiram Eldar, Jun 20 2020: (Start)

Sum_{n>=1} a(n)/n! = 9*e/2 - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 9/(2*e) - 1. (End)

a(n) = A003215(n-1) + A000217(n-1). - Leo Tavares, Jul 19 2022