OFFSET
1,2
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
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 linear recurrences with constant coefficients, signature (3,-3,1).
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)
EXAMPLE
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)
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 *)
PROG
(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
(PARI) a(n)=(7*n^2-7*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Terrel Trotter, Jr., Apr 05 2002
STATUS
approved