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A069099
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Centered heptagonal numbers.
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63
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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) = 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) = 2*a(n-1) - a(n-2) + 7. - Ant King, Jun 17 2012
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)
Sum_{n>=1} a(n)/n! = 9*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 9/(2*e) - 1. (End)
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EXAMPLE
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a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
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)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 8, 22}, 50] (* Harvey P. Dale, Jun 04 2011 *)
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PROG
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(Haskell)
a069099 n = length
[(x, y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n - 1]
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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