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A069099
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Centered heptagonal numbers.
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24
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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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Equals the triangular numbers convolved with [ 1, 5, 1, 0, 0, 0,...] [From Gary W. Adamson & Alexander R. Povolotsky, May 29 2009]
Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and x + y + 1 <= n. [Reinhard Zumkeller, Jan 23 2012]
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
E. Weisstein, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1)
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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) = 7*n+a(n-1)-7 (with a(1)=1) [From Vincenzo Librandi, Aug 08 2010]
G.f.: x*(1+5*x+x^2) / (1-x)^3 . - R. J. Mathar, Feb 04 2011
a(0)=1, a(1)=8, a(2)=22, a(n)=3*a(n-1)-3*a(n-2)+a(n-3) [From Harvey P. Dale, June 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
Contribution 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)
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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.
For n=2, a(2)=7*2+1-7=8; n=3, a(3)=7*3+8-7=22; n=4, a(4)=7*4+22-7=43 [From Vincenzo Librandi, Aug 08 2010]
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MATHEMATICA
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FoldList[#1 + #2 &, 1, 7 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 8, 22}, 50] (* From Harvey P. Dale, June 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]
-- Reinhard Zumkeller, Jan 23 2012
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CROSSREFS
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Cf. A000566 (heptagonal numbers).
Cf. A001263.
Cf. A057655, A001106.
Sequence in context: A058508 A134783 A211529 * A172473 A145067 A112684
Adjacent sequences: A069096 A069097 A069098 * A069100 A069101 A069102
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Terrel Trotter, Jr., Apr 05 2002
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jun 26 2002
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STATUS
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approved
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