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 A069099 Centered heptagonal numbers. 63
 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 (list; graph; refs; listen; history; text; internal format)
 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 sequences related to 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) a(n) = A003215(n-1) + A000217(n-1). - Leo Tavares, Jul 19 2022 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 Cf. A000566 (heptagonal numbers). Cf. A001263, A057655, A001106. Cf. A003215, A000217. Sequence in context: A058508 A134783 A211529 * A172473 A145067 A112684 Adjacent sequences: A069096 A069097 A069098 * A069100 A069101 A069102 KEYWORD nonn,easy,nice AUTHOR Terrel Trotter, Jr., Apr 05 2002 STATUS approved

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