A001106 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.
0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364
Offset: 0
References
- Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
- E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 0..10000 (1000 terms were computed by T. D. Noe)
- S. Barbero, U. Cerruti and N. Murru, Transforming Recurrent Sequences by Using the Binomial and Invert Operators, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
- C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 343
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Eric Weisstein's World of Mathematics, Nonagonal Number.
- Index to sequences related to polygonal numbers
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Crossrefs
Programs
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Haskell
a001106 n = length [(x,y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n] -- Reinhard Zumkeller, Jan 23 2012
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Haskell
a001106 n = n*(7*n-5) `div` 2 -- James Spahlinger, Oct 18 2012
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Mathematica
Table[n(7n - 5)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 9}, 50] (* Harvey P. Dale, Nov 06 2011 *) (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[9], n], {n, 0, 43}] (* Arkadiusz Wesolowski, Aug 27 2016 *) PolygonalNumber[9,Range[0,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 19 2019 *) -
PARI
a(n)=n*(7*n-5)/2 \\ Charles R Greathouse IV, Jun 10 2011
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Python
def aList(): # Intended to compute the initial segment of the sequence, not isolated terms. x, y = 1, 1 yield 0 while True: yield x x, y = x + y + 7, y + 7 A001106 = aList() print([next(A001106) for i in range(49)]) # Peter Luschny, Aug 04 2019
Formula
a(n) = (7*n - 5)*n/2.
G.f.: x*(1+6*x)/(1-x)^3. - Simon Plouffe in his 1992 dissertation.
a(n) = n + 7*A000217(n-1). - Floor van Lamoen, Oct 14 2005
Starting (1, 9, 24, 46, 75, ...) gives the binomial transform of (1, 8, 7, 0, 0, 0, ...). - Gary W. Adamson, Jul 22 2007
Row sums of triangle A131875 starting (1, 9, 24, 46, 75, 111, ...). A001106 = binomial transform of (1, 8, 7, 0, 0, 0, ...). - Gary W. Adamson, Jul 22 2007
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 1, a(2) = 9. - Jaume Oliver Lafont, Dec 02 2008
a(n) = 2*a(n-1) - a(n-2) + 7. - Mohamed Bouhamida, May 05 2010
a(n) = a(n-1) + 7*n - 6 (with a(0) = 0). - Vincenzo Librandi, Nov 12 2010
a(n) = A174738(7n). - Philippe Deléham, Mar 26 2013
a(7*a(n) + 22*n + 1) = a(7*a(n) + 22*n) + a(7*n+1). - Vladimir Shevelev, Jan 24 2014
E.g.f.: x*(2 + 7*x)*exp(x)/2. - Ilya Gutkovskiy, Jul 28 2016
a(n+2) + A000217(n) = (2*n+3)^2. - Ezhilarasu Velayutham, Mar 18 2020
Product_{n>=2} (1 - 1/a(n)) = 7/9. - Amiram Eldar, Jan 21 2021
Sum_{n>=1} 1/a(n) = A244646. - Amiram Eldar, Nov 12 2021
a(n) = A000217(3*n-2) - (n-1)^2. - Charlie Marion, Feb 27 2022
a(n) = 3*A000217(n) + 2*A005563(n-2). In general, if P(k,n) = the n-th k-gonal number, then P(m*k,n) = m*P(k,n) + (m-1)*A005563(n-2). - Charlie Marion, Feb 21 2023
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