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%I M4604 #178 Mar 06 2023 20:44:08
%S 0,1,9,24,46,75,111,154,204,261,325,396,474,559,651,750,856,969,1089,
%T 1216,1350,1491,1639,1794,1956,2125,2301,2484,2674,2871,3075,3286,
%U 3504,3729,3961,4200,4446,4699,4959,5226,5500,5781,6069,6364
%N 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.
%C Sequence found by reading the line from 0, in the direction 0, 9, ... and the parallel line from 1, in the direction 1, 24, ..., in the square spiral whose vertices are the generalized 9-gonal (enneagonal) numbers A118277. Also sequence found by reading the same lines in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - _Omar E. Pol_, Sep 10 2011
%C Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and x+y <= n. - _Reinhard Zumkeller_, Jan 23 2012
%C Partial sums give A007584. - _Omar E. Pol_, Jan 15 2013
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe and William A. Tedeschi, <a href="/A001106/b001106.txt">Table of n, a(n) for n = 0..10000</a> (1000 terms were computed by T. D. Noe)
%H S. Barbero, U. Cerruti and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero/barbero5.html">Transforming Recurrent Sequences by Using the Binomial and Invert Operators</a>, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
%H C. K. Cook and M. R. Bacon, <a href="https://www.fq.math.ca/Papers1/52-4/CookBacon4292014.pdf">Some polygonal number summation formulas</a>, Fib. Q., 52 (2014), 336-343.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=343">Encyclopedia of Combinatorial Structures 343</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonagonalNumber.html">Nonagonal Number</a>.
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (7*n - 5)*n/2.
%F G.f.: x*(1+6*x)/(1-x)^3. - _Simon Plouffe_ in his 1992 dissertation.
%F a(n) = n + 7*A000217(n-1). - _Floor van Lamoen_, Oct 14 2005
%F Starting (1, 9, 24, 46, 75, ...) gives the binomial transform of (1, 8, 7, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 22 2007
%F 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
%F 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
%F a(n) = 2*a(n-1) - a(n-2) + 7. - _Mohamed Bouhamida_, May 05 2010
%F a(n) = a(n-1) + 7*n - 6 (with a(0) = 0). - _Vincenzo Librandi_, Nov 12 2010
%F a(n) = A174738(7n). - _Philippe Deléham_, Mar 26 2013
%F a(7*a(n) + 22*n + 1) = a(7*a(n) + 22*n) + a(7*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F E.g.f.: x*(2 + 7*x)*exp(x)/2. - _Ilya Gutkovskiy_, Jul 28 2016
%F a(n+2) + A000217(n) = (2*n+3)^2. - _Ezhilarasu Velayutham_, Mar 18 2020
%F Product_{n>=2} (1 - 1/a(n)) = 7/9. - _Amiram Eldar_, Jan 21 2021
%F Sum_{n>=1} 1/a(n) = A244646. - _Amiram Eldar_, Nov 12 2021
%F a(n) = A000217(3*n-2) - (n-1)^2. - _Charlie Marion_, Feb 27 2022
%F 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
%t Table[n(7n - 5)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 9}, 50] (* _Harvey P. Dale_, Nov 06 2011 *)
%t (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[9], n], {n, 0, 43}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)
%t PolygonalNumber[9,Range[0,50]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Nov 19 2019 *)
%o (PARI) a(n)=n*(7*n-5)/2 \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Haskell)
%o a001106 n = length [(x,y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n]
%o -- _Reinhard Zumkeller_, Jan 23 2012
%o (Haskell) a001106 n = n*(7*n-5) `div` 2 -- _James Spahlinger_, Oct 18 2012
%o (Python 3)
%o def aList(): # Intended to compute the initial segment of the sequence, not isolated terms.
%o x, y = 1, 1
%o yield 0
%o while True:
%o yield x
%o x, y = x + y + 7, y + 7
%o A001106 = aList()
%o print([next(A001106) for i in range(49)]) # _Peter Luschny_, Aug 04 2019
%Y Cf. A093564 ((7, 1) Pascal, column m=2). Partial sums of A016993.
%Y Cf. A131875, A057655, A069099, A244646.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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