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%I M4690 #188 Sep 12 2022 08:38:02
%S 0,1,10,27,52,85,126,175,232,297,370,451,540,637,742,855,976,1105,
%T 1242,1387,1540,1701,1870,2047,2232,2425,2626,2835,3052,3277,3510,
%U 3751,4000,4257,4522,4795,5076,5365,5662,5967,6280,6601,6930,7267,7612,7965,8326
%N 10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).
%C Write 0, 1, 2, ... in a square spiral, with 0 at the origin and 1 immediately below it; sequence gives numbers on the negative y-axis (see Example section).
%C Number of divisors of 48^(n-1) for n > 0. - _J. Lowell_, Aug 30 2008
%C a(n) is the Wiener index of the graph obtained by connecting two copies of the complete graph K_n by an edge (for n = 3, approximately: |>-<|). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. - _Emeric Deutsch_, Sep 20 2010
%C This sequence does not contain any squares other than 0 and 1. See A188896. - _T. D. Noe_, Apr 13 2011
%C For n > 0: right edge of the triangle A033293. - _Reinhard Zumkeller_, Jan 18 2012
%C Sequence found by reading the line from 0, in the direction 0, 10, ... and the parallel line from 1, in the direction 1, 27, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Jul 18 2012
%C Partial sums give A007585. - _Omar E. Pol_, Jan 15 2013
%C This is also a star pentagonal number: a(n) = A000326(n) + 5*A000217(n-1). - _Luciano Ancora_, Mar 28 2015
%C Also the number of undirected paths in the n-sunlet graph. - _Eric W. Weisstein_, Sep 07 2017
%C After 0, a(n) is the sum of 2*n consecutive integers starting from n-1. - _Bruno Berselli_, Jan 16 2018
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001107/b001107.txt">Table of n, a(n) for n = 0..1000</a>
%H Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, <a href="https://arxiv.org/abs/1807.08899">The Bateman-Horn Conjecture: Heuristics, History, and Applications</a>, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.3 p. 33.
%H Emilio Apricena, <a href="/A035608/a035608.png">A version of the Ulam spiral</a>.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=344">Encyclopedia of Combinatorial Structures 344</a>.
%H Minh Nguyen, <a href="https://aquila.usm.edu/honors_theses/777/">2-adic Valuations of Square Spiral Sequences</a>, Honors Thesis, Univ. of Southern Mississippi (2021).
%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 Leo Tavares, <a href="/A001107/a001107.jpg">Illustration: Conjoined Hexagon/Square Pairs</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarbellGraph.html">Barbell Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DecagonalNumber.html">Decagonal Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphPath.html">Graph Path</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</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) = A033954(-n) = A074377(2*n-1).
%F a(n) = n + 8*A000217(n-1). - _Floor van Lamoen_, Oct 14 2005
%F G.f.: x*(1 + 7*x)/(1 - x)^3.
%F Partial sums of odd numbers 1 mod 8, i.e., 1, 1 + 9, 1 + 9 + 17, ... . - _Jon Perry_, Dec 18 2004
%F 1^3 + 3^3*(n-1)/(n+1) + 5^3*((n-1)*(n-2))/((n+1)*(n+2)) + 7^3*((n-1)*(n-2)*(n-3))/((n+1)*(n+2)*(n+3)) + ... = n*(4*n-3) [Ramanujan]. - Neven Juric, Apr 15 2008
%F Starting (1, 10, 27, 52, ...), this is the binomial transform of [1, 9, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Apr 30 2008
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=1, a(2)=10. - _Jaume Oliver Lafont_, Dec 02 2008
%F a(n) = 8*n + a(n-1) - 7 for n>0, a(0)=0. - _Vincenzo Librandi_, Jul 10 2010
%F a(n) = 8 + 2*a(n-1) - a(n-2). - _Ant King_, Sep 04 2011
%F a(n) = A118729(8*n). - _Philippe Deléham_, Mar 26 2013
%F a(8*a(n) + 29*n+1) = a(8*a(n) + 29*n) + a(8*n + 1). - _Vladimir Shevelev_, Jan 24 2014
%F Sum_{n >= 1} 1/a(n) = Pi/6 + log(2) = 1.216745956158244182494339352... = A244647. - _Vaclav Kotesovec_, Apr 27 2016
%F From _Ilya Gutkovskiy_, Aug 28 2016: (Start)
%F E.g.f.: x*(1 + 4*x)*exp(x).
%F Sum_{n >= 1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 2*log(2) + 2*sqrt(2)*log(1 + sqrt(2)))/6 = 0.92491492293323294695... (End)
%F a(n) = A000217(3*n-2) - A000217(n-2). In general, if P(k,n) be the n-th k-gonal number and T(n) be the n-th triangular number, A000217(n), then P(T(k),n) = T((k-1)*n - (k-2)) - T(k-3)*T(n-2). - _Charlie Marion_, Sep 01 2020
%F Product_{n>=2} (1 - 1/a(n)) = 4/5. - _Amiram Eldar_, Jan 21 2021
%F a(n) = A003215(n-1) + A000290(n) - 1. - _Leo Tavares_, Jul 23 2022
%e On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie along the negative y-axis, as seen in the example below:
%e 99 64--65--66--67--68--69--70--71--72
%e | | |
%e 98 63 36--37--38--39--40--41--42 73
%e | | | | |
%e 97 62 35 16--17--18--19--20 43 74
%e | | | | | | |
%e 96 61 34 15 4---5---6 21 44 75
%e | | | | | | | | |
%e 95 60 33 14 3 *0* 7 22 45 76
%e | | | | | | | | | |
%e 94 59 32 13 2--*1* 8 23 46 77
%e | | | | | | | |
%e 93 58 31 12--11-*10*--9 24 47 78
%e | | | | | |
%e 92 57 30--29--28-*27*-26--25 48 79
%e | | | |
%e 91 56--55--54--53-*52*-51--50--49 80
%e | |
%e 90--89--88--87--86-*85*-84--83--82--81
%e [Edited by _Jon E. Schoenfield_, Jan 02 2017]
%p A001107:=-(1+7*z)/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation
%t LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* _Harvey P. Dale_, May 08 2012 *)
%t Table[PolygonalNumber[RegularPolygon[10], n], {n, 0, 46}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)
%t Table[4 n^2 - 3 n, {n, 0, 49}] (* _Alonso del Arte_, Jan 24 2017 *)
%t PolygonalNumber[10, Range[0, 20]] (* _Eric W. Weisstein_, Sep 07 2017 *)
%t LinearRecurrence[{3, -3, 1}, {1, 10, 27}, {0, 20}] (* _Eric W. Weisstein_, Sep 07 2017 *)
%o (PARI) a(n)=4*n^2-3*n
%o (Magma) [4*n^2-3*n : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 05 2014
%o (Python) a=lambda n: 4*n**2-3*n # _Indranil Ghosh_, Jan 01 2017
%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 + 8, y + 8
%o A001107 = aList()
%o print([next(A001107) for i in range(49)]) # _Peter Luschny_, Aug 04 2019
%Y Cf. A007585, A028994.
%Y Cf. A093565 ((8, 1) Pascal, column m = 2). Partial sums of A017077.
%Y Sequences from spirals: A001107 (this), A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%Y Cf. A003215.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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