This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001107 M4690 #200 Feb 16 2025 08:32:22
%S A001107 0,1,10,27,52,85,126,175,232,297,370,451,540,637,742,855,976,1105,
%T A001107 1242,1387,1540,1701,1870,2047,2232,2425,2626,2835,3052,3277,3510,
%U A001107 3751,4000,4257,4522,4795,5076,5365,5662,5967,6280,6601,6930,7267,7612,7965,8326
%N A001107 10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).
%C A001107 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 A001107 Number of divisors of 48^(n-1) for n > 0. - _J. Lowell_, Aug 30 2008
%C A001107 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 A001107 This sequence does not contain any squares other than 0 and 1. See A188896. - _T. D. Noe_, Apr 13 2011
%C A001107 For n > 0: right edge of the triangle A033293. - _Reinhard Zumkeller_, Jan 18 2012
%C A001107 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 A001107 Partial sums give A007585. - _Omar E. Pol_, Jan 15 2013
%C A001107 This is also a star pentagonal number: a(n) = A000326(n) + 5*A000217(n-1). - _Luciano Ancora_, Mar 28 2015
%C A001107 Also the number of undirected paths in the n-sunlet graph. - _Eric W. Weisstein_, Sep 07 2017
%C A001107 After 0, a(n) is the sum of 2*n consecutive integers starting from n-1. - _Bruno Berselli_, Jan 16 2018
%C A001107 Number of corona of an H0 hexagon with a T(n) triangle. - _Craig Knecht_, Dec 13 2024
%D A001107 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D A001107 Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.
%D A001107 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D A001107 S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
%D A001107 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
%D A001107 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001107 T. D. Noe, <a href="/A001107/b001107.txt">Table of n, a(n) for n = 0..1000</a>
%H A001107 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 A001107 Emilio Apricena, <a href="/A035608/a035608.png">A version of the Ulam spiral</a>.
%H A001107 Yin Choi Cheng, <a href="https://doi.org/10.1016/j.jnt.2024.07.010">Greedy Sidon sets for linear forms</a>, J. Num. Theor. (2024).
%H A001107 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=344">Encyclopedia of Combinatorial Structures 344</a>.
%H A001107 Craig Knecht, <a href="/A001107/a001107.png">Corona of the H0 hexagon with a T(n) triangle</a>.
%H A001107 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 A001107 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 A001107 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001107 Leo Tavares, <a href="/A001107/a001107.jpg">Illustration: Conjoined Hexagon/Square Pairs</a>
%H A001107 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarbellGraph.html">Barbell Graph</a>.
%H A001107 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DecagonalNumber.html">Decagonal Number</a>.
%H A001107 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>.
%H A001107 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a>.
%H A001107 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H A001107 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A001107 a(n) = A033954(-n) = A074377(2*n-1).
%F A001107 a(n) = n + 8*A000217(n-1). - _Floor van Lamoen_, Oct 14 2005
%F A001107 G.f.: x*(1 + 7*x)/(1 - x)^3.
%F A001107 Partial sums of odd numbers 1 mod 8, i.e., 1, 1 + 9, 1 + 9 + 17, ... . - _Jon Perry_, Dec 18 2004
%F A001107 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 A001107 Starting (1, 10, 27, 52, ...), this is the binomial transform of [1, 9, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Apr 30 2008
%F A001107 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 A001107 a(n) = 8*n + a(n-1) - 7 for n>0, a(0)=0. - _Vincenzo Librandi_, Jul 10 2010
%F A001107 a(n) = 8 + 2*a(n-1) - a(n-2). - _Ant King_, Sep 04 2011
%F A001107 a(n) = A118729(8*n). - _Philippe Deléham_, Mar 26 2013
%F A001107 a(8*a(n) + 29*n+1) = a(8*a(n) + 29*n) + a(8*n + 1). - _Vladimir Shevelev_, Jan 24 2014
%F A001107 Sum_{n >= 1} 1/a(n) = Pi/6 + log(2) = 1.216745956158244182494339352... = A244647. - _Vaclav Kotesovec_, Apr 27 2016
%F A001107 From _Ilya Gutkovskiy_, Aug 28 2016: (Start)
%F A001107 E.g.f.: x*(1 + 4*x)*exp(x).
%F A001107 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 A001107 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 A001107 Product_{n>=2} (1 - 1/a(n)) = 4/5. - _Amiram Eldar_, Jan 21 2021
%F A001107 a(n) = A003215(n-1) + A000290(n) - 1. - _Leo Tavares_, Jul 23 2022
%e A001107 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 A001107 99 64--65--66--67--68--69--70--71--72
%e A001107 | | |
%e A001107 98 63 36--37--38--39--40--41--42 73
%e A001107 | | | | |
%e A001107 97 62 35 16--17--18--19--20 43 74
%e A001107 | | | | | | |
%e A001107 96 61 34 15 4---5---6 21 44 75
%e A001107 | | | | | | | | |
%e A001107 95 60 33 14 3 *0* 7 22 45 76
%e A001107 | | | | | | | | | |
%e A001107 94 59 32 13 2--*1* 8 23 46 77
%e A001107 | | | | | | | |
%e A001107 93 58 31 12--11-*10*--9 24 47 78
%e A001107 | | | | | |
%e A001107 92 57 30--29--28-*27*-26--25 48 79
%e A001107 | | | |
%e A001107 91 56--55--54--53-*52*-51--50--49 80
%e A001107 | |
%e A001107 90--89--88--87--86-*85*-84--83--82--81
%e A001107 [Edited by _Jon E. Schoenfield_, Jan 02 2017]
%p A001107 A001107:=-(1+7*z)/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation
%t A001107 LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* _Harvey P. Dale_, May 08 2012 *)
%t A001107 Table[PolygonalNumber[RegularPolygon[10], n], {n, 0, 46}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)
%t A001107 Table[4 n^2 - 3 n, {n, 0, 49}] (* _Alonso del Arte_, Jan 24 2017 *)
%t A001107 PolygonalNumber[10, Range[0, 20]] (* _Eric W. Weisstein_, Sep 07 2017 *)
%t A001107 LinearRecurrence[{3, -3, 1}, {1, 10, 27}, {0, 20}] (* _Eric W. Weisstein_, Sep 07 2017 *)
%o A001107 (PARI) a(n)=4*n^2-3*n
%o A001107 (Magma) [4*n^2-3*n : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 05 2014
%o A001107 (Python) a=lambda n: 4*n**2-3*n # _Indranil Ghosh_, Jan 01 2017
%o A001107 def aList(): # Intended to compute the initial segment of the sequence, not isolated terms.
%o A001107 x, y = 1, 1
%o A001107 yield 0
%o A001107 while True:
%o A001107 yield x
%o A001107 x, y = x + y + 8, y + 8
%o A001107 A001107 = aList()
%o A001107 print([next(A001107) for i in range(49)]) # _Peter Luschny_, Aug 04 2019
%Y A001107 Cf. A007585, A028994.
%Y A001107 Cf. A093565 ((8, 1) Pascal, column m = 2). Partial sums of A017077.
%Y A001107 Sequences from spirals: A001107 (this), A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y A001107 Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y A001107 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 A001107 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 A001107 Cf. A003215.
%K A001107 nonn,easy,nice
%O A001107 0,3
%A A001107 _N. J. A. Sloane_