OFFSET
0,3
COMMENTS
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).
Number of divisors of 48^(n-1) for n > 0. - J. Lowell, Aug 30 2008
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
This sequence does not contain any squares other than 0 and 1. See A188896. - T. D. Noe, Apr 13 2011
For n > 0: right edge of the triangle A033293. - Reinhard Zumkeller, Jan 18 2012
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
Partial sums give A007585. - Omar E. Pol, Jan 15 2013
This is also a star pentagonal number: a(n) = A000326(n) + 5*A000217(n-1). - Luciano Ancora, Mar 28 2015
Also the number of undirected paths in the n-sunlet graph. - Eric W. Weisstein, Sep 07 2017
After 0, a(n) is the sum of 2*n consecutive integers starting from n-1. - Bruno Berselli, Jan 16 2018
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.3 p. 33.
Emilio Apricena, A version of the Ulam spiral.
Yin Choi Cheng, Greedy Sidon sets for linear forms, J. Num. Theor. (2024).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 344.
Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021).
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
Leo Tavares, Illustration: Conjoined Hexagon/Square Pairs
Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Decagonal Number.
Eric Weisstein's World of Mathematics, Graph Path.
Eric Weisstein's World of Mathematics, Sunlet Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = n + 8*A000217(n-1). - Floor van Lamoen, Oct 14 2005
G.f.: x*(1 + 7*x)/(1 - x)^3.
Partial sums of odd numbers 1 mod 8, i.e., 1, 1 + 9, 1 + 9 + 17, ... . - Jon Perry, Dec 18 2004
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
Starting (1, 10, 27, 52, ...), this is the binomial transform of [1, 9, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
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
a(n) = 8*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Jul 10 2010
a(n) = 8 + 2*a(n-1) - a(n-2). - Ant King, Sep 04 2011
a(n) = A118729(8*n). - Philippe Deléham, Mar 26 2013
a(8*a(n) + 29*n+1) = a(8*a(n) + 29*n) + a(8*n + 1). - Vladimir Shevelev, Jan 24 2014
Sum_{n >= 1} 1/a(n) = Pi/6 + log(2) = 1.216745956158244182494339352... = A244647. - Vaclav Kotesovec, Apr 27 2016
From Ilya Gutkovskiy, Aug 28 2016: (Start)
E.g.f.: x*(1 + 4*x)*exp(x).
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)
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
Product_{n>=2} (1 - 1/a(n)) = 4/5. - Amiram Eldar, Jan 21 2021
EXAMPLE
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:
99 64--65--66--67--68--69--70--71--72
| | |
98 63 36--37--38--39--40--41--42 73
| | | | |
97 62 35 16--17--18--19--20 43 74
| | | | | | |
96 61 34 15 4---5---6 21 44 75
| | | | | | | | |
95 60 33 14 3 *0* 7 22 45 76
| | | | | | | | | |
94 59 32 13 2--*1* 8 23 46 77
| | | | | | | |
93 58 31 12--11-*10*--9 24 47 78
| | | | | |
92 57 30--29--28-*27*-26--25 48 79
| | | |
91 56--55--54--53-*52*-51--50--49 80
| |
90--89--88--87--86-*85*-84--83--82--81
[Edited by Jon E. Schoenfield, Jan 02 2017]
MAPLE
A001107:=-(1+7*z)/(z-1)**3; # Simon Plouffe in his 1992 dissertation
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* Harvey P. Dale, May 08 2012 *)
Table[PolygonalNumber[RegularPolygon[10], n], {n, 0, 46}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
Table[4 n^2 - 3 n, {n, 0, 49}] (* Alonso del Arte, Jan 24 2017 *)
PolygonalNumber[10, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 27}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)
PROG
(PARI) a(n)=4*n^2-3*n
(Magma) [4*n^2-3*n : n in [0..50] ]; // Wesley Ivan Hurt, Jun 05 2014
(Python) a=lambda n: 4*n**2-3*n # Indranil Ghosh, Jan 01 2017
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 + 8, y + 8
A001107 = aList()
print([next(A001107) for i in range(49)]) # Peter Luschny, Aug 04 2019
CROSSREFS
Sequences from spirals: A001107 (this), A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
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.
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.
Cf. A003215.
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved