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A001107 10-gonal (or decagonal) numbers: n(4n-3).
(Formerly M4690)
104
0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Write 0,1,2,... in clockwise spiral; sequence gives numbers on negative y axis.

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

REFERENCES

A. 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

Emilio Apricena, A version of the Ulam spiral

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 344

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric W. Weisstein, MathWorld: Decagonal Number

Eric W. Weisstein, MathWorld: Barbell Graph

Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = A033954(-n) = A074377(2n-1).

a(n) = n+8*A000217(n-1). - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 14 2005

G.f.: A(x) = x(1+7x)/(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(4n-3) [Ramanujan]. - Neven Juric, Apr 15 2008

Starting (1, 10, 27, 52,...) = binomial transform of [1, 9, 8, 0, 0, 0,...]. - Gary W. Adamson, Apr 30 2008

a(n) = 3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=10. - Jaume Oliver Lafont, Dec 02 2008

a(n) = 8*n+a(n-1)-7 (with a(0)=0). - Vincenzo Librandi, Jul 10 2010

a(n) = 8+2a(n-1)-a(n-2). - Ant King, Sep 04 2011

a(n) = A118729(8n). - 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

EXAMPLE

Part of the spiral:

16 17 18 19 ...

15 4 5 6 ...

14 3 0 7 ...

13 2 1 8 ...

MAPLE

A001107:=-(1+7*z)/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+8 od: seq(a[n], n=0..46); # Zerinvary Lajos, Feb 18 2008

MATHEMATICA

lst={}; Do[AppendTo[lst, 4*n^2-3*n], {n, 0, 5!}]; lst (* or *) s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 0, 6!, 8}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)

LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* Harvey P. Dale, May 08 2012 *)

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

CROSSREFS

Cf. A007585.

Cf. A093565 ((8, 1) Pascal, column m=2). Partial sums of A017077.

Sequences from spirals: A001107 (this), A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

Cf. n-gonal numbers: A000217, A000290, A000326, A000384, A000566, A000567, A001106, this sequence, A051682, A051624, A051865-A051876.

Sequence in context: A045177 A043887 A161450 * A103135 A220021 A008468

Adjacent sequences:  A001104 A001105 A001106 * A001108 A001109 A001110

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Jul 11 1991

STATUS

approved

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Last modified August 2 04:43 EDT 2014. Contains 245138 sequences.