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A001107
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10-gonal (or decagonal) numbers: n(4n-3).
(Formerly M4690)
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97
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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.
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).
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LINKS
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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 S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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).
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FORMULA
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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 [From Jaume Oliver Lafont, Dec 02 2008]
a(n)=8*n+a(n-1)-7 (with a(0)=0) [From 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
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EXAMPLE
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Part of the spiral:
16 17 18 19 ...
15 4 5 6 ...
14 3 0 7 ...
13 2 1 8 ...
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MAPLE
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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 (zerinvarylajos(AT)yahoo.com), Feb 18 2008
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MATHEMATICA
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lst={}; Do[AppendTo[lst, 4*n^2-3*n], {n, 0, 5!}]; lst...and/or... s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 0, 6!, 8}]; lst [From Vladimir Joseph Stephan Orlovsky, Oct 25 2008]
LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* From Harvey P. Dale, May 08 2012 *)
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PROG
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(PARI) a(n)=4*n^2-3*n
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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