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A001107
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10-gonal (or decagonal) numbers: n(4n-3).
(Formerly M4690)
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90
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 (jhbubby(AT)mindspring.com), Aug 30 2008
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 20 2010: (Start)
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.(End)
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]
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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.
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).
Weisstein, Eric W. "Barbell Graph." http://mathworld.wolfram.com/BarbellGraph.html. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 20 2010]
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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
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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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 (perry(AT)globalnet.co.uk), 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 (qntmpkt(AT)yahoo.com), 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 (joliverlafont(AT)gmail.com), Dec 02 2008]
a(n)=8*n+a(n-1)-7 (with a(0)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jul 10 2010]
a(n)=8+2a(n-1)-a(n-2) [Ant King, Sep 04 2011]
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EXAMPLE
| 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
| A001107:=-(1+7*z)/(z-1)**3; [S. 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
| 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 Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]
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PROG
| (PARI) a(n)=4*n^2-3*n
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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, A000566, A000567, A001106, A051682, A051624, A051865-A051876.
Sequence in context: A045177 A043887 A161450 * A103135 A008468 A179546
Adjacent sequences: A001104 A001105 A001106 * A001108 A001109 A001110
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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