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A001106
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9-gonal (or enneagonal or nonagonal) numbers: n*(7*n-5)/2.
(Formerly M4604)
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66
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0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Sequence found by reading the line from 0, in the direction 0, 9,... and the parallel line from 1, in the direction 1, 24,..., in the square spiral whose vertices vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same lines in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011
Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and x+y <= n. [Reinhard Zumkeller, Jan 23 2012]
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REFERENCES
| A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| William A. Tedeschi, Table of n, a(n) for n=0..10000 [This replaces an earlier b-file computed by T. D. Noe]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 343
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, Nonagonal Number
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| a(n)=(7*n-5)*n/2. G.f.: x*(1+6*x)/(1-x)^3.
a(n)=n+7*A000217(n-1) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 14 2005
Starting (1, 9, 24, 46, 75,...) gives the binomial transform of (1, 8, 7, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 22 2007
Row sums of triangle A131875 starting (1, 9, 24, 46, 75, 111,...). A001106 = binomial transform of (1, 8, 7, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 22 2007
a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=9 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
a(n) = 2*a(n-1) - a(n-2) + 7. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 05 2010]
a(n) = a(n-1)+7*n-6 (with a(0)=0). [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2010]
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MAPLE
| A001106:=-(1+6*z)/(z-1)**3; [Conjectured by 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]+7 od: seq(a[n], n=0..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 2008
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MATHEMATICA
| s=0; lst={s}; Do[s+=n++ +1; AppendTo[lst, s], {n, 0, 6!, 7}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 15 2008]
Table[n(7n-5)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 9}, 50] (* From Harvey P. Dale, Nov 06 2011 *)
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PROG
| (PARI) a(n)=n*(7*n-5)/2 \\ Charles R Greathouse IV, Jun 10 2011
(Haskell)
a001106 n = length [(x, y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n]
-- Reinhard Zumkeller, Jan 23 2012
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CROSSREFS
| Cf. A093564 ((7, 1) Pascal, column m=2). Partial sums of A016993.
Cf. A131875.
Cf. n-gonal numbers: A000217, A000290, A000326, A000566, A000567, A001107, A051682, A051624, A051865-A051876.
Cf. A057655, A069099.
Sequence in context: A063066 A097658 A067725 * A023551 A022787 A079770
Adjacent sequences: A001103 A001104 A001105 * A001107 A001108 A001109
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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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