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A001106 9-gonal (or enneagonal or nonagonal) numbers: n*(7*n-5)/2.
(Formerly M4604)
87
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; text; internal format)
OFFSET

0,3

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 9-gonal (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

Partial sums give A007584. - Omar E. Pol, Jan 15 2013

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe and William A. Tedeschi, Table of n, a(n) for n=0..10000 (1000 terms were computed by T. D. Noe)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 343

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 Weisstein's World of Mathematics, Nonagonal Number

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

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, 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, Jul 22 2007

a(n) = 3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=9. - Jaume Oliver Lafont, Dec 02 2008

a(n) = 2*a(n-1) - a(n-2) + 7. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 05 2010

a(n) = a(n-1)+7*n-6 (with a(0)=0). - Vincenzo Librandi, Nov 12 2010

a(n) = A174738(7n). - Philippe Deléham, Mar 26 2013

a(7*a(n)+22*n+1) =  a(7*a(n)+22*n) + a(7*n+1). - Vladimir Shevelev, Jan 24 2014

MAPLE

A001106:=-(1+6*z)/(z-1)**3; [Conjectured by 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]+7 od: seq(a[n], n=0..43); # Zerinvary Lajos, Feb 18 2008

MATHEMATICA

s=0; lst={s}; Do[s+=n++ +1; AppendTo[lst, s], {n, 0, 6!, 7}]; lst [Vladimir Joseph Stephan Orlovsky, Nov 15 2008]

Table[n(7n-5)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 9}, 50] (* Harvey P. Dale, Nov 06 2011 *)

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

(Haskell) a001106 n = n*(7*n-5) `div` 2 -- James Spahlinger, Oct 18 2012

CROSSREFS

Cf. A093564 ((7, 1) Pascal, column m=2). Partial sums of A016993.

Cf. A131875.

Cf. n-gonal numbers: A000217, A000290, A000326, A000384, A000566, A000567, this sequence, A001107, A051682, A051624, A051865-A051876.

Cf. A057655, A069099.

Sequence in context: A097658 A067725 A213903 * A023551 A022787 A079770

Adjacent sequences:  A001103 A001104 A001105 * A001107 A001108 A001109

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 29 03:09 EST 2014. Contains 250479 sequences.