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A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1. 27
1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Triangular numbers not == 0 (mod 3). - Amarnath Murthy, Nov 13 2005

Shallow diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003

Equals the triangular numbers convolved with [1, 7, 1, 0, 0, 0,...]. [From Gary W. Adamson & Alexander R. Povolotsky, May 29 2009]

a(n) is congruent to 1 (mod 9) for all n. The sequence of digital roots of the a(n) is A000012(n). The sequence of units’ digits of the a(n) is period 20: repeat [1, 0, 8, 5, 1, 6, 0, 3, 5, 6, 6, 5, 3, 0, 6, 1, 5, 8, 0, 1]. - Ant King, Jun 18 2012

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Index entries for two-way infinite sequences

Index entries for sequences related to centered polygonal numbers

Eric Weisstein's World of Mathematics, Marion's Theorem

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

FORMULA

Contribution from Paul Barry, Mar 15 2003: (Start)

a(n) = C(n, 0)+9*C(n, 1)+9*C(n, 2);

binomial transform of (1, 9, 9, 0, 0, 0, .....).

a(n) = (9*n^2-9*n+2)/2 (corrected by Ant King, Jun 17 2012).

G.f. x*(1+7*x+x^2)/(1-x)^3. (End)

a(n) = C(3n, 3)/n = (3n-1)*(3n-2)/2 = a(n-1)+9(n-1) = A060543(n, 3) = A006566(n)/n = A025035(n)/A025035(n-1) = A027468(n-1)+1 = A000217(3n-2).

a(1-n) = a(n).

Narayana transform (A001263) of [1, 9, 0, 0, 0,...]. - Gary W. Adamson, Dec 29 2007

a(n-1) = Pochhammer(4,3*n)/(Pochhammer(2,n)*Pochhammer(n+1,2*n)).

a(n-1) = 1/Hypergeometric([-3*n,3*n+3,1],[3/2,2],3/4). - Peter Luschny, Jan 09 2012

Contribution from Ant King, Jun 18 2012: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

a(n) = 2*a(n-1) - a(n-2) + 9.

a(n) = A000217(n) + 7*A000217(n-1) + A000217(n-2).

sum(n>=1,1/a(n)) = 2*pi/(3*sqrt(3)) = 1.209199576156....

(End)

MAPLE

H := n -> simplify(1/hypergeom([-3*n, 3*n+3, 1], [3/2, 2], 3/4)); A060544 := n -> H(n-1); seq(A060544(i), i=1..19); -- Peter Luschny, Jan 09 2012

MATHEMATICA

Take[Accumulate[Range[150]], {1, -1, 3}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 10, 28}, 50] (* Harvey P. Dale, Mar 11 2013 *)

FoldList[#1 + #2 &, 1, 9 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)

PROG

(PARI) a(n)=(3*n-1)*(3*n-2)/2

(PARI) { for (n=1, 1000, write("b060544.txt", n, " ", (3*n - 1)*(3*n - 2)/2); ) } [From Harry J. Smith, Jul 06 2009]

CROSSREFS

Cf. A001263, A027468, A081266, A190152.

Sequence in context: A177720 A117464 A081273 * A088406 A169879 A054112

Adjacent sequences:  A060541 A060542 A060543 * A060545 A060546 A060547

KEYWORD

easy,nice,nonn

AUTHOR

Henry Bottomley, Apr 02 2001

EXTENSIONS

Additional description from Terry Trotter, Apr 06 2002

STATUS

approved

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Last modified September 20 07:38 EDT 2014. Contains 246994 sequences.