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A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1. 39
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, ...]. - 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

Divide each side of any triangle ABC with area (ABC) into 2n + 1 equal segments by 2n points:A_1, A_2, ..., A_(2n) on side a, and similarly for sides b and c. If the hexagon with area (Hex(n)) delimited by AA_n, AA_(n+1), BB_n, BB_(n+1), CC_n and CC_(n+1) cevians, we have a(n+1) = (ABC)/(Hex(n)) for n >= 1, (see link with java applet). - Ignacio Larrosa Cañestro, Jan 02 2015

For the case n = 1 see the link for Marion's Theorem (actually Marion Walter's Theorem, see the Cugo et al, reference). Also, the generalization considered here has been called there (Ryan) Morgan's Theorem.  - Wolfdieter Lang, Jan 30 2015

REFERENCES

Al Cugo et al., Marion's theorem, The Mathematics Teacher 86 (1993) p. 619.

LINKS

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

Ignacio Larrosa Cañestro, Hexágono y estrella determinados por tres pares de cevianas simétricas, (java applet).

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

Index entries for two-way infinite sequences

Index entries for sequences related to centered polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

FORMULA

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).

From Paul Barry, Mar 15 2003: (Start)

a(n) = C(n-1, 0) + 9*C(n-1, 1) + 9*C(n-1, 2); binomial transform of (1, 9, 9, 0, 0, 0, ...).

a(n) = 9*A000217(n-1) + 1.

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

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

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)) = A248897.

(End)

a(n) = (2*n-1)^2 + (n-1)*n/2. - Ivan N. Ianakiev, Nov 18 2015

a(n) = A101321(9,n-1). - R. J. Mathar, Jul 28 2016

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); ) } \\ Harry J. Smith, Jul 06 2009

(MAGMA) [(2*n-1)^2+(n-1)*n/2: n in [1..50]]; // Vincenzo Librandi, Nov 18 2015

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 Terrel Trotter, Jr., Apr 06 2002

Comment by Ignacio Larrosa Cañestro edited. Formulas by Paul Berry corrected for offset 1. - Wolfdieter Lang, Jan 30 2015

STATUS

approved

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Last modified December 6 06:58 EST 2016. Contains 278775 sequences.