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

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

Offset: 1

Author

Henry Bottomley, Apr 02 2001

Keywords

• easy
• nice
• nonn

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; edited by Wolfdieter Lang, Jan 30 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

Pollock states that every number is the sum of at most 11 terms of this sequence, but note that "1, 10, 28, 35, &c." has a typo (35 should be 55). - Michel Marcus, Nov 04 2017

a(n) is also the number of (nontrivial) paths as well as the Wiener sum index of the (n-1)-alkane graph. - Eric W. Weisstein, Jul 15 2021

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).
• Al Cugo et al., Marion's theorem, The Mathematics Teacher 86 (1993) p. 619.
• John Elias, Illustration of Initial Terms
• F. Pollock, On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders, Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
• Eric Weisstein's World of Mathematics, Alkane Graph
• Eric Weisstein's World of Mathematics, Graph Path
• Eric Weisstein's World of Mathematics, Marion's Theorem
• Eric Weisstein's World of Mathematics, Wiener Sum Index
• 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).

Crossrefs

Cf. A001263, A027468, A081266, A190152.

Programs

GAP

List([1..50],n->(2*n-1)^2+(n-1)*n/2); # Muniru A Asiru, Mar 01 2019

Magma

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

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}] (* Harvey P. Dale, Mar 11 2013 *)
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 *)
Table[(3 n - 1) (3 n - 2)/2, {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[Binomial[3 n - 1, 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[PolygonalNumber[3 n - 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)

PARI

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

Sage

[(3*n-1)*(3*n-2)/2 for n in (1..50)] # G. C. Greubel, Mar 02 2019

Formula

a(n) = C(3*n, 3)/n = (3*n-1)*(3*n-2)/2 = A001504(n-1)/2.

a(n) = a(n-1) + 9*(n-1) = A060543(n, 3) = A006566(n)/n.

a(n) = A025035(n)/A025035(n-1) = A027468(n-1) + 1 = A000217(3*n-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

E.g.f.: (2 + 9*x^2)*exp(x)/2 - 1. - G. C. Greubel, Mar 02 2019

From Amiram Eldar, Jun 20 2020: (Start)

Sum_{n>=1} a(n)/n! = 11*e/2 - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 11/(2*e) - 1. (End)

a(n) = A000567(n) + A005449(n-1) (see illustration in links). - John Elias, Nov 10 2020

a(n) = P(2*n,4)*P(3*n,3)/24 for n>=2, where P(s,k) = ((s - 2)*k^2 - (s - 4)*k)/2 is the k-th s-gonal number. - Lechoslaw Ratajczak, Jul 18 2021

Extensions

Additional description from Terrel Trotter, Jr., Apr 06 2002

Formulas by Paul Berry corrected for offset 1 by Wolfdieter Lang, Jan 30 2015