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 A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1. 56
 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; 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 linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = C(3*n, 3)/n = (3*n-1)*(3*n-2)/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 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], {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 *) PROG (PARI) a(n)=(3*n-1)*(3*n-2)/2 (PARI) for (n=1, 100, 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 (GAP) List([1..50], n->(2*n-1)^2+(n-1)*n/2); # Muniru A Asiru, Mar 01 2019 (Sage) [(3*n-1)*(3*n-2)/2 for n in (1..50)] # G. C. Greubel, Mar 02 2019 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 Formulas by Paul Berry corrected for offset 1 by Wolfdieter Lang, Jan 30 2015 STATUS approved

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Last modified March 28 03:48 EDT 2023. Contains 361577 sequences. (Running on oeis4.)