|
%I #133 Apr 17 2024 04:22:57
%S 1,10,28,55,91,136,190,253,325,406,496,595,703,820,946,1081,1225,1378,
%T 1540,1711,1891,2080,2278,2485,2701,2926,3160,3403,3655,3916,4186,
%U 4465,4753,5050,5356,5671,5995,6328,6670,7021,7381,7750,8128,8515,8911,9316
%N Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.
%C Triangular numbers not == 0 (mod 3). - _Amarnath Murthy_, Nov 13 2005
%C Shallow diagonal of triangular spiral in A051682. - _Paul Barry_, Mar 15 2003
%C Equals the triangular numbers convolved with [1, 7, 1, 0, 0, 0, ...]. - _Gary W. Adamson_ & _Alexander R. Povolotsky_, May 29 2009
%C 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
%C 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
%C 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
%C 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
%C 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
%H T. D. Noe, <a href="/A060544/b060544.txt">Table of n, a(n) for n = 1..1000</a>
%H Ignacio Larrosa Cañestro, <a href="http://www.xente.mundo-r.com/ilarrosa/GeoGebra/Hexagono_cevianas.html">Hexágono y estrella determinados por tres pares de cevianas simétricas</a>, (java applet).
%H Al Cugo et al., <a href="https://www.jstor.org/stable/27968561">Marion's theorem</a>, The Mathematics Teacher 86 (1993) p. 619.
%H John Elias, <a href="/A060544/a060544.png">Illustration of Initial Terms</a>
%H F. Pollock, <a href="https://doi.org/10.1098/rspl.1843.0223">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</a>, Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlkaneGraph.html">Alkane Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MarionsTheorem.html">Marion's Theorem</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerSumIndex.html">Wiener Sum Index</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = C(3*n, 3)/n = (3*n-1)*(3*n-2)/2 = A001504(n-1)/2.
%F a(n) = a(n-1) + 9*(n-1) = A060543(n, 3) = A006566(n)/n.
%F a(n) = A025035(n)/A025035(n-1) = A027468(n-1) + 1 = A000217(3*n-2).
%F a(1-n) = a(n).
%F From _Paul Barry_, Mar 15 2003: (Start)
%F 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, ...).
%F a(n) = 9*A000217(n-1) + 1.
%F G.f.: x*(1 + 7*x + x^2)/(1-x)^3. (End)
%F Narayana transform (A001263) of [1, 9, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 29 2007
%F a(n-1) = Pochhammer(4,3*n)/(Pochhammer(2,n)*Pochhammer(n+1,2*n)).
%F a(n-1) = 1/Hypergeometric([-3*n,3*n+3,1],[3/2,2],3/4). - _Peter Luschny_, Jan 09 2012
%F From _Ant King_, Jun 18 2012: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = 2*a(n-1) - a(n-2) + 9.
%F a(n) = A000217(n) + 7*A000217(n-1) + A000217(n-2).
%F Sum_{n>=1} 1/a(n) = 2*Pi/(3*sqrt(3)) = A248897.
%F (End)
%F a(n) = (2*n-1)^2 + (n-1)*n/2. - _Ivan N. Ianakiev_, Nov 18 2015
%F a(n) = A101321(9,n-1). - _R. J. Mathar_, Jul 28 2016
%F E.g.f.: (2 + 9*x^2)*exp(x)/2 - 1. - _G. C. Greubel_, Mar 02 2019
%F From _Amiram Eldar_, Jun 20 2020: (Start)
%F Sum_{n>=1} a(n)/n! = 11*e/2 - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 11/(2*e) - 1. (End)
%F a(n) = A000567(n) + A005449(n-1) (see illustration in links). - _John Elias_, Nov 10 2020
%F 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
%p 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
%t Take[Accumulate[Range[150]], {1, -1, 3}] (* _Harvey P. Dale_, Mar 11 2013 *)
%t LinearRecurrence[{3, -3, 1}, {1, 10, 28}, 50] (* _Harvey P. Dale_, Mar 11 2013 *)
%t FoldList[#1 + #2 &, 1, 9 Range @ 50] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t Table[(3 n - 1) (3 n - 2)/2, {n, 20}] (* _Eric W. Weisstein_, Jul 15 2021 *)
%t Table[Binomial[3 n - 1, 2], {n, 20}] (* _Eric W. Weisstein_, Jul 15 2021 *)
%t Table[PolygonalNumber[3 n - 2], {n, 20}] (* _Eric W. Weisstein_, Jul 15 2021 *)
%o (PARI) a(n)=(3*n-1)*(3*n-2)/2
%o (PARI) for (n=1, 100, write("b060544.txt", n, " ", (3*n - 1)*(3*n - 2)/2); ) \\ _Harry J. Smith_, Jul 06 2009
%o (Magma) [(2*n-1)^2+(n-1)*n/2: n in [1..50]]; // _Vincenzo Librandi_, Nov 18 2015
%o (GAP) List([1..50],n->(2*n-1)^2+(n-1)*n/2); # _Muniru A Asiru_, Mar 01 2019
%o (Sage) [(3*n-1)*(3*n-2)/2 for n in (1..50)] # _G. C. Greubel_, Mar 02 2019
%Y Cf. A001263, A027468, A081266, A190152.
%K easy,nice,nonn
%O 1,2
%A _Henry Bottomley_, Apr 02 2001
%E Additional description from _Terrel Trotter, Jr._, Apr 06 2002
%E Formulas by Paul Berry corrected for offset 1 by _Wolfdieter Lang_, Jan 30 2015
|
