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%I M4362 #429 Feb 22 2024 11:22:29
%S 1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919,1027,
%T 1141,1261,1387,1519,1657,1801,1951,2107,2269,2437,2611,2791,2977,
%U 3169,3367,3571,3781,3997,4219,4447,4681,4921,5167,5419,5677,5941,6211,6487,6769
%N Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).
%C The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C Crystal ball sequence for A_2 lattice. - _Michael Somos_, Jun 03 2012
%C Sixth spoke of hexagonal spiral (cf. A056105-A056109).
%C Number of ordered integer triples (a,b,c), -n <= a,b,c <= n, such that a+b+c=0. - _Benoit Cloitre_, Jun 14 2003
%C Also the number of partitions of 6n into at most 3 parts, A001399(6n). - _R. K. Guy_, Oct 20 2003
%C Also, a(n) is the number of partitions of 6(n+1) into exactly 3 distinct parts. - _William J. Keith_, Jul 01 2004
%C Number of dots in a centered hexagonal figure with n+1 dots on each side.
%C Values of second Bessel polynomial y_2(n) (see A001498).
%C First differences of cubes (A000578). - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%C Final digits of Hex numbers (hex(n) mod 10) are periodic with palindromic period of length 5 {1, 7, 9, 7, 1}. Last two digits of Hex numbers (hex(n) mod 100) are periodic with palindromic period of length 100. - _Alexander Adamchuk_, Aug 11 2006
%C All divisors of a(n) are congruent to 1, modulo 6. Proof: If p is an odd prime different from 3 then 3n^2 + 3n + 1 = 0 (mod p) implies 9(2n + 1)^2 = -3 (mod p), whence p = 1 (mod 6). - _Nick Hobson_, Nov 13 2006
%C For n>=1, a(n) is the side of Outer Napoleon Triangle whose reference triangle is a right triangle with legs (3a(n))^(1/2) and 3n(a(n))^(1/2). - Tom Schicker (tschicke(AT)email.smith.edu), Apr 25 2007
%C Number of triples (a,b,c) where 0<=(a,b)<=n and c=n (at least once the term n). E.g., for n = 1: (0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,0,1), (1,1,0), (1,1,1), so a(1)=7. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007
%C Equals the triangular numbers convolved with [1, 4, 1, 0, 0, 0, ...]. - _Gary W. Adamson_ and _Alexander R. Povolotsky_, May 29 2009
%C From Terry Stickels, Dec 07 2009: (Start)
%C Also the maximum number of viewable cubes from any one static point while viewing a cube stack of identical cubes of varying magnitude.
%C For example, viewing a 2 X 2 X 2 stack will yield 7 maximum viewable cubes.
%C If the stack is 3 X 3 X 3, the maximum number of viewable cubes from any one static position is 19, and so on.
%C The number of cubes in the stack must always be the same number for width, length, height (at true regular cubic stack) and the maximum number of visible cubes can always be found by taking any cubic number and subtracting the number of the cube that is one less.
%C Examples: 125 - 64 = 61, 64 - 27 = 37, 27 - 8 = 19. (End)
%C The sequence of digital roots of the a(n) is period 3: repeat [1,7,1]. - _Ant King_, Jun 17 2012
%C The average of the first n (n>0) centered hexagonal numbers is the n-th square. - _Philippe Deléham_, Feb 04 2013
%C A002024 is the following array A read along antidiagonals:
%C 1, 2, 3, 4, 5, 6, ...
%C 2, 3, 4, 5, 6, 7, ...
%C 3, 4, 5, 6, 7, 8, ...
%C 4, 5, 6, 7, 8, 9, ...
%C 5, 6, 7, 8, 9, 10, ...
%C 6, 7, 8, 9, 10, 11, ...
%C and a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - _R. J. Mathar_, Jun 30 2013
%C a(n) is the sum of the terms in the n+1 X n+1 matrices minus those in n X n matrices in an array formed by considering A158405 an array (the beginning terms in each row are 1,3,5,7,9,11,...). - _J. M. Bergot_, Jul 05 2013
%C The formula also equals the product of the three distinct combinations of two consecutive numbers: n^2, (n+1)^2, and n*(n+1). - _J. M. Bergot_, Mar 28 2014
%C The sides of any triangle ABC are divided into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_2n in side a, and also on the sides b and c cyclically. If A'B'C' is the triangle delimited by AA_n, BB_n and CC_n cevians, we have (ABC)/(A'B'C') = a(n) (see Java applet link). - _Ignacio Larrosa Cañestro_, Jan 02 2015
%C a(n) is the maximal number of parts into which (n+1) triangles can intersect one another. - _Ivan N. Ianakiev_, Feb 18 2015
%C ((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = ((2^m-1)(2n+1))^t mod a(n), where m any positive integer, and t = 0(mod 6). - _Alzhekeyev Ascar M_, Oct 07 2016
%C ((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = a(n) - (((2^m-1)(2n+1))^t mod a(n)), where m any positive integer, and t = 3(mod 6). - _Alzhekeyev Ascar M_, Oct 07 2016
%C (3n+1)^(a(n)-1) mod a(n) = (3n+2)^(a(n)-1) mod a(n) = 1. If a(n) not prime, then always strong pseudoprime. - _Alzhekeyev Ascar M_, Oct 07 2016
%C Every positive integer is the sum of 8 hex numbers (zero included), at most 3 of which are greater than 1. - _Mauro Fiorentini_, Jan 01 2018
%C Area enclosed by the segment of Archimedean spiral between n*Pi/2 and (n+1)*Pi/2 in Pi^3/48 units. - _Carmine Suriano_, Apr 10 2018
%C This sequence contains all numbers k such that 12*k - 3 is a square. - _Klaus Purath_, Oct 19 2021
%C The continued fraction expansion of sqrt(3*a(n)) is [3n+1; {1, 1, 2n, 1, 1, 6n+2}]. For n = 0, this collapses to [1; {1, 2}]. - _Magus K. Chu_, Sep 12 2022
%D M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 18.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003215/b003215.txt">Table of n, a(n) for n = 0..1000</a>
%H G. L. Alexanderson and John E. Wetzel, <a href="http://dx.doi.org/10.1016/0095-8956(71)90014-1">Dissections of a tetrahedron</a>, J. Combinatorial Theory Ser. B 11 (1971), 58--66. MR0303412 (46 #2549). See p. 58.
%H B. T. Bennett and R. B. Potts, <a href="http://dx.doi.org/10.1017/S144678870000505X">Arrays and brooks</a>, J. Austral. Math. Soc., 7 (1967), 23-31 (see p. 30).
%H B. T. Bennett and R. B. Potts, <a href="/A002047/a002047_1.pdf">Arrays and brooks</a>, J. Austral. Math. Soc., 7 (1967), 23-31. [Annotated scanned copy]
%H Aran Bingham, <a href="https://scholarworks.uno.edu/td/1959">Commutative n-ary Arithmetic</a>, University of New Orleans Theses and Dissertations, Paper 1959, 2015.
%H Henry Bottomley, <a href="/A003215/a003215.gif">Illustration of initial terms</a>
%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H M. Gardner & N. J. A. Sloane, <a href="/A003154/a003154.pdf">Correspondence, 1973-74</a>
%H R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H Md. Towhidul Islam, <a href="/A003215/a003215.png">Extending triangle</a>
%H G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sa&i=20">On a Conjecture of Moessner and a General Problem</a>, Bull. Soc. Math. Grèce, Nouvelle Série - vol. 2, fasc. 1-2, pp. 23-30 (1961).
%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 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H Vladimir Pletser, <a href="https://doi.org/10.20944/preprints202402.0545.v1">Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems</a>, Preprints.org, 2024. See p. 20.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexNumber.html">Hex Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NexusNumber.html">Nexus Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OuterNapoleonTriangle.html">Outer Napoleon Triangle</a>.
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Aa#A2">Index entries for sequences related to A2 = hexagonal = triangular lattice</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*n*(n+1) + 1, n >= 0 (see the name).
%F a(n) = (n+1)^3 - n^3 = a(-1-n).
%F G.f.: (1 + 4*x + x^2) / (1 - x)^3. - _Simon Plouffe_ in his 1992 dissertation
%F a(n) = 6*A000217(n) + 1.
%F a(n) = a(n-1) + 6*n = 2a(n-1) - a(n-2) + 6 = 3*a(n-1) - 3*a(n-2) + a(n-3) = A056105(n) + 5n = A056106(n) + 4*n = A056107(n) + 3*n = A056108(n) + 2*n = A056108(n) + n.
%F n-th partial arithmetic mean is n^2. - _Amarnath Murthy_, May 27 2003
%F a(n) = 1 + Sum_{j=0..n} (6*j). E.g., a(2)=19 because 1+ 6*0 + 6*1 + 6*2 = 19. - Xavier Acloque, Oct 06 2003
%F The sum of the first n hexagonal numbers is n^3. That is, Sum_{n>=1} (3*n*(n-1) + 1) = n^3. - Edward Weed (eweed(AT)gdrs.com), Oct 23 2003
%F a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 2 1 0 / 3 3 1]. M^n * [1 1 1] = [1 2n+1 a(n)]. E.g., a(4) = 61, right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 9 61] = [1 2n+1 a(4)]. - _Gary W. Adamson_, Dec 22 2004
%F Row sums of triangle A130298. - _Gary W. Adamson_, Jun 07 2007
%F a(n) = 3*n^2 + 3*n + 1. Proof: 1) If n occurs once, it may be in 3 positions; for the two other ones, n terms are independently possible, then we have 3*n^2 different triples. 2) If the term n occurs twice, the third one may be placed in 3 positions and have n possible values, then we have 3*n more different triples. 3) The term n may occurs 3 times in one way only that gives the formula. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007
%F Binomial transform of [1, 6, 6, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 6, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 29 2007
%F a(n) = (n-1)*A000166(n) + (n-2)*A000166(n-1) = (n-1)floor(n!*e^(-1)+1) + (n-2)*floor((n-1)!*e^(-1)+1) (with offset 0). - _Gary Detlefs_, Dec 06 2009
%F a(n) = A028896(n) + 1. - _Omar E. Pol_, Oct 03 2011
%F a(n) = integral( (sin((n+1/2)x)/sin(x/2))^3, x=0..Pi)/Pi. - _Yalcin Aktar_, Dec 03 2011
%F Sum_{n>=0} 1/a(n) = Pi/sqrt(3)*tanh(Pi/(2*sqrt(3))) = 1.305284153013581... - _Ant King_, Jun 17 2012
%F a(n) = A000290(n) + A000217(2n+1). - _Ivan N. Ianakiev_, Sep 24 2013
%F a(n) = A002378(n+1) + A056220(n) = A005408(n) + 2*A005449(n) = 6*A000217(n) + 1. - _Ivan N. Ianakiev_, Sep 26 2013
%F a(n) = 6*A000124(n) - 5. - _Ivan N. Ianakiev_, Oct 13 2013
%F a(n) = A239426(n+1) / A239449(n+1) = A215630(2*n+1,n+1). - _Reinhard Zumkeller_, Mar 19 2014
%F a(n) = A243201(n) / A002061(n + 1). - _Mathew Englander_, Jun 03 2014
%F a(n) = A101321(6,n). - _R. J. Mathar_, Jul 28 2016
%F E.g.f.: (1 + 6*x + 3*x^2)*exp(x). - _Ilya Gutkovskiy_, Jul 28 2016
%F a(n) = (A001844(n) + A016754(n))/2. - _Bruce J. Nicholson_, Aug 06 2017
%F a(n) = A045943(2n+1). - _Miquel Cerda_, Jan 22 2018
%F a(n) = 3*Integral_{x=n..n+1} x^2 dx. - _Carmine Suriano_, Apr 10 2018
%F a(n) = A287326(A000124(n), 1). - _Kolosov Petro_, Oct 22 2018
%F From _Amiram Eldar_, Jun 20 2020: (Start)
%F Sum_{n>=0} a(n)/n! = 10*e.
%F Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 2/e. (End)
%F G.f.: polylog(-3, x)*(1-x)/x. See the _Simon Plouffe_ formula above, and the g.f. of the rows of A008292 by _Vladeta Jovovic_, Sep 02 2002. - _Wolfdieter Lang_, May 08 2021
%F a(n) = T(n-1)^2 - 2*T(n)^2 + T(n+1)^2, n >= 1, T = triangular number A000217. - _Klaus Purath_, Oct 11 2021
%F a(n) = 1 + 2*Sum_{j=n..2n} j. - _Klaus Purath_, Oct 19 2021
%F a(n) = A069099(n+1) - A000217(n). - _Klaus Purath_, Nov 03 2021
%F From _Leo Tavares_, Dec 03 2021: (Start)
%F a(n) = A005448(n) + A140091(n);
%F a(n) = A001844(n) + A002378(n);
%F a(n) = A005891(n) + A000217(n);
%F a(n) = A000290(n) + A000384(n+1);
%F a(n) = A060544(n-1) + 3*A000217(n);
%F a(n) = A060544(n-1) + A045943(n).
%F a(2*n+1) = A154105(n).
%F (End)
%e G.f. = 1 + 7*x + 19*x^2 + 37*x^3 + 61*x^4 + 91*x^5 + 127*x^6 + 169*x^7 + 217*x^8 + ...
%e From _Omar E. Pol_, Aug 21 2011: (Start)
%e Illustration of initial terms:
%e .
%e . o o o o
%e . o o o o o o o o
%e . o o o o o o o o o o o o
%e . o o o o o o o o o o o o o o o o
%e . o o o o o o o o o o o o
%e . o o o o o o o o
%e . o o o o
%e .
%e . 1 7 19 37
%e .
%e (End)
%e From _Klaus Purath_, Dec 03 2021: (Start)
%e (1) a(19) is not a prime number, because besides a(19) = a(9) + P(29), a(19) = a(15) + P(20) = a(2) + P(33) is also true.
%e (2) a(25) is prime, because except for a(25) = a(12) + P(38) there is no other equation of this pattern. (End)
%p A003215:=n->3*n*(n+1)+1; seq(A003215(n), n=0..100); # _Wesley Ivan Hurt_, Mar 28 2014
%t FoldList[#1 + #2 &, 1, 6 Range@ 50] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t LinearRecurrence[{3, -3, 1}, {1, 7, 19}, 47] (* _Robert G. Wilson v_, Jul 06 2013 *)
%o (PARI) {a(n) = 3*n*(n+1) + 1};
%o (Haskell)
%o a003215 n = 3 * n * (n + 1) + 1 -- _Reinhard Zumkeller_, Oct 22 2011
%o (Maxima) makelist(3*n*(n+1)+1, n, 0, 30); /* _Martin Ettl_, Nov 12 2012 */
%o (Magma) [3*n*(n+1)+1: n in [0..50]]; // _G. C. Greubel_, Nov 04 2017
%o (Python) [3*n*(n+1)+1 for n in range(47)] # _Michael S. Branicky_, Jan 07 2021
%Y Cf. A000124, A000166, A000217, A000290, A000578 (the cubes, or partial sums), A001263, A001498, A002061, A002378, A002407 (primes), A003514, A005408, A005449, A005891, A028896, A048766, A056105, A056106, A056107, A056108, A056109, A063496, A056220, A130298, A132111 (second diagonal), A158405, A215630, A239449, A243201.
%Y Column k=3 of A080853, and column k=2 of A047969.
%Y See also A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
%Y Cf. A287326(A000124(n), 1).
%Y Cf. A008292.
%Y Cf. A005448, A140091, A001844, A000384, A060544, A045943.
%Y Cf. A154105.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Partially edited by _Joerg Arndt_, Mar 11 2010
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