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4-dimensional figurate numbers: a(n) = n*binomial(n+2, 3).
(Formerly M4506 N1907)
114

%I M4506 N1907 #143 Sep 08 2022 08:44:30

%S 1,8,30,80,175,336,588,960,1485,2200,3146,4368,5915,7840,10200,13056,

%T 16473,20520,25270,30800,37191,44528,52900,62400,73125,85176,98658,

%U 113680,130355,148800,169136,191488,215985,242760,271950,303696,338143,375440,415740

%N 4-dimensional figurate numbers: a(n) = n*binomial(n+2, 3).

%C a(n) is 1/6 the number of colorings of a 2 X 2 hexagonal array with n+2 colors. - _R. H. Hardin_, Feb 23 2002

%C a(n) is the sum of all numbers that cannot be written as t*(n+1) + u*(n+2) for nonnegative integers t,u (see Schuh). - _Floor van Lamoen_, Oct 09 2002

%C a(n) is the total number of rectangles (including squares) contained in a stepped pyramid of n rows (or of base 2n-1) of squares. A stepped pyramid of squares of base 2*6 - 1 = 11, for instance, has the following vertices:

%C ..........X.X

%C ........X.X.X.X

%C ......X.X.X.X.X.X

%C ....X.X.X.X.X.X.X.X

%C ..X.X.X.X.X.X.X.X.X.X

%C X.X.X.X.X.X.X.X.X.X.X.X

%C X.X.X.X.X.X.X.X.X.X.X.X - _Lekraj Beedassy_, Sep 02 2003

%C Partial sums of A002412. - _Jonathan Vos Post_, Mar 16 2006

%C a(n) equals -1 times the coefficient of x^3 of the characteristic polynomial of the (n + 2) X (n + 2) matrix with 2's along the main diagonal and 1's everywhere else (see Mathematica code below). - _John M. Campbell_, May 28 2011

%C a(n) is the n-th antidiagonal sum of the convolution array A213750. - _Clark Kimberling_, Jun 20 2012

%C Convolution of A000027 with A000384 (excluding 0). - _Bruno Berselli_, Dec 06 2012

%C The sequence is the binomial transform of (1, 7, 15, 13, 4, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 31 2015

%C Also the number of 3-cycles in the (n+2)-triangular graph. - _Eric W. Weisstein_, Aug 14 2017

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.

%D K. -W. Lau, Solution to Problem 2495, Journal of Recreational Mathematics 2002-3 31(1) 79-80.

%D Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%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="/A002417/b002417.txt">Table of n, a(n) for n = 1..1000</a>

%H Teofil Bogdan and Mircea Dan Rus, <a href="https://arxiv.org/abs/2007.13472">Counting the lattice rectangles inside Aztec diamonds and square biscuits</a>, arXiv:2007.13472 [math.CO], 2020.

%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>

%H A. F. Y. Zhao, <a href="http://www.emis.de/journals/JIS/VOL17/Zhao/zhao3.html">Pattern Popularity in Multiply Restricted Permutations</a>, Journal of Integer Sequences, 17 (2014), #14.10.3.

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n^2*(n+1)*(n+2)/6.

%F G.f.: x*(1+3*x)/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = C(n+2, 2)*n^2/3. - _Paul Barry_, Jun 26 2003

%F a(n) = C(n+3, n)*C(n+1, 1). - _Zerinvary Lajos_, Apr 27 2005

%F a(n) = (binomial(n+3,n-1) - binomial(n+2,n-2))*(binomial(n+1,n-1) - binomial(n,n-2)). - _Zerinvary Lajos_, May 12 2006

%F a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5), n>5. - _Wesley Ivan Hurt_, Aug 01 2015

%F G.f.: x*hypergeometric2F1(2,4;1;x). - _R. J. Mathar_, Aug 09 2015

%F a(n) = A080852(4,n-1). - _R. J. Mathar_, Jul 28 2016

%F Sum_{n>=1} 1/a(n) = Pi^2/2 - 15/4. - _Jaume Oliver Lafont_, Jul 13 2017

%F E.g.f.: x*(6 + 18*x + 9*x^2 + x^3)*exp(x)/3!. - _G. C. Greubel_, Jul 03 2019

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi^2 + 27 - 48*log(2))/4. - _Amiram Eldar_, Jun 28 2020

%F a(n) = A000332(n+3) + 3*A000332(n+2). - _Mircea Dan Rus_, Jul 29 2020

%p seq(n^2*(n+1)*(n+2)/6, n=1..50);

%t Table[n Binomial[n+2, 3], {n, 40}]

%t Table[-Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + 1 &, {n+2, n+2}], x], x^3], {n, 40}] (* _John M. Campbell_, May 28 2011 *)

%t Nest[Accumulate, Range[1, 170, 4], 3] (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2012 *)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {1, 8, 30, 80, 175}, 40] (* _Harvey P. Dale_, Jan 11 2014 *)

%t Table[n Pochhammer[n, 3]/6, {n, 40}] (* or *) CoefficientList[Series[ (1+3x)/(1-x)^5, {x,0,40}], x] (* _Eric W. Weisstein_, Aug 14 2017 *)

%o (PARI) a(n)=n^2*(n+1)*(n+2)/6 \\ _Charles R Greathouse IV_, Jun 10 2011

%o (Magma) /* A000027 convolved with A000384 (excluding 0): */ A000384:=func<n | n*(2*n-1)>; [&+[(n-i+1)*A000384(i): i in [1..n]]: n in [1..40]]; // _Bruno Berselli_, Dec 06 2012

%o (Magma) [n*Binomial(n+2,3):n in [1..40]]; // _Vincenzo Librandi_, Aug 02 2015

%o (Sage) [n*binomial(n+2,3) for n in (1..40)] # _G. C. Greubel_, Jul 03 2019

%o (GAP) List([1..40], n-> n^2*(n+1)*(n+2)/6 ) # _G. C. Greubel_, Jul 03 2019

%Y Bisection of A002624.

%Y a(n) = A093561(n+3, 4).

%Y Cf. A000027, A000384, A002412, A062196, A213750, A000332.

%Y Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

%Y Cf. A151974 (number of 4-cycles in the triangular graph), A290939 (5-cycles), A290940 (6-cycles).

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E Edited and extended by _Floor van Lamoen_, Oct 09 2002