A002417 4-dimensional figurate numbers: a(n) = n*binomial(n+2, 3). (Formerly M4506 N1907)

1, 8, 30, 80, 175, 336, 588, 960, 1485, 2200, 3146, 4368, 5915, 7840, 10200, 13056, 16473, 20520, 25270, 30800, 37191, 44528, 52900, 62400, 73125, 85176, 98658, 113680, 130355, 148800, 169136, 191488, 215985, 242760, 271950, 303696, 338143, 375440, 415740

Offset

1,2

Comments

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

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

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:

..........X.X

........X.X.X.X

......X.X.X.X.X.X

....X.X.X.X.X.X.X.X

..X.X.X.X.X.X.X.X.X.X

X.X.X.X.X.X.X.X.X.X.X.X

X.X.X.X.X.X.X.X.X.X.X.X - Lekraj Beedassy, Sep 02 2003

Partial sums of A002412. - Jonathan Vos Post, Mar 16 2006

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

a(n) is the n-th antidiagonal sum of the convolution array A213750. - Clark Kimberling, Jun 20 2012

Convolution of A000027 with A000384 (excluding 0). - Bruno Berselli, Dec 06 2012

The sequence is the binomial transform of (1, 7, 15, 13, 4, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015

Also the number of 3-cycles in the (n+2)-triangular graph. - Eric W. Weisstein, Aug 14 2017

References

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

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

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

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Teofil Bogdan and Mircea Dan Rus, Counting the lattice rectangles inside Aztec diamonds and square biscuits, arXiv:2007.13472 [math.CO], 2020.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Johnson Graph

Eric Weisstein's World of Mathematics, Triangular Graph

A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.

Index to sequences related to pyramidal numbers

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

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

G.f.: x*(1+3*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation

a(n) = C(n+2, 2)*n^2/3. - Paul Barry, Jun 26 2003

a(n) = C(n+3, n)*C(n+1, 1). - Zerinvary Lajos, Apr 27 2005

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

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

G.f.: x*hypergeometric2F1(2,4;1;x). - R. J. Mathar, Aug 09 2015

a(n) = A080852(4,n-1). - R. J. Mathar, Jul 28 2016

Sum_{n>=1} 1/a(n) = Pi^2/2 - 15/4. - Jaume Oliver Lafont, Jul 13 2017

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

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

a(n) = A000332(n+3) + 3*A000332(n+2). - Mircea Dan Rus, Jul 29 2020

Maple

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

Mathematica

Table[n Binomial[n+2, 3], {n, 40}]
Table[-Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + 1 &, {n+2, n+2}], x], x^3], {n, 40}] (* John M. Campbell, May 28 2011 *)
Nest[Accumulate, Range[1, 170, 4], 3] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 8, 30, 80, 175}, 40] (* Harvey P. Dale, Jan 11 2014 *)
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 *)

Prog

(PARI) a(n)=n^2*(n+1)*(n+2)/6 \\ Charles R Greathouse IV, Jun 10 2011
(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
(Magma) [n*Binomial(n+2, 3):n in [1..40]]; // Vincenzo Librandi, Aug 02 2015
(Sage) [n*binomial(n+2, 3) for n in (1..40)] # G. C. Greubel, Jul 03 2019
(GAP) List([1..40], n-> n^2*(n+1)*(n+2)/6 ) # G. C. Greubel, Jul 03 2019

Crossrefs

Bisection of A002624.

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

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

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

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

Sequence in context: A063489 A299284 A348461 * A126858 A232772 A213776

Adjacent sequences: A002414 A002415 A002416 * A002418 A002419 A002420

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Edited and extended by Floor van Lamoen, Oct 09 2002

Status

approved