|
|
A002417
|
|
4-dimensional figurate numbers: a(n) = n*binomial(n+2, 3).
(Formerly M4506 N1907)
|
|
114
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
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
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
|
|
|
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) = (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
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
|
|
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 *)
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
|
(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
|
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|