%I A002417 M4506 N1907
%S A002417 1,8,30,80,175,336,588,960,1485,2200,3146,4368,5915,7840,10200,13056,
%T A002417 16473,20520,25270,30800,37191,44528,52900,62400,73125,85176,98658,
%U A002417 113680,130355,148800,169136,191488,215985,242760,271950,303696,338143
%N A002417 4-dimensional figurate numbers: n*C(n+2,3).
%C A002417 a(n) is 1/6 the number of colorings of a 2 X 2 hexagonal array with n+2 
               colors. - Ron Hardin (rhhardin(AT)att.net), Feb 23 2002
%C A002417 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 (fvlamoen(AT)hotmail.com), 
               Oct 09 2002
%C A002417 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 A002417 ..........X.X
%C A002417 ........X.X.X.X
%C A002417 ......X.X.X.X.X.X
%C A002417 ....X.X.X.X.X.X.X.X
%C A002417 ..X.X.X.X.X.X.X.X.X.X
%C A002417 X.X.X.X.X.X.X.X.X.X.X.X
%C A002417 X.X.X.X.X.X.X.X.X.X.X.X - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 
               02 2003
%D A002417 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002417 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002417 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A002417 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 195.
%D A002417 Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift 
               voor Wiskunde, 52 (1964-1965) 193-198.
%D A002417 K. -W. Lau, Solution to Problem 2495, Journal of Recreational Mathematics 
               2002-3 31(1) 79-80.
%H A002417 T. D. Noe, <a href="b002417.txt">Table of n, a(n) for n=1..1000</a>
%H A002417 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002417 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002417 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A002417 n^2*(n+1)*(n+2)/6. G.f.: x*(1+3*x)/(1-x)^5.
%F A002417 a(n)=C(n+2, 2)n^2/3 - Paul Barry (pbarry(AT)wit.ie), Jun 26 2003
%F A002417 C(n+3, n)*C(n+1, 1) - Zerinvary Lajos (zlaja(AT)freemail.hu), Apr 27 
               2005
%F A002417 Partial sums of A002412, hexagonal pyramidal numbers, or greengrocer's 
               numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 16 2006
%F A002417 (binomial(n+3,n-1)-binomial(n+2,n-2))*(binomial(n+1,n-1)-binomial(n,n-2)). 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2006
%p A002417 seq(n^2*(n+1)*(n+2)/6,n=1..50);
%p A002417 a:=n->sum(j*(j+1)*n/2,j=0..n):seq(a(n),n=1..37); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 06 2007
%p A002417 a:=n->1/6*sum(sum (2*binomial(n,2),j=2..n),k=0..n): seq(a(n),n=2..38); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007
%p A002417 A002417:=-(1+3*z)/(z-1)^5; [S. Plouffe in his 1992 dissertation.]
%p A002417 a:=n->(sum((numbperm(n,3)), j=3..n)):seq(a(n)/6, n=2..39); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2008
%p A002417 a:=n->(sum((numbcomp(n,4)), j=4..n)):seq(a(n), n=4..40); [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
%p A002417 a:=n->add(binomial(n,2)+add(binomial(n,2), j=1..n),j=2..n)/3:seq(a(n), 
               n=2..35); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 
               27 2008]
%Y A002417 Bisection of A002624.
%Y A002417 a(n)= A093561(n+3, 4), (4, 1)-Pascal column.
%Y A002417 Cf. A062196.
%Y A002417 Cf. A002412.
%Y A002417 Sequence in context: A055832 A100175 A063489 this_sequence A126858 A113751 
               A107233
%Y A002417 Adjacent sequences: A002414 A002415 A002416 this_sequence A002418 A002419 
               A002420
%K A002417 easy,nice,nonn
%O A002417 1,2
%A A002417 N. J. A. Sloane (njas(AT)research.att.com).
%E A002417 Edited and extended by Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 
               09 2002
%E A002417 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009