1,2

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

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

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 (blekraj(AT)yahoo.com), Sep 02 2003

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

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

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.

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

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

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

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

Index entries for sequences related to linear recurrences with constant coefficients

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

n^2*(n+1)*(n+2)/6. G.f.: x*(1+3*x)/(1-x)^5.

a(n)=C(n+2, 2)n^2/3 - Paul Barry (pbarry(AT)wit.ie), Jun 26 2003

C(n+3, n)*C(n+1, 1) - Zerinvary Lajos (zlaja(AT)freemail.hu), Apr 27 2005

Partial sums of A002412, hexagonal pyramidal numbers, or greengrocer's numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 16 2006

(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

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

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

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

A002417:=-(1+3*z)/(z-1)^5; [S. Plouffe in his 1992 dissertation.]

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

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]

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]

Bisection of A002624.

a(n)= A093561(n+3, 4), (4, 1)-Pascal column.

Cf. A062196.

Cf. A002412.

Sequence in context: A055832 A100175 A063489 this_sequence A126858 A113751 A107233

Adjacent sequences: A002414 A002415 A002416 this_sequence A002418 A002419 A002420

easy,nice,nonn

N. J. A. Sloane (njas(AT)research.att.com).

Edited and extended by Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 09 2002

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009