1,2

a(n) is 1/6 the number of colorings of a 2 X 2 hexagonal array with n+2 colors. - Ron Hardin (rhh(AT)cadence.com), 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

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

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 (jvospost2(AT)yahoo.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; [Conjectured by 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

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

njas

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