0,3

a(n)=n^2(n+1)/2 is half the number of colorings of three points on a line with n+1 colors. - Ron Hardin (rhh(AT)cadence.com), Feb 23 2002

a(n) = number of (n+6)-bit binary sequences with exactly 7 1's none of which is isolated. A 1 is isolated if its immediate neighbor(s) are 0. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004

Also as a(n)=(1/6)*(3*n^3+3*n^2), n>0: structured trigonal prism numbers (Cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.

Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 18 2005

If Y is a 3-subset of an n-set X then, for n>=5, a(n-4) is the number of 5-subsets of X having at least two elements in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Nov 23 2007

a(n-1), n>=2, is the number of ways to have n identical objects in m=2 of alltogether n distinguishable boxes (n-2 boxes stay empty). W. Lang, Nov 13 2007.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/5).

T. D. Noe, Table of n, a(n) for n=0..1000

P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 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.

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Wiener Index

Average of n^2 and n^3.

a(n) = sum of n smallest multiples of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 20 2002

G.f.: x(1+2x)/(1-x)^4 - Paul Barry (pbarry(AT)wit.ie), Mar 21 2003

a(n)=sum{k=0..n, n(n-k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003

a(n) = whole numbers * triangular numbers - Xavier Acloque Oct 27 2003

a(n) = (1/2)*Sum[Sum[(i+j),{i, 1, n}],{j, 1, n}] = (1/2)*(n^2+n^3) = (1/2)*A011379(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 13 2006

sum (n*j,j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 12 2006

a(n)=sum(sum(k, j=1..n),k=1..n), n>=0 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

Row sums of triangle A127739, triangle A132118; and binomial transform of [1, 5, 7, 3, 0, 0, 0,...] = (1, 6, 18, 40, 75,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 10 2007

a(3)=18 because 4 identical balls can be put into m=2 of n=4 distinguishable boxes in binomial(4,2)*(2!/(1!*1!)+ 2!/(2!) ) = 6*(2+1) =18 ways. The m=2 part partitions of 4, namely (1,3) and (2^2) specify the filling of each of the 6 possible two box choices. W. Lang, Nov 13 2007.

Epi:=(r, n)->stirling2(r, n): [seq (Epi(n+1, n), n=0..41)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 05 2006

seq((n-1)*binomial(n, 2), n=1..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 28 2007

a:=n->sum(sum(k, j=1..n), k=1..n): seq(a(n), n=0..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

A002411:=(1+2*z)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=n^2*(n+1)/2

Cf. A015223, A015224, A014799, A014800.

Cf. A011379, A127739, A132118.

A006002(n)=-a(-1-n).

a(n)= A093560(n+2, 3), (3, 1)-Pascal column.

A row or column of A132191.

Second column of triangle A103371.

Sequence in context: A129863 A035489 A122061 this_sequence A023658 A059834 A015224

Adjacent sequences: A002408 A002409 A002410 this_sequence A002412 A002413 A002414

nonn,easy,nice

njas