1,2

These are Hogben's central polygonal numbers

2

.P

3 n

Also the sum of three consecutive triangular numbers (A000217), i.e.; a(4) = 19 = T4 + T3 + T2 = 10 + 6 + 3. - Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 27 2001

For n>2 sigma(a(n)) gives the sum pertaining to the magic square of order n. E.g. for n = 5 we have sigma(a(n)) = 1+4+10+19+31= 65. In general sigma( a(n)) = n(n^2 +1)/2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 22 2001

Binomial transform of (1,3,3,0,0,0,.....). - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

a(n) is the difference of two tetrahedral(or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-2)(n-1)(n)/6. - Alexander Adamchuk (alex(AT)kolmogorov.com), May 20 2006

Partial sums are A006003(n) = n(n^2+1)/2. Finite differences are a(n+1) - a(n) = A008585(n) = 3n. - Alexander Adamchuk (Smith(AT)xxx.yyy.com), Jun 03 2006

If X is an n-set and Y a fixed 3-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007

Equals (1, 2, 3,...) convolved with (1, 2, 3, 3, 3,...). a(4) = 19 = (1, 2, 3, 4) dot (3, 3, 2, 1) = (3 + 6 + 6 + 4). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009]

Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0,...]. [From Gary W. Adamson & Alexander Povolotsky (qntmpkt(AT)yahoo.com), May 29 2009]

Except for the first term, a(n)=3*n+a(n-1), (with a(1)=4) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]

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

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 22.

R. Reed, The Lemming Simulation Problem, Math. in School, 3 (#6, Nov. 1974), 5-6.

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

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

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Two Enumerative Functions

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.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

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

Index entries for sequences related to centered polygonal numbers

Expansion of (1-x^3 )/(1-x)^4.

a(n)=C(n+3, 3)-C(n, 3)=C(n, 0)+3C(n, 1)+3C(n, 2). - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

a(n) = 1+ sum (3*n) - Xavier Acloque Oct 25 2003

a(n) = T(n) + S(n-1) = A000217(n) + A000290(n-1) = (3*A016754(n) + 5)/8. - Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 05 2005

Euler transform of length 3 sequence [ 4, 0, -1]. - Michael Somos Sep 23 2006

a(1-n)=a(n). - Michael Somos Sep 23 2006

binomial(n+1,n-1)+binomial(n,n-2)+binomial(n-1,n-3). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 03 2006

Row sums of triangle A134482. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 27 2007

Narayana transform (A001263) * [1, 3, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2007

a(n)=3a(n-1)-3a(n-2)+a(n-3), a(1)=1, a(2)=4, a(3)=10 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]

a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. [From Omar E. Pol (info(AT)polprimos.com), Dec 27 2008]

a(n) = 3n(n+1)/2 + 1, with offset 0. [From Omar E. Pol (info(AT)polprimos.com), Nov 08 2009]

a(n)=3*n+a(n-1)-3 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

For n=2, a(2)=3*2+1-3=4; n=3, a(3)=3*3+4-3=10; n=4, a(4)=3*4+10-3=19 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

A005448 := n->(3*n^2+3*n+2)/2;

A005448:=-(1+z+z**2)/(z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]

s=1; lst={s}; Do[s+=n; AppendTo[lst, s], {n, 3, 7!, 3}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 13 2008]

(PARI) {a(n)=3*(n^2-n)/2+1} /* Michael Somos Sep 23 2006 */

Cf. A045943, A001844.

Cf. A000292.

Cf. A006003 - partial sums. Cf. A008585 - finite differences.

Cf. A134482.

Cf. A001263.

Cf. A000217. [From Omar E. Pol (info(AT)polprimos.com), Dec 27 2008]

Sequence in context: A152946 A025720 A022793 this_sequence A037040 A007077 A009895

Adjacent sequences: A005445 A005446 A005447 this_sequence A005449 A005450 A005451

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy, Dec 12, 1974

More terms from Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007