A064808 a(n) is the (n+1)st (n+2)-gonal number.

1, 3, 9, 22, 45, 81, 133, 204, 297, 415, 561, 738, 949, 1197, 1485, 1816, 2193, 2619, 3097, 3630, 4221, 4873, 5589, 6372, 7225, 8151, 9153, 10234, 11397, 12645, 13981, 15408, 16929, 18547, 20265, 22086, 24013, 26049, 28197, 30460, 32841, 35343, 37969, 40722

Offset: 0

Author

Floor van Lamoen, Oct 22 2001

Keywords

• nonn
• easy

Comments

Sum of n terms of the arithmetic progression with first term 1 and common difference n-1. - Amarnath Murthy, Aug 04 2005

a(n) is the sum of (n+1)-th row terms of triangle A144693. - Gary W. Adamson, Sep 19 2008

See also A131685(k) = smallest positive number m such that c(i) = m*(i^1 + 1)*(i^2 + 2)* ... *(i^k+ k) / k! takes integral values for all i>=0: For k=2, A131685(k)=1, which implies that this is a well-defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Justin Crum, Cyrus Cheng, David A. Ham, Lawrence Mitchell, Robert C. Kirby, Joshua A. Levine, and Andrew Gillette, Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake, arXiv:2104.12986 [math.NA], 2021.
• Index to divisibility sequences
• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Main diagonal of A057145.

Row sums of A076110.

Cf. A144693. - Gary W. Adamson, Sep 19 2008

Cf. A000096, A002378, A006002, A006003.

Programs

Magma

[(n+1)*(n^2+2)/2 : n in [0..50]]; // Wesley Ivan Hurt, Feb 21 2015

Maple

A064808:=n->(n+1)*(n^2+2)/2: seq(A064808(n), n=0..50); # Wesley Ivan Hurt, Feb 21 2015

Mathematica

Table[(n + 1) (n^2 + 2)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Feb 21 2015 *)

PARI

a(n) = { (n + 1)*(n^2 + 2)/2 } \\ Harry J. Smith, Sep 26 2009

Formula

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

From Paul Barry, Nov 18 2005: (Start)

a(n) = Sum_{k=0..n} Sum_{j=0..n} (k-(k-1)*C(0, j-k)).

a(n) = A006002(n) - A000096(n-2). (End)

G.f.: (1 - x + 3x^2)/(1 - x)^4. - R. J. Mathar, Jul 07 2009

a(n) = A006003(n+1) - A002378(n). - Rick L. Shepherd, Feb 21 2015

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Feb 21 2015

a(n) = A057145(n+2,n+1). - R. J. Mathar, Jul 28 2016