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a(n) is the (n+1)st (n+2)-gonal number.
21

%I #62 Sep 08 2022 08:45:04

%S 1,3,9,22,45,81,133,204,297,415,561,738,949,1197,1485,1816,2193,2619,

%T 3097,3630,4221,4873,5589,6372,7225,8151,9153,10234,11397,12645,13981,

%U 15408,16929,18547,20265,22086,24013,26049,28197,30460,32841,35343,37969,40722

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

%C Sum of n terms of the arithmetic progression with first term 1 and common difference n-1. - _Amarnath Murthy_, Aug 04 2005

%C a(n) is the sum of (n+1)-th row terms of triangle A144693. - _Gary W. Adamson_, Sep 19 2008

%C 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

%H Harry J. Smith, <a href="/A064808/b064808.txt">Table of n, a(n) for n = 0..1000</a>

%H Justin Crum, Cyrus Cheng, David A. Ham, Lawrence Mitchell, Robert C. Kirby, Joshua A. Levine, and Andrew Gillette, <a href="https://arxiv.org/abs/2104.12986">Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake</a>, arXiv:2104.12986 [math.NA], 2021.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F From _Paul Barry_, Nov 18 2005: (Start)

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

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

%F G.f.: (1 - x + 3x^2)/(1 - x)^4. - _R. J. Mathar_, Jul 07 2009

%F a(n) = A006003(n+1) - A002378(n). - _Rick L. Shepherd_, Feb 21 2015

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, Feb 21 2015

%F a(n) = A057145(n+2,n+1). - _R. J. Mathar_, Jul 28 2016

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

%t Table[(n + 1) (n^2 + 2)/2, {n, 0, 50}] (* _Wesley Ivan Hurt_, Feb 21 2015 *)

%o (PARI) { for (n=0, 1000, write("b064808.txt", n, " ", (n + 1)*(n^2 + 2)/2) ) } \\ _Harry J. Smith_, Sep 26 2009

%o (Magma) [(n+1)*(n^2+2)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Feb 21 2015

%Y Main diagonal of A057145.

%Y Row sums of A076110.

%Y Cf. A144693. - _Gary W. Adamson_, Sep 19 2008

%Y Cf. A000096, A002378, A006002, A006003.

%K nonn,easy

%O 0,2

%A _Floor van Lamoen_, Oct 22 2001