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%I #52 May 03 2024 17:11:58
%S 1,9,26,52,87,131,184,246,317,397,486,584,691,807,932,1066,1209,1361,
%T 1522,1692,1871,2059,2256,2462,2677,2901,3134,3376,3627,3887,4156,
%U 4434,4721,5017,5322,5636,5959,6291,6632,6982,7341,7709,8086,8472,8867,9271
%N Diagonal of triangular spiral in A051682.
%C Binomial transform of (1, 8, 9, 0, 0, 0, ...).
%H Vincenzo Librandi, <a href="/A081267/b081267.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjić, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Richard J. Mathar, <a href="https://arxiv.org/abs/2404.18806">Bivariate generating functions enumerating non-bonding dominoes on rectangular boards</a>, arXiv:2404.18806 (2024) Table 2.
%H Richard J. Mathar, <a href="https://arxiv.org/abs/2404.18806">Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards</a>, arXiv:2404.18806 [math.CO], 2024. See p. 7.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = C(n, 0) + 8*C(n, 1) + 9*C(n, 2).
%F a(n) = (9*n^2 + 7*n + 2)/2.
%F G.f.: (1 + 6*x + 2*x^2)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 3 1 0 / 5 3 1]. M^n * [1 1 1] = [1 3n+1 a(n)]. - _Gary W. Adamson_, Dec 22 2004
%F a(n) = 9*n + a(n-1) - 1 with n > 0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = (n+1)*A000326(n+1) - (n)*A000326(n). - _Bruno Berselli_, Dec 10 2012
%F a(n) = A050509(n) - A050509(n-1). - _Bill McEachen_, Nov 01 2020
%F E.g.f.: exp(x)*(2 + 16*x + 9*x^2)/2. - _Stefano Spezia_, Dec 25 2022
%t LinearRecurrence[{3,-3,1},{1,9,26},50] (* _Harvey P. Dale_, Aug 13 2014 *)
%t CoefficientList[Series[(1 + 6 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 14 2014 *)
%o (Magma) [(9*n^2 + 7*n + 2)/2: n in [0..50]]; // _Vincenzo Librandi_, Aug 14 2014
%o (PARI) a(n)=(9*n^2+7*n+2)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000326, A050509, A051682, A081268.
%Y Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 15 2003
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