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%I #88 Feb 26 2022 04:24:23
%S 0,15,35,60,90,125,165,210,260,315,375,440,510,585,665,750,840,935,
%T 1035,1140,1250,1365,1485,1610,1740,1875,2015,2160,2310,2465,2625,
%U 2790,2960,3135,3315,3500,3690,3885,4085,4290,4500,4715,4935,5160,5390,5625,5865
%N a(n) = 5*n*(n+5)/2.
%C Numbers of the form n*t(n+5,h)-(n+5)*t(n,h), where t(k,h) = k*(k+2*h+1)/2 for any h. Likewise:
%C A000217(n) = n*t(n+1,h)-(n+1)*t(n,h),
%C A005563(n) = n*t(n+2,h)-(n+2)*t(n,h),
%C A140091(n) = n*t(n+3,h)-(n+3)*t(n,h),
%C A067728(n) = n*t(n+4,h)-(n+4)*t(n,h) (n>0),
%C A140681(n) = n*t(n+6,h)-(n+6)*t(n,h).
%C This is the case r=7 in the formula:
%C u(r,n) = (P(r, P(n+r, r+6)) - P(n+r, P(r, r+6))) / ((r+5)*(r+6)/2)^2, where P(s, m) is the m-th s-gonal number.
%C Also, a(k) is a square for k = (5/2)*(A078986(n)-1).
%C Sum of reciprocals of a(n), for n>0: 137/750.
%C Also, numbers h such that 8*h/5+25 is a square.
%C The table given below as example gives the dimensions D(h, n) of the irreducible SU(3) multiplets (h,n). See the triangle A098737 with offset 0, and the comments there, also with a link and the Coleman reference. - _Wolfdieter Lang_, Dec 18 2020
%H Bruno Berselli, <a href="/A212331/b212331.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: 5*x*(3-2*x)/(1-x)^3.
%F a(n) = a(-n-5) = 5*A055998(n).
%F E.g.f.: (5/2)*x*(x + 6)*exp(x). - _G. C. Greubel_, Jul 21 2017
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/25 - 47/750. - _Amiram Eldar_, Feb 26 2022
%e From the first and second comment derives the following table:
%e ----------------------------------------------------------------
%e h \ n | 0 1 2 3 4 5 6 7 8 9 10
%e ------|---------------------------------------------------------
%e 0 | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... (A000217)
%e 1 | 0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, ... (A005563)
%e 2 | 0, 6, 15, 27, 42, 60, 81, 105, 132, 162, 195, ... (A140091)
%e 3 | 0, 10, 24, 42, 64, 90, 120, 154, 192, 234, 280, ... (A067728)
%e 4 | 0, 15, 35, 60, 90, 125, 165, 210, 260, 315, 375, ... (A212331)
%e 5 | 0, 21, 48, 81, 120, 165, 216, 273, 336, 405, 480, ... (A140681)
%e 6 | 0, 28, 63, 105, 154, 210, 273, 343, 420, 504, 595, ...
%e 7 | 0, 36, 80, 132, 192, 260, 336, 420, 512, 612, 720, ...
%e 8 | 0, 45, 99, 162, 234, 315, 405, 504, 612, 729, 855, ...
%e 9 | 0, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, ...
%e with the formula n*(h+1)*(h+n+1)/2. See also A098737.
%t Table[(5/2) n (n + 5), {n, 0, 46}]
%o (Magma) [5*n*(n+5)/2: n in [0..46]];
%o (PARI) a(n)=5*n*(n+5)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000217, A000537, A005563, A055998, A067728, A098737, A140091, A140681.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, May 30 2012
%E Extended by _Bruno Berselli_, Aug 05 2015
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