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A212331 a(n) = 5*n*(n+5)/2.

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