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 A212331 a(n) = 5*n*(n+5)/2. 6
 0, 15, 35, 60, 90, 125, 165, 210, 260, 315, 375, 440, 510, 585, 665, 750, 840, 935, 1035, 1140, 1250, 1365, 1485, 1610, 1740, 1875, 2015, 2160, 2310, 2465, 2625, 2790, 2960, 3135, 3315, 3500, 3690, 3885, 4085, 4290, 4500, 4715, 4935, 5160, 5390, 5625, 5865 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Numbers of the form n*t(n+5,h)-(n+5)*t(n,h), where t(k,h) = k(k+2h+1)/2 for any h. Likewise: A000217(n) = n*t(n+1,h)-(n+1)*t(n,h), A005563(n) = n*t(n+2,h)-(n+2)*t(n,h), A140091(n) = n*t(n+3,h)-(n+3)*t(n,h), A067728(n) = n*t(n+4,h)-(n+4)*t(n,h) (n>0), A140681(n) = n*t(n+6,h)-(n+6)*t(n,h). This is the case r=7 in the formula: 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. Also, a(k) is a square for k = (5/2)*(A078986(n)-1). Sum of reciprocals of a(n), for n>0: 137/750. Also, numbers h such that 8*h/5+25 is a square. LINKS Bruno Berselli, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA G.f.: 5*x*(3-2*x)/(1-x)^3. a(n) = a(-n-5) = 5*A055998(n). E.g.f.: (5/2)*x*(x + 6)*exp(x). - G. C. Greubel, Jul 21 2017 EXAMPLE From the first and second comment derives the following table: ---------------------------------------------------------------- h \ n | 0   1    2    3    4    5    6    7    8    9    10 ------|--------------------------------------------------------- 0     | 0,  1,   3,   6,  10,  15,  21,  28,  36,  45,   55, ...  (A000217) 1     | 0,  3,   8,  15,  24,  35,  48,  63,  80,  99,  120, ...  (A005563) 2     | 0,  6,  15,  27,  42,  60,  81, 105, 132, 162,  195, ...  (A140091) 3     | 0, 10,  24,  42,  64,  90, 120, 154, 192, 234,  280, ...  (A067728) 4     | 0, 15,  35,  60,  90, 125, 165, 210, 260, 315,  375, ...  (A212331) 5     | 0, 21,  48,  81, 120, 165, 216, 273, 336, 405,  480, ...  (A140681) 6     | 0, 28,  63, 105, 154, 210, 273, 343, 420, 504,  595, ... 7     | 0, 36,  80, 132, 192, 260, 336, 420, 512, 612,  720, ... 8     | 0, 45,  99, 162, 234, 315, 405, 504, 612, 729,  855, ... 9     | 0, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, ... with the formula n*(h+1)*(h+n+1)/2. See also A098737. MATHEMATICA Table[(5/2) n (n + 5), {n, 0, 46}] PROG (MAGMA) [5*n*(n+5)/2: n in [0..46]]; (PARI) a(n)=5*n*(n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015 CROSSREFS Cf. A000217, A000537, A005563, A067728, A098737, A140091, A140681. Sequence in context: A162280 A290781 A063532 * A067930 A257836 A146688 Adjacent sequences:  A212328 A212329 A212330 * A212332 A212333 A212334 KEYWORD nonn,easy AUTHOR Bruno Berselli, May 30 2012 EXTENSIONS Extended by Bruno Berselli, Aug 05 2015 STATUS approved

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Last modified January 16 11:20 EST 2019. Contains 319188 sequences. (Running on oeis4.)