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A269457 a(n) = 5*(n + 1)*(n + 4)/2.

%I A269457 #44 Dec 25 2024 04:23:24
%S A269457 10,25,45,70,100,135,175,220,270,325,385,450,520,595,675,760,850,945,
%T A269457 1045,1150,1260,1375,1495,1620,1750,1885,2025,2170,2320,2475,2635,
%U A269457 2800,2970,3145,3325,3510,3700,3895,4095,4300,4510,4725,4945,5170,5400,5635
%N A269457 a(n) = 5*(n + 1)*(n + 4)/2.
%C A269457 More generally, the ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2 is (k*(k - 1)/2 + (k*(3 - k)/2)*x)/(1 - x)^3 (see links section).
%H A269457 Ilya Gutkovskiy, <a href="/A269457/a269457.pdf">Sequences of the form k*(n + 1)*(n - 1 + k)/2</a>.
%H A269457 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A269457 G.f.: 5*(2 - x)/(1 - x)^3.
%F A269457 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A269457 a(n) = Sum_{k=0..n} 5*(k + 2) = Sum_{k=0..n} A008587(k + 2).
%F A269457 Sum_{n>=0} 1/a(n) = 11/45 = 0.24444444444... = A040002.
%F A269457 a(n) = 5*A000096(n+1).
%F A269457 a(n) = A055998(2*n+2) + A055998(n+1). - _Bruno Berselli_, Sep 23 2016
%F A269457 E.g.f.: 5*exp(x)*(4 + 6*x + x^2)/2. - _Elmo R. Oliveira_, Dec 24 2024
%e A269457 a(0) = 0 + 1 + 2 + 3 + 4 = 10;
%e A269457 a(1) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 = 25;
%e A269457 a(2) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 45, etc.
%t A269457 Table[5 (n + 1) ((n + 4)/2), {n, 0, 45}]
%t A269457 Table[Sum[5 (k + 2), {k, 0, n}], {n, 0, 45}]
%t A269457 LinearRecurrence[{3, -3, 1}, {10, 25, 45}, 46]
%o A269457 (Magma) [5*(n+1)*(n+4)/2: n in [0..50]]; // _Vincenzo Librandi_, Feb 28 2016
%o A269457 (PARI) a(n) = 5*(n + 1)*(n + 4)/2; \\ _Michel Marcus_, Feb 29 2016
%o A269457 (PARI) Vec(5*(2-x)/(1-x)^3 + O(x^100)) \\ _Altug Alkan_, Mar 04 2016
%Y A269457 Cf. A000096, A008587, A028895, A040002, A045943, A054000, A055998, A067728.
%K A269457 nonn,easy
%O A269457 0,1
%A A269457 _Ilya Gutkovskiy_, Feb 27 2016