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A226492 a(n) = n*(11*n-5)/2.

%I A226492 #55 Dec 28 2024 10:18:46
%S A226492 0,3,17,42,78,125,183,252,332,423,525,638,762,897,1043,1200,1368,1547,
%T A226492 1737,1938,2150,2373,2607,2852,3108,3375,3653,3942,4242,4553,4875,
%U A226492 5208,5552,5907,6273,6650,7038,7437,7847,8268,8700,9143,9597,10062,10538,11025,11523
%N A226492 a(n) = n*(11*n-5)/2.
%C A226492 Sequences of numbers of the form n*(n*k - k + 6)/2:
%C A226492 . k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;
%C A226492 . k=11: a(n);
%C A226492 . k=12: A094159;
%C A226492 . k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;
%C A226492 . k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;
%C A226492 . k=15: A152773;
%C A226492 . k=16: A139272;
%C A226492 . k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;
%C A226492 . k=18: A152751;
%C A226492 . k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;
%C A226492 . k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;
%C A226492 . k=21: A152759;
%C A226492 . k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;
%C A226492 . k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;
%C A226492 . k=24: A152767;
%C A226492 . k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;
%C A226492 . k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;
%C A226492 . k=27: A153783;
%C A226492 . k=28: A195021;
%C A226492 . k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;
%C A226492 . k=30: A153448;
%C A226492 . k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;
%C A226492 . k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;
%C A226492 . k=33: A153875.
%C A226492 Also:
%C A226492 a(n) - n         = A180223(n);
%C A226492 a(n) + n         = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;
%C A226492 a(n) - 2*n       = A051865(n);
%C A226492 a(n) + 2*n       = A022268(n);
%C A226492 a(n) - 3*n       = A152740(n-1);
%C A226492 a(n) + 3*n       = A022269(n);
%C A226492 a(n) - 4*n       = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;
%C A226492 a(n) + 4*n       = A254963(n);
%C A226492 a(n) - n*(n-1)/2 = A147874(n+1);
%C A226492 a(n) + n*(n-1)/2 = A094159(n) (case k=12);
%C A226492 a(n) - n*(n-1)   = A062741(n) (see above, this is the case k=9);
%C A226492 a(n) + n*(n-1)   = n*(13*n-7)/2 (case k=13);
%C A226492 a(n) - n*(n+1)/2 = A135706(n);
%C A226492 a(n) + n*(n+1)/2 = A033579(n);
%C A226492 a(n) - n*(n+1)   = A051682(n);
%C A226492 a(n) + n*(n+1)   = A186030(n);
%C A226492 a(n) - n^2       = A062708(n);
%C A226492 a(n) + n^2       = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.
%C A226492 Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...
%H A226492 Bruno Berselli, <a href="/A226492/b226492.txt">Table of n, a(n) for n = 0..1000</a>
%H A226492 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A226492 G.f.: x*(3+8*x)/(1-x)^3.
%F A226492 a(n) + a(-n) = A033584(n).
%F A226492 From _Elmo R. Oliveira_, Dec 27 2024: (Start)
%F A226492 E.g.f.: exp(x)*x*(6 + 11*x)/2.
%F A226492 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
%F A226492 a(n) = n + A180223(n). (End)
%t A226492 Table[n (11 n - 5)/2, {n, 0, 50}]
%t A226492 CoefficientList[Series[x (3 + 8 x) / (1 - x)^3, {x, 0, 45}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A226492 LinearRecurrence[{3,-3,1},{0,3,17},50] (* _Harvey P. Dale_, Jan 14 2019 *)
%o A226492 (Magma) [n*(11*n-5)/2: n in [0..50]];
%o A226492 (Magma) I:=[0,3,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..46]]; // _Vincenzo Librandi_, Aug 18 2013
%o A226492 (PARI) a(n)=n*(11*n-5)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A226492 Cf. sequences in Comments lines.
%Y A226492 First differences are in A017425.
%Y A226492 Cf. A033584, A180223.
%K A226492 nonn,easy
%O A226492 0,2
%A A226492 _Bruno Berselli_, Jun 11 2013