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A254963 a(n) = n*(11*n + 3)/2.

%I A254963 #25 Dec 15 2024 19:58:10
%S A254963 0,7,25,54,94,145,207,280,364,459,565,682,810,949,1099,1260,1432,1615,
%T A254963 1809,2014,2230,2457,2695,2944,3204,3475,3757,4050,4354,4669,4995,
%U A254963 5332,5680,6039,6409,6790,7182,7585,7999,8424,8860,9307,9765,10234,10714,11205,11707
%N A254963 a(n) = n*(11*n + 3)/2.
%C A254963 This sequence provides the first differences of A254407 and the partial sums of A017473.
%C A254963 Also:
%C A254963 a(n) - n         = A022269(n);
%C A254963 a(n) + n         = n*(11*n+5)/2: 0, 8, 27, 57, 98, 150, 213, 287, ...;
%C A254963 a(n) - 2*n       = A022268(n);
%C A254963 a(n) + 2*n       = n*(11*n+7)/2: 0, 9, 29, 60, 102, 155, 219, 294, ...;
%C A254963 a(n) - 3*n       = n*(11*n-3)/2: 0, 4, 19, 45, 82, 130, 189, 259, ...;
%C A254963 a(n) + 3*n       = A211013(n);
%C A254963 a(n) - 4*n       = A226492(n);
%C A254963 a(n) + 4*n       = A152740(n);
%C A254963 a(n) - 5*n       = A180223(n);
%C A254963 a(n) + 5*n       = n*(11*n+13)/2: 0, 12, 35, 69, 114, 170, 237, 315, ...;
%C A254963 a(n) - 6*n       = A051865(n);
%C A254963 a(n) + 6*n       = n*(11*n+15)/2: 0, 13, 37, 72, 118, 175, 243, 322, ...;
%C A254963 a(n) - 7*n       = A152740(n-1) with A152740(-1) = 0;
%C A254963 a(n) + 7*n       = n*(11*n+17)/2: 0, 14, 39, 75, 122, 180, 249, 329, ...;
%C A254963 a(n) - n*(n-1)/2 = A168668(n);
%C A254963 a(n) + n*(n-1)/2 = A049453(n);
%C A254963 a(n) - n*(n+1)/2 = A202803(n);
%C A254963 a(n) + n*(n+1)/2 = A033580(n).
%H A254963 Bruno Berselli, <a href="/A254963/b254963.txt">Table of n, a(n) for n = 0..1000</a>
%H A254963 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A254963 G.f.: x*(7 + 4*x)/(1 - x)^3.
%F A254963 From _Elmo R. Oliveira_, Dec 15 2024: (Start)
%F A254963 E.g.f.: exp(x)*x*(14 + 11*x)/2.
%F A254963 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A254963 Table[n (11 n + 3)/2, {n, 0, 50}]
%t A254963 LinearRecurrence[{3,-3,1},{0,7,25},50] (* _Harvey P. Dale_, Mar 25 2018 *)
%o A254963 (PARI) vector(50, n, n--; n*(11*n+3)/2)
%o A254963 (Sage) [n*(11*n+3)/2 for n in (0..50)]
%o A254963 (Magma) [n*(11*n+3)/2: n in [0..50]];
%o A254963 (Maxima) makelist(n*(11*n+3)/2, n, 0, 50);
%Y A254963 Cf. A008729 and A218530 (seventh column); A017473, A254407.
%Y A254963 Cf. similar sequences of the type 4*n^2 + k*n*(n+1)/2: A055999 (k=-7, n>6), A028552 (k=-6, n>2), A095794 (k=-5, n>1), A046092 (k=-4, n>0), A000566 (k=-3), A049450 (k=-2), A022264 (k=-1), A016742 (k=0), A022267 (k=1), A202803 (k=2), this sequence (k=3), A033580 (k=4).
%Y A254963 Cf. A069125: (2*n+1)^2 + 3*n*(n+1)/2; A147875: n^2 + 3*n*(n+1)/2.
%Y A254963 Cf. A022268, A022269, A049453, A051865, A152740, A168668, A180223, A211013, A226492.
%K A254963 nonn,easy
%O A254963 0,2
%A A254963 _Bruno Berselli_, Feb 11 2015