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a(n) = n*(11*n + 3)/2.
15

%I #22 Sep 08 2022 08:46:11

%S 0,7,25,54,94,145,207,280,364,459,565,682,810,949,1099,1260,1432,1615,

%T 1809,2014,2230,2457,2695,2944,3204,3475,3757,4050,4354,4669,4995,

%U 5332,5680,6039,6409,6790,7182,7585,7999,8424,8860,9307,9765,10234,10714,11205

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

%C This sequence provides the first differences of A254407 and the partial sums of A017473.

%C Also:

%C a(n) - n = A022269(n);

%C a(n) + n = n*(11*n+5)/2: 0, 8, 27, 57, 98, 150, 213, 287, ...;

%C a(n) - 2*n = A022268(n);

%C a(n) + 2*n = n*(11*n+7)/2: 0, 9, 29, 60, 102, 155, 219, 294, ...;

%C a(n) - 3*n = n*(11*n-3)/2: 0, 4, 19, 45, 82, 130, 189, 259, ...;

%C a(n) + 3*n = A211013(n);

%C a(n) - 4*n = A226492(n);

%C a(n) + 4*n = A152740(n);

%C a(n) - 5*n = A180223(n);

%C a(n) + 5*n = n*(11*n+13)/2: 0, 12, 35, 69, 114, 170, 237, 315, ...;

%C a(n) - 6*n = A051865(n);

%C a(n) + 6*n = n*(11*n+15)/2: 0, 13, 37, 72, 118, 175, 243, 322, ...;

%C a(n) - 7*n = A152740(n-1) with A152740(-1) = 0;

%C a(n) + 7*n = n*(11*n+17)/2: 0, 14, 39, 75, 122, 180, 249, 329, ...;

%C a(n) - n*(n-1)/2 = A168668(n);

%C a(n) + n*(n-1)/2 = A049453(n);

%C a(n) - n*(n+1)/2 = A202803(n);

%C a(n) + n*(n+1)/2 = A033580(n).

%H Bruno Berselli, <a href="/A254963/b254963.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: x*(7 + 4*x)/(1 - x)^3.

%t Table[n (11 n + 3)/2, {n, 0, 50}]

%t LinearRecurrence[{3,-3,1},{0,7,25},50] (* _Harvey P. Dale_, Mar 25 2018 *)

%o (PARI) vector(50, n, n--; n*(11*n+3)/2)

%o (Sage) [n*(11*n+3)/2 for n in (0..50)]

%o (Magma) [n*(11*n+3)/2: n in [0..50]];

%o (Maxima) makelist(n*(11*n+3)/2, n, 0, 50);

%Y Cf. A008729 and A218530 (seventh column); A017473, A254407.

%Y 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 Cf. A069125: (2*n+1)^2 + 3*n*(n+1)/2; A147875: n^2 + 3*n*(n+1)/2.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Feb 11 2015