A281151 - OEIS

%I #29 Sep 08 2022 08:46:18

%S 0,1,4,9,16,24,33,44,57,72,88,105,124,145,168,192,217,244,273,304,336,

%T 369,404,441,480,520,561,604,649,696,744,793,844,897,952,1008,1065,

%U 1124,1185,1248,1312,1377,1444,1513,1584,1656,1729,1804,1881,1960,2040,2121,2204,2289

%N a(n) = floor(4*n*(n+1)/5).

%H Bruno Berselli, <a href="/A281151/b281151.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).

%F O.g.f.: x*(1 + x^2)*(1 + x)^2/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).

%F a(n) = a(-n-1) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) = a(n-5) + 8*(n-2).

%F a(5*k+r) = 20*k^2 + 4*(2*r+1)*k + r^2, where 0 <= r <= 4. Example: for r=3, a(5*k+3) = (2*k+1)*(10*k+9), which gives: 9, 57, 145, 273, 441, 649 etc. Also, a(n) belongs to A047462, in fact: for r = 0 or 4, a(n) == 0 (mod 8); for r = 1 or 3, a(n) == 1 (mod 8); for r = 2, a(n) == 4 (mod 8).

%F a(n) = a(-n) + A047462(n).

%F a(n) = n^2 - floor((n-2)^2/5).

%t Table[Floor[4 n (n + 1)/5], {n, 0, 60}]

%o (PARI) vector(60, n, n--; floor(4*n*(n+1)/5))

%o (Python) [int(4*n*(n+1)/5) for n in range(60)]

%o (Sage) [floor(4*n*(n+1)/5) for n in range(60)]

%o (Maxima) makelist(floor(4*n*(n+1)/5), n, 0, 60);

%o (Magma) [4*n*(n+1) div 5: n in [0..60]];

%Y Subsequence of A047462.

%Y Partial sums of A047486.

%Y Cf. A184005: n^2 - floor((n-2)^2/4).

%Y Cf. sequences with formula floor(k*n*(n+1)/(k+1)): A000217 (k=1), A143978 (k=2), A281026 (k=3), this sequence (k=4), A194275 (k=5).

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Jan 16 2017