A294259 - OEIS

%I #33 Sep 22 2025 16:01:28

%S 0,1,4,15,43,100,201,364,610,963,1450,2101,2949,4030,5383,7050,9076,

%T 11509,14400,17803,21775,26376,31669,37720,44598,52375,61126,70929,

%U 81865,94018,107475,122326,138664,156585,176188,197575,220851,246124,273505,303108,335050,369451

%N a(n) = n*(n^3 + 2*n^2 - 5*n + 10)/8.

%C a(n) is even for n in A047481.

%C Also, a(n) is divisible by 5 if and only if n belongs to A047218.

%H Bruno Berselli, <a href="/A294259/b294259.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F O.g.f.: x*(1 - x + 5*x^2 - 2*x^3)/(1 - x)^5.

%F E.g.f.: x*(8 + 8*x + 8*x^2 + x^3)*exp(x)/8.

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.

%F a(n) = 2*n + Sum_{i=0..n} i*(i^2 - 3)/2.

%e After 0:

%e 1   =                     -(0) + (1);

%e 4   =                 -(0 + 1) + (2 + 2*3/2);

%e 15  =             -(0 + 1 + 2) + (3 + 4 + 5 + 3*4/2);

%e 43  =         -(0 + 1 + 2 + 3) + (4 + 5 + 6 + 7 + 8 + 9 + 4*5/2);

%e 100 =     -(0 + 1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + ... + 14 + 5*6/2);

%e 201 = -(0 + 1 + 2 + 3 + 4 + 5) + (6 + 7 + 8 + 9 + ... + 20 + 6*7/2), etc.

%p a := n -> n*(n*(n*(n+2)-5)+10)/8: seq(a(n),n=0..41); # _Peter Luschny_, Nov 06 2017

%t Table[n (n^3 + 2 n^2 - 5 n + 10)/8, {n, 0, 50}]

%t LinearRecurrence[{5,-10,10,-5,1},{0,1,4,15,43},50] (* _Harvey P. Dale_, Jan 08 2024 *)

%o (PARI) vector(50, n, n--; n*(n^3+2*n^2-5*n+10)/8)

%o (SageMath) [n*(n^3+2*n^2-5*n+10)/8 for n in range(50)]

%o (Maxima) makelist(n*(n^3+2*n^2-5*n+10)/8, n, 0, 50);

%o (Magma) [n*(n^3+2*n^2-5*n+10)/8: n in [0..50]];

%o (GAP) List([0..50], n -> n*(n^3+2*n^2-5*n+10)/8);

%Y Cf. A000217, A002817, A176145.

%Y Cf. A101374: the sums in the Example section end in squares.

%Y Subsequence of A047207.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Oct 30 2017