A281026 a(n) = floor(3*n*(n+1)/4).
0, 1, 4, 9, 15, 22, 31, 42, 54, 67, 82, 99, 117, 136, 157, 180, 204, 229, 256, 285, 315, 346, 379, 414, 450, 487, 526, 567, 609, 652, 697, 744, 792, 841, 892, 945, 999, 1054, 1111, 1170, 1230, 1291, 1354, 1419, 1485, 1552, 1621, 1692, 1764, 1837, 1912, 1989, 2067, 2146
Offset: 0
Links
- Bruno Berselli, Table of n, a(n) for n = 0..1000
- Bruno Berselli, Illustration of the initial terms.
- Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).
Crossrefs
Programs
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Magma
[3*n*(n+1) div 4: n in [0..60]];
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Maple
A281026:=n->floor(3*n*(n+1)/4): seq(A281026(n), n=0..100); # Wesley Ivan Hurt, Jan 13 2017
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Mathematica
Table[Floor[3 n (n + 1)/4], {n, 0, 60}] LinearRecurrence[{3,-4,4,-3,1},{0,1,4,9,15},60] (* Harvey P. Dale, Jun 04 2023 *) -
Maxima
makelist(floor(3*n*(n+1)/4), n, 0, 60);
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PARI
vector(60, n, n--; floor(3*n*(n+1)/4))
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Python
[int(3*n*(n+1)/4) for n in range(60)]
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Sage
[floor(3*n*(n+1)/4) for n in range(60)]
Formula
O.g.f.: x*(1 + x + x^2)/((1 + x^2)*(1 - x)^3).
E.g.f.: -(1 - 6*x - 3*x^2)*exp(x)/4 - (1 + i)*(i - exp(2*i*x))*exp(-i*x)/8, where i=sqrt(-1).
a(n) = a(-n-1) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) = a(n-4) + 6*n - 9.
a(n) = 3*n*(n+1)/4 + (i^(n*(n+1)) - 1)/4. Therefore:
a(4*k+r) = 12*k^2 + 3*(2*r+1)*k + r^2, where 0 <= r <= 3.
a(n) = n^2 - floor((n-1)*(n-2)/4).
a(n) = A011865(3*n+2).
Comments