A081250 Numbers k such that A081249(m)/m^2 has a local minimum for m = k.
1, 3, 11, 33, 101, 303, 911, 2733, 8201, 24603, 73811, 221433, 664301, 1992903, 5978711, 17936133, 53808401, 161425203, 484275611, 1452826833, 4358480501, 13075441503, 39226324511, 117678973533, 353036920601, 1059110761803
Offset: 0
Examples
11 is a term since A081249(10)/10^2 = 11/100 = 0.110, A081249(11)/11^2 = 13/121 = 0.107, A081249(12)/12^2 = 16/144 = 0.111.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
- Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
- Index entries for linear recurrences with constant coefficients, signature (3,1,-3).
Programs
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GAP
List([0..30], n-> (5*3^n +(-1)^n -2)/4); # G. C. Greubel, Jul 14 2019
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Magma
[Floor(3^n*5/4): n in [0..30]]; // Vincenzo Librandi, Jun 10 2011
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Maple
a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+2 od: seq(a[n], n=0..30); # Zerinvary Lajos, Apr 28 2008
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Mathematica
Floor[5*3^Range[0, 30]/4] (* Wesley Ivan Hurt, Mar 30 2017 *)
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PARI
vector(30, n, n--; (5*3^n +(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019
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Sage
[(5*3^n +(-1)^n -2)/4 for n in (0..30)] # G. C. Greubel, Jul 14 2019
Formula
a(n) = floor(3^n*5/4).
G.f.: x*(1+x^2)/((1-x)*(1+x)*(1-3*x)).
a(n) = 3*a(n-1) + 1*a(n-2) - 3*a(n-3).
a(n) = (5*3^n + (-1)^n - 2)/4. - Paul Barry, May 19 2003
a(n) = a(n-2) + 10*3^(n-2) for n > 1.
a(n+2) - a(n) = A005052(n).
a(2*n) = Sum_{j=1..n+1} A062107(2*j).
a(2*n+1) = Sum_{j=1..n+1} A062107(2*j+1).
With a leading 0, this is a(n) = (5*3^n - 6 + 4*0^n - 3*(-1)^n)/12, the binomial transform of A084183. - Paul Barry, May 19 2003
Convolution of 3^n and {1, 0, 2, 0, 2, 0, ...}. a(n) = Sum_{k=0..n} ((1 + (-1)^k) - 0^k)*3^(n-k) = Sum_{k=0..n} ((1 + (-1)^(n-k)) - 0^(n-k))3^k. - Paul Barry, Jul 19 2004
a(n) = 2*a(n-1) + 3*a(n-2) + 2, a(0)=1, a(1)=3. - Zerinvary Lajos, Apr 28 2008
Extensions
Offset changed from 1 to 0 by Vincenzo Librandi, Jun 10 2011
Comments