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A008854
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Numbers that are congruent to {0, 1, 4} mod 5.
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31
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0, 1, 4, 5, 6, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 31, 34, 35, 36, 39, 40, 41, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61, 64, 65, 66, 69, 70, 71, 74, 75, 76, 79, 80, 81, 84, 85, 86, 89, 90, 91, 94, 95, 96, 99, 100, 101, 104, 105, 106, 109
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OFFSET
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1,3
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COMMENTS
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n^3 and n have the same last digit.
Partial sums of (0, 1, 3, 1, 1, 3, 1, 1, 3, 1, ...). - Gary W. Adamson, Jun 19 2008
Row sum of a triangle where every "triple" contains 1,2,2. - Craig Knecht, Jul 30 2015
Nonnegative m such that floor(k*m^2/5) = k*floor(m^2/5), where k = 2, 3 or 4. - Bruno Berselli, Dec 03 2015
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 459.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Craig Knecht, Triangle where every "triple" contains 1,2,2.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
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FORMULA
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a(n) = -1 + Sum_{k=1..n} (-(k mod 3)+5*((k+1) mod 3)+11*((k+2) mod 3))/9. - Paolo P. Lava, Sep 03 2010
G.f.: x^2*(1+3*x+x^2) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = A047217(n+1)-1. - R. J. Mathar, Aug 04 2015
E.g.f: (5/3)*(x-1)*exp(x) + (2/3)*exp(-x/2)*cos(sqrt(3)*x/2) + (2/9)*exp(-x/2)*sin(sqrt(3)*x/2) + 1. - Robert Israel, Aug 04 2015
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (15*n-15+6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 5k-1, a(3k-1) = 5k-4, a(3k-2) = 5k-5. (End)
a(n) = 5*n/3 - 2*(n mod 3)/3 - 1. - Ammar Khatab, Aug 26 2020
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/5 - arccoth(3/sqrt(5))/sqrt(5). - Amiram Eldar, Dec 10 2021
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MAPLE
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for n to 1000 do if n^3 - n mod 10 = 0 then print(n); fi; od;
A008854:=n->(15*n-15+6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A008854(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016
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MATHEMATICA
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Select[Range[0, 150], MemberQ[{0, 1, 4}, Mod[#, 5]] &] (* or *) LinearRecurrence[{1, 0, 1, -1}, {0, 1, 4, 5}, 91] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
CoefficientList[Series[x (1 + 3 x + x^2) / ((1 + x + x^2) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
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PROG
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(PARI) concat(0, Vec(x^2*(1+3*x+x^2)/((1+x+x^2)*(x-1)^2) + O(x^100))) \\ Altug Alkan, Dec 03 2015
(MAGMA) [n : n in [0..150] | n mod 5 in [0, 1, 4]]; // Wesley Ivan Hurt, Jun 14 2016
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CROSSREFS
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Cf. A047217.
Sequence in context: A064931 A177103 A114454 * A062726 A159629 A328173
Adjacent sequences: A008851 A008852 A008853 * A008855 A008856 A008857
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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