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 A008854 Numbers that are congruent to {0, 1, 4} mod 5. 31
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 REFERENCES 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. LINKS 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). FORMULA 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 From Peter Bala, Aug 04 2022: (Start) a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 4 with a(1) = 0, a(2) = 1 and a(3) = 4. a(2*n) = a(n) + a(n+1); a(2*n+1) = a(n) + a(n+2). Cf. A047222 and A042965. (End) MAPLE for n to 1000 do if n^3 - n mod 10 = 0 then print(n); fi; od; MATHEMATICA 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 *) PROG (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 (PARI) a(n) = vecsum(divrem(5*n-7, 3)); \\ Kevin Ryde, Aug 08 2022 (Magma) [n : n in [0..150] | n mod 5 in [0, 1, 4]]; // Wesley Ivan Hurt, Jun 14 2016 (Python) def a(n): return sum(divmod(5*n-7, 3)) print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Aug 08 2022 after Kevin Ryde CROSSREFS Cf. A047217, A047222, A042965. Sequence in context: A064931 A177103 A114454 * A062726 A159629 A328173 Adjacent sequences: A008851 A008852 A008853 * A008855 A008856 A008857 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified September 24 12:42 EDT 2023. Contains 365579 sequences. (Running on oeis4.)