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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)

MAPLE

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

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

(MAGMA) [n : n in [0..150] | n mod 5 in [0, 1, 4]]; // Wesley Ivan Hurt, Jun 14 2016

CROSSREFS

Cf. A047217.

Sequence in context: A064931 A177103 A114454 * A062726 A159629 A223138

Adjacent sequences:  A008851 A008852 A008853 * A008855 A008856 A008857

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 26 02:35 EDT 2019. Contains 323579 sequences. (Running on oeis4.)