A008854 Numbers that are congruent to {0, 1, 4} mod 5.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A047217, A047222, A042965.

Programs

Magma

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

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 *)

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

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

Formula

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)