A008851 Congruent to 0 or 1 mod 5.

0, 1, 5, 6, 10, 11, 15, 16, 20, 21, 25, 26, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51, 55, 56, 60, 61, 65, 66, 70, 71, 75, 76, 80, 81, 85, 86, 90, 91, 95, 96, 100, 101, 105, 106, 110, 111, 115, 116, 120, 121, 125, 126, 130, 131, 135, 136, 140, 141, 145, 146, 150, 151

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers k that have the same last digit as k^2.

References

• L. E. Dickson, History of the Theory of Numbers, I, p. 459.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001622, A003226, A045953, A046831, A046851, A086457.

Programs

Haskell

a008851 n = a008851_list !! (n-1)
a008851_list = [10*n + m | n <- [0..], m <- [0,1,5,6]]
-- Reinhard Zumkeller, Jul 27 2011

Magma

[n: n in [0..200] | n mod 5 in {0, 1}]; // Vincenzo Librandi, Nov 17 2014

Maple

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+5 od: seq(a[n], n=0..61); # Zerinvary Lajos, Mar 16 2008

Mathematica

Select[Range[0, 151], MemberQ[{0, 1}, Mod[#, 5]] &] (* T. D. Noe, Mar 31 2013 *)

PARI

a(n) = 5*(n\2)+bitand(n,1); /* Joerg Arndt, Mar 31 2013 */

PARI

a(n) = floor((5/3)*floor(3*(n-1)/2)); /* Joerg Arndt, Mar 31 2013 */

Formula

a(n) = 5*n - a(n-1) - 9, n >= 2. - Vincenzo Librandi, Nov 18 2010 [Corrected for offset by David Lovler, Oct 10 2022]

G.f.: x^2*(1+4*x) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 07 2011

a(n+1) = Sum_{k>=0} A030308(n,k)*A146523(k). - Philippe Deléham, Oct 17 2011

a(n) = floor((5/3)*floor(3*(n-1)/2)). - Clark Kimberling, Jul 04 2012

a(n) = (10*n - 13 - 3*(-1)^n)/4. - Robert Israel, Nov 17 2014 [Corrected by David Lovler, Sep 21 2022]

E.g.f.: 4 + ((10*x - 13)*exp(x) - 3*exp(-x))/4. - David Lovler, Sep 11 2022

Sum_{n>=2} (-1)^n/a(n) = sqrt(1+2/sqrt(5))*Pi/10 + log(phi)/(2*sqrt(5)) + log(5)/4, where phi is the golden ratio (A001622). - Amiram Eldar, Oct 12 2022

Extensions

Offset corrected by Reinhard Zumkeller, Jul 27 2011