A047219 Numbers that are congruent to {1, 3} mod 5.

1, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28, 31, 33, 36, 38, 41, 43, 46, 48, 51, 53, 56, 58, 61, 63, 66, 68, 71, 73, 76, 78, 81, 83, 86, 88, 91, 93, 96, 98, 101, 103, 106, 108, 111, 113, 116, 118, 121, 123, 126, 128, 131, 133, 136, 138, 141, 143, 146, 148

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A001844(N) = N^2 + (N+1)^2 is divisible by 5 if and only if N=a(n), n>=1. E.g., n=2: 5|(3^2 + 4^2). But 7^2 + 8^2 is not congruent to 0 (mod 5). - Wolfdieter Lang, May 09 2012

The number of partitions of 5*(n-1) into at most 2 parts. - Colin Barker, Mar 31 2015

The maximum possible number of 6-cycles in an outerplanar graph on n+5 vertices. - Stephen Bartell, Jul 10 2025

Links

• David Lovler, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A000566, A001622, A212160, A212161, A001844.

Bisections give: A016861 (odd part), A016885 (even part).

Programs

Mathematica

Flatten[#+{1,3}&/@(5*Range[0,30])] (* Harvey P. Dale, Dec 22 2013 *)

PARI

a(n)=(5*n-3)\2 \\ Charles R Greathouse IV, Oct 28 2011

Formula

a(n) = floor((5*n-3)/2). - Santi Spadaro, Jul 24 2001, corrected by Gary Detlefs, Oct 28 2011

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

a(n) = 2*n + floor((n-1)/2) - 1. - Arkadiusz Wesolowski, Sep 19 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1/2 + sqrt(5)/10)*Pi/5 + log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021

E.g.f.: 2 + ((10*x - 7)*exp(x) - exp(-x))/4. - David Lovler, Aug 23 2022

a(n) = a(n-2) + 5 for n >= 3. - Rémi Guillaume, Nov 23 2024