A306277 Numbers congruent to 1 or 8 mod 10.

1, 8, 11, 18, 21, 28, 31, 38, 41, 48, 51, 58, 61, 68, 71, 78, 81, 88, 91, 98, 101, 108, 111, 118, 121, 128, 131, 138, 141, 148, 151, 158, 161, 168, 171, 178, 181, 188, 191, 198, 201, 208, 211, 218, 221, 228, 231, 238, 241, 248, 251, 258, 261, 268, 271, 278, 281, 288, 291, 298, 301, 308, 311, 318, 321

Offset: 1

Author

Davis Smith, Feb 02 2019

Keywords

• nonn
• easy
• base

Comments

A007310(a(n)+1) is always a multiple of 5.

a(1) = 1, a(n+1) = a(n)+7 when n is odd, a(n+1) = a(n)+3 when n is even.

a(n) mod 6 follows the following pattern: 1,2,5,0,3,4,1,2,5,0,3,4, and so on.

A020639(A007310(a(n)+1)) = 5.

Links

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

Crossrefs

Cf. A001622, A020639, A007310, A000034.

Cf. A017281, A017365 (bisections).

One less than A273669.

Programs

Maple

seq(seq(10*i+j, j=[1, 8]), i=0..350);

Mathematica

Select[Range[350], MemberQ[{1, 8}, Mod[#, 10]] &]

PARI

for(n=1, 350, if((n%10==1) || (n%10==8), print1(n, ", ")))

PARI

Vec(x*(1 + 7*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 09 2019

Formula

a(n) = 5*n - 2*A000034(n+1).

a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.

a(n) = A273669(n) - 1. - Antti Karttunen, Feb 07 2019

G.f.: x*(1 + 7*x + 2*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Feb 09 2019

E.g.f.: 2 + (5*x - 3)*exp(x) + exp(-x). - David Lovler, Sep 07 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = (5+sqrt(5))^(3/2)*phi*Pi/(100*sqrt(2)) + log(phi)/(2*sqrt(5)) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023