A253195 Numbers congruent to 5 or 8 mod 9.

5, 8, 14, 17, 23, 26, 32, 35, 41, 44, 50, 53, 59, 62, 68, 71, 77, 80, 86, 89, 95, 98, 104, 107, 113, 116, 122, 125, 131, 134, 140, 143, 149, 152, 158, 161, 167, 170, 176, 179, 185, 188, 194, 197, 203, 206, 212, 215, 221, 224, 230, 233, 239, 242, 248, 251

Offset: 1

Author

Arkadiusz Wesolowski, Mar 24 2015

Keywords

• nonn
• easy

Comments

These numbers cannot be written as the sum of two triangular numbers.

Links

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

Crossrefs

Subsequence of A014132 and of A020757.

Programs

Magma

[n: n in [0..251] | n mod 9 in {5, 8}];

Mathematica

LinearRecurrence[{1, 1, -1}, {5, 8, 14}, 56]
Select[Range[300],MemberQ[{5,8},Mod[#,9]]&] (* Harvey P. Dale, Mar 17 2020 *)

PARI

Vec(x*(5 + 3*x + x^2)/((1 + x)*(1 - x)^2) + O(x^80)) \\ Michel Marcus, Mar 25 2015

Formula

a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4.

G.f.: x*(5 + 3*x + x^2)/((1 + x)*(1 - x)^2).

a(n) = a(n-2) + 9.

a(n) = 9*n - a(n-1) - 5.

a(n) = 4*n + 2*ceiling(n/2) - floor(n/2) - 1.

a(n) = (9*n - (3/2)*(1 + (- 1)^n) + 1)/2.

E.g.f.: 1 + ((18*x - 1)*exp(x) - 3*exp(-x))/4. - David Lovler, Sep 06 2022