A301451 Numbers congruent to {1, 7} mod 9.

1, 7, 10, 16, 19, 25, 28, 34, 37, 43, 46, 52, 55, 61, 64, 70, 73, 79, 82, 88, 91, 97, 100, 106, 109, 115, 118, 124, 127, 133, 136, 142, 145, 151, 154, 160, 163, 169, 172, 178, 181, 187, 190, 196, 199, 205, 208, 214, 217, 223, 226, 232, 235, 241, 244, 250, 253, 259, 262, 268

Offset: 1

Author

Bruno Berselli, Mar 21 2018

Keywords

• nonn
• easy

Comments

First bisection of A056991, second bisection of A242660.

The squares of the terms of A174396 are the squares of this sequence.

Links

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

Crossrefs

Cf. A274406: numbers congruent to {0, 8} mod 9.

Cf. A193910: numbers congruent to {2, 6} mod 9.

Subsequence of A016777, A026225, A029739, A033627, A047236, A047259, A055047, A055054, A056991, A153053, A187318, A242660.

Programs

GAP

a := [1,7,10];; for n in [4..60] do a[n] := a[n-1] + a[n-2] - a[n-3]; od; a;

Magma

&cat [[9*n+1, 9*n+7]: n in [0..40]];

Mathematica

Table[2 (2 n - 1) + (2 n - 3 (1 - (-1)^n))/4, {n, 1, 60}]
{#+1,#+7}&/@(9*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,7,10},60] (* Harvey P. Dale, Nov 08 2020 *)

PARI

Vec(x*(1 + 6*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 22 2018

Python

[2*(2*n-1)+(2*n-3*(1-(-1)**n))/4 for n in range(1,70)]

Sage

[n for n in (1..300) if n % 9 in (1,7)]

Formula

O.g.f.: x*(1 + 6*x + 2*x^2)/((1 + x)*(1 - x)^2).

E.g.f.: (3 + 8*exp(x) - 11*exp(2*x) + 18*x*exp(2*x))*exp(-x)/4.

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

a(n) = 2*(2*n - 1) + (2*n - 3*(1 - (-1)^n))/4. Therefore, for n even a(n) = (9*n - 4)/2, otherwise a(n) = (9*n - 7)/2.

a(2n+1) = A017173(n). a(2n) = A017245(n-1). - R. J. Mathar, Feb 28 2019