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

1,2

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).

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

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 *)

Prog

(GAP) a := [1, 7, 10];; for n in [4..60] do a[n] := a[n-1] + a[n-2] - a[n-3]; od; a;
(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)]
(Magma) &cat [[9*n+1, 9*n+7]: n in [0..40]];
(PARI) Vec(x*(1 + 6*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 22 2018

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.

Sequence in context: A083390 A234093 A287567 * A033817 A286873 A218128

Adjacent sequences: A301448 A301449 A301450 * A301452 A301453 A301454

Keyword

nonn,easy

Author

Bruno Berselli, Mar 21 2018

Status

approved