A306331 Numbers congruent to 6 or 31 mod 38.

6, 31, 44, 69, 82, 107, 120, 145, 158, 183, 196, 221, 234, 259, 272, 297, 310, 335, 348, 373, 386, 411, 424, 449, 462, 487, 500, 525, 538, 563, 576, 601, 614, 639, 652, 677, 690, 715, 728, 753, 766, 791, 804, 829, 842, 867, 880, 905

Offset: 1

Author

Davis Smith, Feb 07 2019

Keywords

• nonn
• easy

Comments

A007310(a(n) + 1) is always a multiple of 19.

A020639(A007310(a(n) + 1)) = 5, 7, 11, 13, 17, or 19.

It equals 5 when n is a term in A273669.

It equals 7 when n is congruent to 3 or 12 (mod 14) but not a term in A273669.

It equals 11 when n is congruent to 4 or 19 (mod 22) but not a case where it equals 5 or 7.

It equals 13 when n is congruent to 5 or 22 (mod 26) (one more than a term in A306285) but not a case where it equals 5, 7, or 11.

It equals 17 when n is congruent to 6 or 29 (mod 34) but not a case where it equals 5, 7, 11, or 13.

For all other cases, it equals 19.

a(n) and (n - 1) have the same remainder (mod 6) (see A010875).

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. A005408, A005843, A007310, A010875, A020639, A040031, A273669, A306285.

Programs

Maple

seq(seq(38*i+j, j=[6, 31]), i=0..200);

Mathematica

Select[Range[200], MemberQ[{6, 31}, Mod[#, 38]] &]
Union[38Range[30] - 32, 38Range[30] - 7] (* Alonso del Arte, Feb 08 2019 *)

PARI

for(n=6, 905, if((n%38==6) || (n%38==31), print1(n, ", ")))

PARI

Vec(x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 09 2019

Scala

(6 to 1108 by 38).union(31 to 1133 by 38).sorted // Alonso del Arte, Feb 08 2019

Formula

G.f.: x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Feb 09 2019

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

a(n) = 19*n - 10 + 3*(-1)^n. - Wesley Ivan Hurt, Mar 10 2019

a(n) = 19*n - 13 when n is odd and 19*n - 7 when n is even.

a(n) = 19*n - (A040031(n + 1) + 1).

E.g.f.: 7 + (19*x - 10)*exp(x) + 3*exp(-x). - David Lovler, Sep 10 2022