A306285 Numbers congruent to 4 or 21 mod 26.

4, 21, 30, 47, 56, 73, 82, 99, 108, 125, 134, 151, 160, 177, 186, 203, 212, 229, 238, 255, 264, 281, 290, 307, 316, 333, 342, 359, 368, 385, 394, 411, 420, 437, 446, 463, 472, 489, 498, 515, 524, 541, 550, 567, 576, 593, 602, 619, 628, 645, 654, 671, 680, 697, 706, 723, 732, 749, 758, 775, 784, 801, 810, 827, 836, 853, 862

Offset: 1

Author

Davis Smith, Feb 03 2019

Keywords

• nonn
• easy

Comments

A007310(a(n)+1) is always a multiple of 13.

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

a(n) mod 4 = A010873(n).

A020639(A007310(a(n)+1)) = 5 when n is congruent to 2 or 9 (mod 10) (n is a term in A273669). It equals 7 when n is congruent to 3 or 12 (mod 14) but not congruent to 2 or 9 (mod 10). It equals 11 when n is congruent to 4 or 19 (mod 22) but not congruent to 2 or 9 (mod 10) and not congruent to 3 or 12 (mod 14). Otherwise, it is 13.

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. A007310, A010720, A020639, A010873, A273669.

Programs

Maple

seq(seq(26*i+j, j=[4, 21]), i=0..200);

Mathematica

Select[Range[200], MemberQ[{4, 21}, Mod[#, 26]] &]

PARI

for(n=1, 1000, if((n%26==4) || (n%26==21), print1(n, ", ")))

PARI

Vec(x*(4 + 17*x + 5*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 08 2019

Formula

a(n) = 13*n - A010720(n+1).

From Colin Barker, Feb 08 2019: (Start)

G.f.: x*(4 + 17*x + 5*x^2) / ((1 - x)^2*(1 + x)).

a(n) = 13*n - 5 for n even.

a(n) = 13*n - 9 for n odd.

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

E.g.f.: 5 + (13*x - 7)*exp(x) + 2*exp(-x). - David Lovler, Sep 09 2022