A271508 Numbers that are congruent to {1,4} mod 10.

1, 4, 11, 14, 21, 24, 31, 34, 41, 44, 51, 54, 61, 64, 71, 74, 81, 84, 91, 94, 101, 104, 111, 114, 121, 124, 131, 134, 141, 144, 151, 154, 161, 164, 171, 174, 181, 184, 191, 194, 201, 204, 211, 214, 221, 224, 231, 234, 241, 244, 251, 254, 261, 264, 271, 274

Offset: 1

Author

Wesley Ivan Hurt, Apr 08 2016

Keywords

• nonn
• easy

Comments

Numbers ending in 1 or 4, Union of A017281 and A017317.

a(n+3) gives the sum of 5 consecutive terms of A004442 starting at A004442(n) for n>0. (i.e., a(4) = 14 = 0+3+2+5+4 = Sum_{i=0..4} A004442(n+i)).

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001622, A004442, A017281, A017317, A042948, A047241.

Programs

Magma

[5*n-5-(-1)^n : n in [1..100]];

Maple

A271508:=n->5*n-5-(-1)^n: seq(A271508(n), n=1..100);

Mathematica

Table[5 n - 5 - (-1)^n, {n, 60}] (* or *)
Select[Range[0, 300], MemberQ[{1, 4}, Mod[#, 10]] &]

PARI

my(x='x+O('x^99)); Vec(x*(1+3*x+6*x^2)/((-1+x)^2*(1+x))) \\ Altug Alkan, Apr 09 2016

Formula

G.f.: x*(1+3*x+6*x^2)/((-1+x)^2*(1+x)).

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

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

a(n) = -n + 2*A047241(n).

a(n+1) = n + 1 + 2*A042948(n).

Shifted bisections: a(2n+2) = A017317(n), a(2n+1) = A017281(n).

E.g.f.: 5*(x-1)*exp(x) - exp(-x). - G. C. Greubel, Apr 08 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1+2/sqrt(5))*Pi/10 + log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023