A174396 Numbers congruent to {1,4,5,8} mod 9.

1, 4, 5, 8, 10, 13, 14, 17, 19, 22, 23, 26, 28, 31, 32, 35, 37, 40, 41, 44, 46, 49, 50, 53, 55, 58, 59, 62, 64, 67, 68, 71, 73, 76, 77, 80, 82, 85, 86, 89, 91, 94, 95, 98, 100, 103, 104, 107, 109, 112, 113, 116, 118, 121, 122, 125, 127, 130, 131, 134, 136

Offset: 1

Author

Gary Detlefs, Mar 18 2010

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Programs

Magma

[(18*n-9+3*(-1)^n-2*(-1)^((2*n+1-(-1)^n) div 4))/8 : n in [1..100]]; // Wesley Ivan Hurt, Oct 17 2015

Maple

seq(3*(n - floor(n/4)) + (-1)^floor(n/2), n=0..100);

Mathematica

CoefficientList[Series[(1 + 3 x + x^2 + 3 x^3 + x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Oct 17 2015 *)
RecurrenceTable[{a[1] == 1, a[2] == 4, a[3] == 5, a[4] == 8, a[5] == 10 , a[n+5] == a[n+4] + a[n+1] - a[n] }, a, {n, 1, 100}] (* G. C. Greubel, Oct 18 2015 *)

Formula

a(n) = 3*(n-1-floor((n-1)/4)) + (-1)^floor((n-1)/2).

From Wesley Ivan Hurt, Oct 17 2015: (Start)

G.f.: x*(1+3*x+x^2+3*x^3+x^4)/((x-1)^2*(1+x+x^2+x^3)).

a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.

a(n) = (18*n-9+3*(-1)^n-2*(-1)^((2*n+1-(-1)^n)/4))/8. (End)

E.g.f.: (1/8)*(2*sin(x) - 2*cos(x) + 18*x*exp(x) + 3*exp(-x) - 9*exp(x) + 8). - G. C. Greubel, Oct 18 2015

Extensions

Formula corrected by Gary Detlefs, Mar 19 2010