A305189 a(n) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 + ... + (up to n).

1, 2, 5, 9, 25, 31, 38, 87, 96, 106, 206, 218, 231, 400, 415, 431, 687, 705, 724, 1085, 1106, 1128, 1612, 1636, 1661, 2286, 2313, 2341, 3125, 3155, 3186, 4147, 4180, 4214, 5370, 5406, 5443, 6812, 6851, 6891, 8491, 8533, 8576, 10425, 10470, 10516, 12632

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Formula

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

From Colin Barker, Sep 16 2018: (Start)

G.f.: x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3).

a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.

(End)

Example

a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 + 3 = 5;
a(4) = 1*2 + 3 + 4 = 9;
a(5) = 1*2 + 3 + 4*5 = 25;
a(6) = 1*2 + 3 + 4*5 + 6 = 31;
a(7) = 1*2 + 3 + 4*5 + 6 + 7 = 38;
a(8) = 1*2 + 3 + 4*5 + 6 + 7*8 = 87;
a(9) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 = 96;
a(10) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10 = 106;
a(11) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;
a(12) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 = 218; etc.

Maple

seq(coeff(series((x*(1+x+3*x^2+x^3+13*x^4-3*x^5-2*x^6+4*x^7))/((1-x)^4*(1+x+x^2)^3), x, n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Sep 16 2018

Mathematica

Table[3*Floor[n/3]*(Floor[n/3] + 1)/2 + Floor[(n + 1)/3]*(3*Floor[(n + 1)/3]^2 - 1) + n*(Floor[(n - 1)/3] - Floor[(n - 2)/3]), {n, 50}]
LinearRecurrence[{1, 0, 3, -3, 0, -3, 3, 0, 1, -1 }, {1, 2, 5, 9, 25, 31, 38, 87, 96, 106}, 50] (* Stefano Spezia, Sep 16 2018 *)

Prog

(PARI) Vec(x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018

Crossrefs

Cf. A093361, A228958, A319014.

Sequence in context: A320798 A364267 A006405 * A344394 A243559 A334077

Adjacent sequences: A305186 A305187 A305188 * A305190 A305191 A305192

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Sep 15 2018

Status

approved