A264938 a(n) = n*(2*n-1) + floor(n/3).

0, 1, 6, 16, 29, 46, 68, 93, 122, 156, 193, 234, 280, 329, 382, 440, 501, 566, 636, 709, 786, 868, 953, 1042, 1136, 1233, 1334, 1440, 1549, 1662, 1780, 1901, 2026, 2156, 2289, 2426, 2568, 2713, 2862, 3016, 3173, 3334, 3500, 3669, 3842, 4020, 4201, 4386, 4576, 4769

Offset

0,3

Comments

Sequence extended to the left:

..., 133, 102, 76, 53, 34, 20, 9, 2, 0, 1, 6, 16, 29, 46, 68, 93, ...

Conjecture: after 0, a(n) provides the first bisection of A264041.

Conjecture: 2, 9, 20, 34, 53, 76, 102, 133, ... is A248121.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

a(n) = a(n-3) + 12*n - 20 for n>2.

From Colin Barker, Dec 02 2015: (Start)

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.

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

(End)

a(n) = A000217(2n-1) + A002264(n).

a(n) + a(-n) = 3*A256320(n).

a(n +8) - a(n -7) = 20*A016777(n).

a(n+16) - a(n-14) = 20*A016969(n).

a(n+23) - a(n-22) = 20*A017197(n).

a(n+31) - a(n-29) = 20*A017641(n).

Generalization of the previous four formulas:

a(n+30*k +8) - a(n-30*k -7) = 20*(4*k+1)*(3*n+1).

a(n+30*k+16) - a(n-30*k-14) = 20*(2*k+1)*(6*n+5).

a(n+30*k+24) - a(n-30*k-21) = 20*(4*k+3)*(3*n+4).

a(n+30*k+32) - a(n-30*k-28) = 20*(2*k+2)*(6*n+11).

E.g.f.: (6*x^2+4*x-1)*exp(x)/3 + (cos(sqrt(3)*x/2)/3 +sqrt(3)*sin(sqrt(3)*x/2)/9)*exp(-x/2). - Robert Israel, Dec 02 2015

Maple

seq(n*(2*n-1) + floor(n/3), n=0..100); # Robert Israel, Dec 02 2015

Mathematica

Table[n (2 n - 1) + Floor[n/3], {n, 0, 50}] (* Vincenzo Librandi, Dec 02 2015 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 6, 16, 29}, 60] (* Harvey P. Dale, Oct 13 2020 *)

Prog

(PARI) concat(0, Vec(x*(1+x)^2*(1+2*x)/((1-x)^3*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Dec 02 2015
(PARI) a(n) = n*(2*n-1) + n\3; \\ Altug Alkan, Dec 01 2015
(Magma) [n*(2*n-1)+Floor(n/3): n in [0..60]]; // Vincenzo Librandi, Dec 02 2015

Crossrefs

Second bisection of A260708.

Cf. A000217, A002264, A016777, A016969, A017197, A017641, A099837, A248121, A256320, A260699, A264041.

Cf. A171452: A000217(n-1) + A002264(n).

Sequence in context: A346620 A032422 A164052 * A350011 A168472 A054000

Adjacent sequences: A264935 A264936 A264937 * A264939 A264940 A264941

Keyword

nonn,easy

Author

Paul Curtz, Nov 29 2015

Extensions

Edited by Bruno Berselli, Dec 01 2015

Status

approved