A332410 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

0, 1, 3, 6, 11, 17, 24, 32, 41, 52, 64, 77, 91, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 355, 385, 416, 448, 481, 516, 552, 589, 627, 666, 707, 749, 792, 836, 881, 928, 976, 1025, 1075, 1126, 1179

Offset: 0

Author

Paul Curtz, Feb 11 2020

Keywords

• nonn
• easy

Comments

This sequence occurs twice as a linear spoke in the hexagonal spiral constructed from A002266:

17 17 17 17 17 18 18

16 11 11 11 11 12 12 18

16 11 6 6 7 7 7 12 18

16 10 6 3 3 3 3 7 12 18

16 10 6 3 1 1 1 4 7 12 19

16 10 6 2 0 0 0 1 4 8 13 19

15 10 5 2 0 0 1 4 8 13 19

15 10 5 2 2 2 4 8 13 19

15 9 5 5 5 4 8 13 19

15 9 9 9 9 8 13 20

15 14 14 14 14 14 20

a(-1-n) = 0, 1, 4, 8, 13, 19, 26, 35, 45, ... also occurs twice in the same spiral.

Difference table:

0, 1, 3, 6, 11, 17, 24, 32, 41, 52, ... = a(n)

1, 2, 3, 5, 6, 7, 8, 9, 11, 12, ... = A047256(n+1)

1, 1, 2, 1, 1, 1, 1, 2, 1, 1, ... = A130782.

There is no linear spoke with three copies in this spiral. Compare with the spiral illustrated in sequence A330707 and constructed from A002265 where the same spokes occur three times: A006578, A001859 and A077043, essentially. Strictly, three times from 1, 1, 1 for A006578, from 2, 2, 2 for A001859 and from 7, 7, 7 for A077043.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Crossrefs

Cf. A002266, A008602, A047256, A130782, A330707.

Cf. A001859, A006578, A077043.

Programs

Mathematica

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {0, 1, 3, 6, 11, 17, 24}, 45] (* Amiram Eldar, Feb 12 2020 *)

PARI

concat(0, Vec(x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^50))) \\ Colin Barker, Feb 11 2020, Apr 24 2020

Formula

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

G.f.: x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Feb 11 2020