A319390 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=2, a(2)=3, a(3)=6, a(4)=8.

1, 2, 3, 6, 8, 13, 16, 23, 27, 36, 41, 52, 58, 71, 78, 93, 101, 118, 127, 146, 156, 177, 188, 211, 223, 248, 261, 288, 302, 331, 346, 377, 393, 426, 443, 478, 496, 533, 552, 591, 611, 652, 673, 716, 738, 783, 806, 853, 877, 926, 951, 1002

Offset: 0

Author

Paul Curtz, Sep 18 2018

Keywords

• nonn
• easy

Comments

The bisections A104249(n) = 1, 3, 8, ... and A143689(n+1) = 2, 6, 13, 23, ... are in the following hexagonal spiral:

29--28--28--27--27

/ \

29 17--17--16--16 26

/ / \ \

30 18 9---8---8 15 26

/ / / \ \ \

30 18 9 3---3 7 15 25

/ / / / \ \ \ \

31 19 10 4 1 2 7 14 25

/ / / / / / / /

19 10 4 1---2 6 14 24

\ \ \ / / /

20 11 5---5---6 13 24

\ \ / /

20 11--12--12--13 23

\ /

21--21--22--22--23

.

a(n) mod 9 = A140265(n) mod 9.

Links

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

Crossrefs

Cf. A001318, A104249, A143689, A004526, A140265, A026741.

Programs

Mathematica

LinearRecurrence[{1,2,-2,-1,1},{1,2,3,6,8},100] (* Paolo Xausa, Nov 13 2023 *)

PARI

Vec((1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jun 05 2019

Formula

a(2n) = (3*n^2 + n + 2)/2. a(2n+1) = (3*n^2 + 5*n + 4)/2.

a(-n) = a(n).

a(n) = a(n-1) + A026741(n).

G.f.: (1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jun 05 2019

a(n) = 1 + A001318(n). - Peter Bala, Feb 04 2021

E.g.f.: ((8 + 7*x + 3*x^2)*cosh(x) + (9 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Feb 05 2021