A276819 a(n) = (9*n^2 - n)/2 + 1.

1, 5, 18, 40, 71, 111, 160, 218, 285, 361, 446, 540, 643, 755, 876, 1006, 1145, 1293, 1450, 1616, 1791, 1975, 2168, 2370, 2581, 2801, 3030, 3268, 3515, 3771, 4036, 4310, 4593, 4885, 5186, 5496, 5815, 6143, 6480, 6826, 7181, 7545, 7918, 8300, 8691, 9091, 9500, 9918, 10345, 10781, 11226, 11680, 12143, 12615

Offset: 0

Author

Yuriy Sibirmovsky, Sep 18 2016

Keywords

• nonn
• easy

Comments

Diagonal of triangular spiral in A051682. The other 5 diagonals are given by A140064, A117625, A081267, A064225, A006137. See the link as well.

First differences are given by A017209.

72*a(n) - 71 is a perfect square. - Klaus Purath, Jan 14 2022

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Yuriy Sibirmovsky, Six diagonals of the triangular spiral.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002378, A003215, A006137, A017209, A051682, A064225, A081267, A117625, A140064.

Programs

Mathematica

Table[(9*n^2-n)/2+1, {n,0,100}]

PARI

Vec((1+2*x+6*x^2)/(1-x)^3 + O(x^60)) \\ Colin Barker, Sep 18 2016

PARI

a(n) = (9*n^2 - n)/2 + 1; \\ Altug Alkan, Sep 18 2016

Formula

a(n) = (9*n^2 - n)/2 + 1.

a(n) = a(n-1) + 9*n - 5 with a(0) = 1.

From Colin Barker, Sep 18 2016: (Start)

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

G.f.: (1 + 2*x + 6*x^2)/(1 - x)^3. (End)

From Klaus Purath, Jan 14 2022: (Start)

a(n) = A006137(n) - n.

A003215(a(n)) - A003215(a(n)-3) = A002378(9*n-1). (End)

E.g.f.: exp(x)*(2 + 8*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022