A137930 The sum of the principal diagonals of an n X n spiral.

0, 1, 10, 25, 56, 101, 170, 261, 384, 537, 730, 961, 1240, 1565, 1946, 2381, 2880, 3441, 4074, 4777, 5560, 6421, 7370, 8405, 9536, 10761, 12090, 13521, 15064, 16717, 18490, 20381, 22400, 24545, 26826, 29241, 31800, 34501, 37354, 40357, 43520, 46841, 50330

Offset

0,3

Comments

n X n spirals of the form: (Examples of n = 3, 4)

7...8...9

6...1...2

5...4...3

and

7...8...9...10

6...1...2...11

5...4...3...12

16..15..14..13

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = mod(n^(n+1),n+1) + floor(n/2)^2*(6-4(-1)^n) + [16*floor(n/2)^3 + floor(n/2)*(14-12(-1)^n)]/3

Interweave A114254 and A137931.

Empirical G.f.: x*(1+7*x-3*x^2+3*x^3)/((1-x)^4*(1+x)). - Colin Barker, Jan 12 2012

From Robert Israel, Jun 25 2019: (Start)

Empirical G.f. confirmed using G.f.'s of A114254 and A137931.

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

E.g.f.: (x*(15 + 15*x + 4*x^2)*cosh(x) - (9 - 15*x - 15*x^2 - 4*x^3)*sinh(x))/6. - Stefano Spezia, May 17 2026

Example

a(1) = mod(1^(1+1),1+1) + floor(1/2)^2*(6-4(-1)^1) + [16*floor(1/2)^3 + floor(1/2)*(14-12(-1)^1)]/3 = 1
a(2) = mod(2^(2+1),2+1) + floor(2/2)^2*(6-4(-1)^2) + [16*floor(2/2)^3 + floor(2/2)*(14-12(-1)^2)]/3 = 10

Maple

f:= n -> 2*n^3/3 + n^2/2 + 4*n/3 + 3*((-1)^n -1)/4:
map(f, [$0..100]); # Robert Israel, Jun 25 2019

Mathematica

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 10, 25, 56}, 50] (* Paolo Xausa, May 16 2026 *)

Prog

(PARI) a(n) = (4*n^3 + 3*n^2 + 8*n - n%2*9)/6 \\ Charles R Greathouse IV, May 16 2026

Crossrefs

Cf. A114254, A137931.

Sequence in context: A283243 A263310 A063424 * A251310 A251194 A071289

Adjacent sequences: A137927 A137928 A137929 * A137931 A137932 A137933

Keyword

nonn,easy

Author

William A. Tedeschi, Feb 29 2008

Status

approved