A325958 Sum of the corners of a 2n+1 X 2n+1 square spiral.

24, 76, 160, 276, 424, 604, 816, 1060, 1336, 1644, 1984, 2356, 2760, 3196, 3664, 4164, 4696, 5260, 5856, 6484, 7144, 7836, 8560, 9316, 10104, 10924, 11776, 12660, 13576, 14524, 15504, 16516, 17560, 18636, 19744, 20884, 22056, 23260, 24496, 25764, 27064, 28396

Offset

1,1

Comments

The 3 X 3 and 5 X 5 spirals are

.

7---8---9

|

6 1---2

| |

5---4---3

.

with corners 7 + 9 + 5 + 3 = 24

and

.

21--22--23--24--25

|

20 7---8---9--10

| | |

19 6 1---2 11

| | | |

18 5---4---3 12

| |

17--16--15--14--13

.

with corners 21 + 25 + 17 + 13 = 76.

An issue arises when considering a 1 X 1 spiral. For ease, a 1 X 1 spiral happens to have no corners so the corresponding value might be considered as undefined (namely, undefined for n = 0).

However, from a theoretical perspective if n is allowed to be 0, meaning that a 1 X 1 spiral can have corners, the formulas below that include A114254 might need reconsideration. With the current formula a(n) = 16*n^2 + 4*n + 4, a(0) = 4, meaning that a 1 X 1 spiral (with value 1) has 4 corners with value 1, giving sum 4. This might pave the way to a discussion, considered parallel with A114254. With the given equations, a 1 X 1 spiral happens to have a corner sum of 4. However, a 1 X 1 spiral has a diagonal sum of 1, from A114254. This seems as to be a contradiction; namely, the first term of A114254 should at least be 4 in this case, as corners constitute a subset of diagonal elements.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021).

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

Formula

a(n) = A114254(n) - A114254(n-1).

a(n) = 16*n^2 + 4*n + 4.

From Colin Barker, Sep 10 2019: (Start)

G.f.: 4*x*(6 + x + x^2) / (1 - x)^3.

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

(End)

E.g.f.: -4 + 4*exp(x)*(1 + 5*x + 4*x^2). - Stefano Spezia, Sep 11 2019

Example

For n=1 (our first value) namely for a 3 X 3 spiral, we get a(1) = 24.
For n=2, for a 5 X 5 spiral, we get a(2) = 76.

Mathematica

Table[ 16n^2+4n+4, {n, 42}] (* Metin Sariyar, Sep 14 2019 *)

Prog

(PARI) a(n) = 16*n^2 + 4*n + 4;
(PARI) Vec(4*x*(6 + x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Sep 10 2019

Crossrefs

Cf. A114254.

Sequence in context: A233883 A291630 A195027 * A211574 A211588 A211596

Adjacent sequences: A325955 A325956 A325957 * A325959 A325960 A325961

Keyword

nonn,easy

Author

Yigit Oktar, Sep 10 2019

Status

approved