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A325958 Sum of the corners of a 2n+1 X 2n+1 square spiral. 2
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 (list; graph; refs; listen; history; text; internal format)
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
Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021).
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
KEYWORD
nonn,easy
AUTHOR
Yigit Oktar, Sep 10 2019
STATUS
approved

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Last modified May 18 16:47 EDT 2024. Contains 372664 sequences. (Running on oeis4.)