The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A308064 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A308064

Number of triangles with perimeter n whose side lengths are square numbers.
(history; published version)

#11 by Wesley Ivan Hurt at Tue May 12 21:51:45 EDT 2020

STATUS

editing

approved

#10 by Wesley Ivan Hurt at Tue May 12 21:50:17 EDT 2020

FORMULA

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * A010052c(i) * A010052c(k) * c(n-i-k), where c(n) is the characteristic function of squares (A010052(n-i-k).

STATUS

approved

editing

#9 by Peter Luschny at Wed Jan 01 16:58:34 EST 2020

STATUS

proposed

approved

#8 by Robert Israel at Wed Jan 01 16:45:48 EST 2020

STATUS

editing

proposed

#7 by Robert Israel at Wed Jan 01 16:44:00 EST 2020

LINKS

Robert Israel, <a href="/A308064/b308064.txt">Table of n, a(n) for n = 1..10000</a>

MAPLE

N:= 100:

V:= Vector(N):

for a from 1 to floor(sqrt(N/3)) do

for b from a to floor(sqrt((N-a^2)/2)) do

R:= map(c -> a^2 + b^2 + c^2, [$b .. floor(sqrt(min(a^2+b^2-1, N-a^2-b^2)))]);

V[R]:= map(`+`, V[R], 1);

od od:

convert(V, list); # Robert Israel, Jan 01 2020

STATUS

approved

editing

#6 by Susanna Cuyler at Sat May 11 10:12:42 EDT 2019

STATUS

proposed

approved

#5 by Wesley Ivan Hurt at Sat May 11 05:06:58 EDT 2019

STATUS

editing

proposed

#4 by Wesley Ivan Hurt at Fri May 10 23:19:03 EDT 2019

NAME

Number of triangles with perimeter n whose sides side lengths are square numbers.

#3 by Wesley Ivan Hurt at Fri May 10 22:28:31 EDT 2019

NAME

allocated for Wesley Ivan HurtNumber of triangles with perimeter n whose sides are square numbers.

DATA

0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0

OFFSET

1,99

FORMULA

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * A010052(i) * A010052(k) * A010052(n-i-k).

MATHEMATICA

Table[Sum[Sum[(Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[k]] - Floor[Sqrt[k - 1]]) (Floor[Sqrt[n - k - i]] - Floor[Sqrt[n - k - i - 1]])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

CROSSREFS

Cf. A010052.

KEYWORD

allocated

nonn

AUTHOR

Wesley Ivan Hurt, May 10 2019

STATUS

approved

editing

#2 by Wesley Ivan Hurt at Fri May 10 22:28:31 EDT 2019

NAME

allocated for Wesley Ivan Hurt

KEYWORD

recycled

allocated