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Zero together with the partial sums of A195019.

The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives A008587. The vertices on the main diagonal are the numbers A024966 = (3+4)*A000217 = 7*A000217, where both 3 and 4 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 3, while the distance "b" between nearest edges that are parallel to the initial edge is 4, so the distance "c" between nearest vertices on the same axis is 5 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(3^2+4^2) = sqrt(9+16) = sqrt(25) = 5.

Let an array have m(0,n)=m(n,0)=n*(n-1)/2 and m(n,n)=n*(n+1)/2. The first n+1 terms in row(n) are the numbers in the closed interval m(0,n) to m(n,n). The terms in column(n) are the same from m(n,0) to m(n,n). The first few antidiagonals are 0; 0,0; 1,1,1; 3,2,2,3; 6,4,3,4,6; 10,7,5,5,7,10. a(n) is the difference between the sum of the terms in the n+1 X n+1 matrices and those in the n X n matrices. - J. M. Bergot, Jul 05 2013 [The first five rows are: 0,0,1,3,6; 0,1,2,4,7; 1,2,3,5,8; 3,4,5,6,9; 6,7,8,9,10]

Zero together with partial sums of A195033.

The only primes in the sequence are 41 and 83 since a(n) = (1/2)*((2*n+(-1)^n+3)/4)*((82*n-43*(-1)^n+43)/4). - Bruno Berselli, Oct 12 2011

The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 29 (Cf. A195819). The vertices on the main diagonal are the numbers A195038 = (21+20)*A000217 = 41*A000217, where both 21 and 20 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 21, while the distance "b" between nearest edges that are parallel to the initial edge is 20, so the distance "c" between nearest vertices on the same axis is 29 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(21^2+20^2) = sqrt(441+400) = sqrt(841) = 29. - Omar E. Pol, Oct 12 2011

Zero together with partial sums of A195031.

The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives multiples of 13 (cf. A008595). The vertices on the main diagonal are the numbers A195037 = (5+12)*A000217 = 17*A000217, where both 5 and 12 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 5, while the distance "b" between nearest edges that are parallel to the initial edge is 12, so the distance "c" between nearest vertices on the same axis is 13 because from the Pythagorean theorem we can write c = (a^2 + b^2)^(1/2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. - Omar E. Pol, Oct 12 2011

First differences of A195036.

a(n) is also the length of the n-th edge of a square spiral in which the first two edges are the legs of the primitive Pythagorean triple [15, 8, 17]. Zero together with partial sums give A195036; the vertices of the spiral.

Related to the primitive Pythagorean triple [15, 8, 17].

Sequence found by reading the line from 0, in the direction 0, 23, ..., and the same line from 0, in the direction 0, 69, ..., in the Pythagorean spiral whose edges have length A195035 and whose vertices are the numbers A195036. This is the main diagonal of the square spiral.