A195019 Multiples of 3 and of 4 interleaved: a(2*n-1) = 3*n, a(2*n) = 4*n.

3, 4, 6, 8, 9, 12, 12, 16, 15, 20, 18, 24, 21, 28, 24, 32, 27, 36, 30, 40, 33, 44, 36, 48, 39, 52, 42, 56, 45, 60, 48, 64, 51, 68, 54, 72, 57, 76, 60, 80, 63, 84, 66, 88, 69, 92, 72, 96, 75, 100, 78, 104, 81, 108, 84, 112, 87, 116, 90, 120, 93, 124, 96, 128

Offset: 1

Author

Omar E. Pol, Sep 07 2011, Sep 12 2011

Keywords

• nonn
• easy

Comments

First differences of A195020.

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 [3, 4, 5]. The spiral contains infinitely many Pythagorean triples in which the hypotenuses are the positives A008587. Zero together with partial sums give A195020; the vertices of the spiral.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean triangles and triples
• Eric Weisstein's World of Mathematics, Pythagorean Triple
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A008585, A008586, A026741, A195020-A195029, A195031, A195033, A195035.

Programs

Magma

[((n-3)*(-1)^n+7*n+3)/4: n in [1..60]]; // Vincenzo Librandi, Sep 12 2011

Mathematica

Table[((n-3)*(-1)^n + 7*n + 3)/4, {n,1,50}] (* G. C. Greubel, Aug 19 2017 *)

PARI

a(n)=(n+1)\2*(4-n%2)  \\ M. F. Hasler, Sep 08 2011

Formula

pair(3*n, 4*n).

a(2*n-1) = 3*n, a(2*n) = 4*n. - M. F. Hasler, Sep 08 2011

G.f.: x*(3+4*x) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Sep 09 2011

From Bruno Berselli, Sep 12 2011: (Start)

a(n) = ((n-3)*(-1)^n + 7*n + 3)/4.

a(n) + a(n+1) = A047355(n+2). (End)

E.g.f.: (1/4)*((3 + 7*x)*exp(x) - (3 + x)*exp(-x)). - G. C. Greubel, Aug 19 2017