A195014 Vertex number of a square spiral whose edges have length A195013.

0, 2, 5, 9, 15, 21, 30, 38, 50, 60, 75, 87, 105, 119, 140, 156, 180, 198, 225, 245, 275, 297, 330, 354, 390, 416, 455, 483, 525, 555, 600, 632, 680, 714, 765, 801, 855, 893, 950, 990, 1050, 1092, 1155, 1199, 1265, 1311, 1380, 1428, 1500, 1550, 1625, 1677

Offset: 0

Author

Omar E. Pol, Sep 09 2011

Keywords

• nonn
• easy

Comments

Zero together with the partial partial sums of A195013.

Second bisection is 2, 9, 21, 38, 60, 87, 119, ...: A005476. - Omar E. Pol, Sep 25 2011

Number of pairs (x,y) with even x in {0,...,n}, odd y in {0,...,3n}, and xClark Kimberling, Jul 02 2012

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A005476, A028895, A195013, A195020.

Programs

Magma

[(10*n^2 + 18*n + 3 + (2*n - 3)*(-1)^n)/16 : n in [0..50]]; // Vincenzo Librandi, Oct 26 2014

Mathematica

LinearRecurrence[{1,2,-2,-1,1},{0,2,5,9,15},60] (* Harvey P. Dale, May 20 2019 *)

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

G.f.: f(x)/g(x), where f(x) = 2*x + 3*x^2 and g(x) = (1+x)^2 * (1-x)^3. - Clark Kimberling, Jul 02 2012

a(n) = (10*n^2 + 18*n + 3 + (2*n - 3)*(-1)^n)/16. - Luce ETIENNE, Aug 11 2014