A123596 Squares alternating with triangular numbers.

0, 0, 1, 1, 4, 3, 9, 6, 16, 10, 25, 15, 36, 21, 49, 28, 64, 36, 81, 45, 100, 55, 121, 66, 144, 78, 169, 91, 196, 105, 225, 120, 256, 136, 289, 153, 324, 171, 361, 190, 400, 210, 441, 231, 484, 253, 529, 276, 576, 300, 625, 325, 676, 351, 729, 378, 784, 406, 841, 435

Offset

0,5

Comments

Rearrangement of A054686.

For n >= 2, a(n) is the number of distinct positive slopes of least squares regression lines fitted to n points (j,y_j), 1 <= j <= n, where all y_j are 0 or 1. The total number of distinct slopes is 2*a(n)+1 (a(n) positive, a(n) negative, and the zero slope). - Pontus von Brömssen, Mar 10 2024

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

a(2*n) = n^2, a(2*n+1) = (n^2+n)/2.

From R. J. Mathar, Feb 12 2010: (Start)

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

G.f.: x^2*(1+x+x^2)/((1-x)^3*(1+x)^3). (End)

a(n) = (3*n^2-1+(n^2+1)*(-1)^n)/16. - Luce ETIENNE, May 30 2015

Mathematica

CoefficientList[Series[x^2*(1+x+x^2)/((1-x)^3*(1+x)^3), {x, 0, 50}], x] (* or *) Table[(3*n^2-1+(n^2+1)*(-1)^n)/16, {n, 0, 50}] (* G. C. Greubel, Oct 26 2017 *)
With[{nn=30}, Riffle[Range[0, nn]^2, Accumulate[Range[0, nn]]]] (* or *) LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 0, 1, 1, 4, 3}, 60] (* Harvey P. Dale, Feb 11 2020 *)

Prog

(PARI) {a(n) = if(n%2, (n^2-1)/8, n^2/4)} \\ Michael Somos, Nov 18 2006
(Magma) [(3*n^2-1+(n^2+1)*(-1)^n)/16: n in [0..10]]; // G. C. Greubel, Oct 26 2017

Crossrefs

Cf. A002620, A054686, A124093.

Sequence in context: A103825 A073238 A010655 * A309460 A200361 A222471

Adjacent sequences: A123593 A123594 A123595 * A123597 A123598 A123599

Keyword

nonn

Author

Peter Hansen (babyskbaby(AT)web.de), Nov 14 2006

Extensions

Edited by Michael Somos, and several other correspondents, Nov 14 2005

Status

approved