A147685 Squares and centered square numbers interleaved.

0, 1, 1, 5, 4, 13, 9, 25, 16, 41, 25, 61, 36, 85, 49, 113, 64, 145, 81, 181, 100, 221, 121, 265, 144, 313, 169, 365, 196, 421, 225, 481, 256, 545, 289, 613, 324, 685, 361, 761, 400, 841, 441, 925, 484, 1013, 529, 1105, 576, 1201, 625, 1301, 676, 1405, 729, 1513

Offset

0,4

Comments

This could be called the inverse Motzkin transform of A109188 since the substitution x -> x/(1+x+x^2) in the independent variable of the g.f. A109188(x) yields this sequence here.

Links

Vincenzo Librandi, 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) = A000290(n), a(2*n+1) = A001844(n).

O.g.f.: x*(1+x+x^2)*(1+x^2)/((1-x)^3*(1+x)^3).

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6), n>5.

Euler transform of length 4 sequence [ 1, 4, -1, -1]. - Michael Somos, Aug 07 2014

a(2n+1) = a(2n) + a(2n+2) for all n in Z. - Michael Somos, Aug 07 2014

A120328(n-1) = 3*n^2 + 2 = a(2*n + 1) - a(2*n)+ a(2*n - 1) for all n in Z. - Michael Somos, Aug 07 2014

a(n) = n^2*(1+(-1)^n)/8+(n^2+1)*(1-(-1)^n)/4. - Wesley Ivan Hurt, Sep 06 2015

Example

G.f. = x + x^2 + 5*x^3 + 4*x^4 + 13*x^5 + 9*x^6 + 25*x^7 + 16*x^8 + 41*x^9 + ...

Maple

A147685:=n->n^2*(1+(-1)^n)/8+(n^2+1)*(1-(-1)^n)/4: seq(A147685(n), n=0..70); # Wesley Ivan Hurt, Sep 06 2015

Mathematica

CoefficientList[Series[x (1 + x + x^2) (1 + x^2)/((1 - x)^3 (1 + x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2014 *)

Prog

(PARI) {a(n) = if( n%2, (n^2 + 1) / 2, n^2 / 4)}; /* Michael Somos, Aug 07 2014 */

Crossrefs

Cf. A000290, A001844, A109188, A120328.

Sequence in context: A019068 A215947 A226555 * A353151 A078930 A329162

Adjacent sequences: A147682 A147683 A147684 * A147686 A147687 A147688

Keyword

easy,nonn

Author

R. J. Mathar, Nov 10 2008

Status

approved