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A147685 Squares and centered square numbers interleaved. 2
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 (list; graph; refs; listen; history; text; internal format)
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 * A078930 A094414 A158867

Adjacent sequences:  A147682 A147683 A147684 * A147686 A147687 A147688

KEYWORD

easy,nonn

AUTHOR

R. J. Mathar, Nov 10 2008

STATUS

approved

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Last modified March 19 13:08 EDT 2019. Contains 321330 sequences. (Running on oeis4.)