A124093 Triangular numbers alternating with squares.

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

Offset: 0

Author

N. J. A. Sloane, based on a suggestion from Robert G. Wilson v, Nov 27 2006

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A123596. Rearrangement of A054686.

Programs

Maple

a:=proc(n) if n mod 2 = 0 then n*(n+2)/8 else (n-1)^2/4 fi end: seq(a(n),n=0..70); # Emeric Deutsch, Nov 29 2006

Mathematica

tr=Table[{k(k+1)/2,k^2},{k,0,100}]//Flatten (Seidov)
With[{nn=30},Riffle[Accumulate[Range[0,nn]],Range[0,nn]^2]] (* Harvey P. Dale, Jul 13 2014 *)

PARI

concat([0,0], Vec(-x^2*(x^3+x+1)/((x-1)^3*(x+1)^3) + O(x^100))) \\ Colin Barker, May 28 2015

Formula

a(n) = n(n+2)/8 if n is even; a(n) = (n-1)^2/4 if n is odd (n>=0). - Emeric Deutsch, Nov 29 2006

a(n) = (3*n^2-2*n+2-(n^2-6*n+2)*(-1)^n)/16. - Luce ETIENNE, May 28 2015

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>5. - Colin Barker, May 28 2015

G.f.: -x^2*(x^3+x+1) / ((x-1)^3*(x+1)^3). - Colin Barker, May 28 2015

Extensions

More terms from Zak Seidov, Nov 28 2006

More terms from Emeric Deutsch, Nov 29 2006