A000463 n followed by n^2.

1, 1, 2, 4, 3, 9, 4, 16, 5, 25, 6, 36, 7, 49, 8, 64, 9, 81, 10, 100, 11, 121, 12, 144, 13, 169, 14, 196, 15, 225, 16, 256, 17, 289, 18, 324, 19, 361, 20, 400, 21, 441, 22, 484, 23, 529, 24, 576, 25, 625, 26, 676, 27, 729, 28, 784, 29, 841, 30, 900, 31, 961, 32, 1024, 33, 1089, 34, 1156, 35, 1225, 36, 1296

Offset: 1

Author

Dominick Cancilla

Keywords

• nonn
• easy
• look

Comments

Eigensequence of a triangle with nonnegative integers interlaced with zeros (1, 0, 2, 0, 3, ...) as the right and left borders, with the rest zeros. - Gary W. Adamson, Aug 01 2016

Examples

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A188652 (first differences), A188653 (second differences), A159693 (partial sums), A000290 (squares).

Programs

Haskell

a000463 n = a000463_list !! (n-1)
a000463_list = concatMap (\x -> [x,x^2]) [1..]
-- Reinhard Zumkeller, Apr 13 2011

Magma

&cat[ [ n, n^2 ]: n in [1..36] ]; // Klaus Brockhaus, Apr 20 2009

Maple

seq(seq(n^k, k=1..2), n=1..36); # Zerinvary Lajos, Jun 29 2007

Mathematica

Array[{#, #^2} &, 36, 0] // Flatten
Riffle[Range[40], Range[40]^2] (* Bruno Berselli, Jul 15 2013 *)
a[ n_] := If[ OddQ @ n, (n + 1) / 2, n^2 / 4]; (* Michael Somos, May 28 2014 *)

PARI

{a(n) = if( n%2, (n + 1) / 2, n^2 / 4)}; /* Michael Somos, May 28 2014 */

Formula

a(n) = ((((-1)^(n+1))+1)/4)(n+1) - ((((-1)^(n+1))-1)/8)n^2 - Sam Alexander

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

a(n) = if(n mod 2, (n+1)/2, (n/2)^2). - Gerald Hillier, Sep 25 2008

a(n) = floor((n+1) / 2) ^ (2 - n mod 2). - Reinhard Zumkeller, Aug 15 2011

E.g.f.: (x + 2)*(sinh(x) + x*cosh(x))/4. - Ilya Gutkovskiy, Aug 02 2016

Extensions

Square of 14 corrected by Sean A. Irvine, Oct 25 2010