A154958 Antidiagonal sums of number triangle A154957 regarded as a lower triangular infinite matrix.

1, 1, 2, 1, 2, 1, 3, 2, 4, 2, 4, 2, 5, 3, 6, 3, 6, 3, 7, 4, 8, 4, 8, 4, 9, 5, 10, 5, 10, 5, 11, 6, 12, 6, 12, 6, 13, 7, 14, 7, 14, 7, 15, 8, 16, 8, 16, 8, 17, 9, 18, 9, 18, 9, 19, 10, 20, 10, 20, 10, 21, 11, 22, 11, 22, 11, 23, 12, 24, 12, 24, 12, 25, 13, 26, 13, 26, 13, 27, 14, 28, 14, 28

Offset: 0

Author

Paul Barry, Jan 18 2009

Keywords

• easy
• nonn

Examples

			G.f. = 1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 4*x^8 + 2*x^9 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-2,1,1,-1).

Crossrefs

Cf. A002264, A004523, A026741, A029578, A065423.

Programs

Mathematica

CoefficientList[Series[1/((x - 1)^2 (x + 1)^2 (x^2 - x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 22 2014 *)
LinearRecurrence[{1,1,-2,1,1,-1},{1,1,2,1,2,1},90] (* Harvey P. Dale, Aug 26 2016 *)

PARI

{a(n) = if( n<-5, -a(-6-n), if( n<0, 0, polcoeff( 1 / (1 - x - x^2 + 2*x^3 - x^4 - x^5 + x^6) + x * O(x^n), n)))}; /* Michael Somos, Mar 21 2014 */

Formula

a(2n) = A004523(n+2); a(2n+1) = floor((n+3)/3) = A002264(n+3).

G.f.: 1/((x-1)^2*(x+1)^2*(x^2-x+1)). - Philippe Deléham, Mar 21 2014

a(n) = a(n-1) + a(n-2) - 2*a(n-3) + a(n-4) + a(n-5) - a(n-6), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 1, a(4) = 2, a(5) = 1. - Philippe Deléham, Mar 21 2014

Euler transform of length 6 sequence [ 1, 1, -1, 0, 0, 1]. - Michael Somos, Mar 21 2014

a(-6-n) = -a(n). - Michael Somos, Mar 21 2014

a(3*n) = A026741(n+1). a(3*n + 1) = A029578(n+2). a(3*n + 2) = A065423(n+3). - Michael Somos, Mar 21 2014