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

0,3

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).

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A139024 A217317 A331128 * A025806 A025802 A145706

Adjacent sequences: A154955 A154956 A154957 * A154959 A154960 A154961

Keyword

easy,nonn

Author

Paul Barry, Jan 18 2009

Status

approved