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

%I A154958 #17 May 09 2024 09:08:14
%S A154958 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,
%T A154958 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,
%U A154958 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
%N A154958 Antidiagonal sums of number triangle A154957 regarded as a lower triangular infinite matrix.
%H A154958 Vincenzo Librandi, <a href="/A154958/b154958.txt">Table of n, a(n) for n = 0..10000</a>
%H A154958 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-2,1,1,-1).
%F A154958 a(2n) = A004523(n+2); a(2n+1) = floor((n+3)/3) = A002264(n+3).
%F A154958 G.f.: 1/((x-1)^2*(x+1)^2*(x^2-x+1)). - _Philippe Deléham_, Mar 21 2014
%F A154958 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
%F A154958 Euler transform of length 6 sequence [ 1, 1, -1, 0, 0, 1]. - _Michael Somos_, Mar 21 2014
%F A154958 a(-6-n) = -a(n). - _Michael Somos_, Mar 21 2014
%F A154958 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
%e A154958 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 + ...
%t A154958 CoefficientList[Series[1/((x - 1)^2 (x + 1)^2 (x^2 - x + 1)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Mar 22 2014 *)
%t A154958 LinearRecurrence[{1,1,-2,1,1,-1},{1,1,2,1,2,1},90] (* _Harvey P. Dale_, Aug 26 2016 *)
%o A154958 (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 */
%Y A154958 Cf. A002264, A004523, A026741, A029578, A065423.
%K A154958 easy,nonn
%O A154958 0,3
%A A154958 _Paul Barry_, Jan 18 2009