Search: id:A118654
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%I A118654
%S A118654 1,1,0,1,1,1,1,3,2,1,1,7,4,3,2,1,15,8,7,5,3,1,31,16,15,11,8,5,1,63,32,
%T A118654 31,23,18,13,8,1,127,64,63,47,38,29,21,13,1,255,128,127,95,78,61,47,34,
%U A118654 21,1,511,256,255,191,158,125,99,76,55,34
%N A118654 Matrix, a(n,k) = 2^n(Fibonacci(k)) - Fibonacci(k-2), read by anti-diagonals.
%C A118654 Inverse binomial transform (by columns) of A090888.
%F A118654 a(n,k) = 2^n(Fibonacci(k)) - Fibonacci(k-2). a(n,k) = (2^n-2)Fibonacci(k) 
               + Fibonacci(k+1). a(n,0) = 1; a(n,1) = 2^n - 1; a(n,k) = a(n,k-1) 
               + a(n,k-2), for k > 1. a(0,k) = Fibonacci(k-1); a(1,k) = Fibonacci(k+1); 
               a(n,k) = 3a(n-1,k) - 2a(n-2,k), for n > 1. a(n,k) = 2a(n-1,k) + Fibonacci(k-2), 
               for n > 0. O.g.f. (by rows) = (1+(-2+2^n)x)/(1-x-x^2).
%F A118654 Sum[a(n-k, k), {k, 0, n}] = A119587(n+1). - Ross La Haye (rlahaye(AT)new.rr.com), 
               May 31 2006 Ross
%e A118654 a(2,3) = 7 because 2^2(Fibonacci(3)) - Fibonacci(3-2) = 4*2 - 1 = 7.
%e A118654 {1}; {1,0}; {1,1,1}; {1,3,2,1}; {1,7,4,3,2}; {1,15,8,7,5,3}; {1,31,16,
               15,11,8,5}; {1,63,32,31,23,18,13,8}.
%Y A118654 Cf. a(n, k) = A109754(2^n-2, k+1) = A101220(2^n-2, 0, k+1), for n > 0. 
               Rows: a(0, k) = A000045(k-1), for k > 0; a(1, k) = A000045(k+1); 
               a(2, k) = A000032(k+1); a(3, k) = A022097(k); a(4, k) = A022105(k); 
               a(5, k) = A022401(k). Columns: a(n, 1) = A000225(n); a(n, 2) = A000079(n); 
               a(n, 3) = A000225(n+1); a(n, 4) = A055010(n+1); a(n, 5) = A051633(n); 
               a(n, 6) = A036563(n+3).
%Y A118654 Sequence in context: A113185 A132069 A073201 this_sequence A111760 A078424 
               A092742
%Y A118654 Adjacent sequences: A118651 A118652 A118653 this_sequence A118655 A118656 
               A118657
%K A118654 nonn,tabl
%O A118654 0,8
%A A118654 Ross La Haye (rlahaye(AT)new.rr.com), May 17 2006

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