|
%I #22 Oct 22 2021 23:47:17
%S 1,1,1,1,2,2,1,3,3,1,1,4,4,4,3,1,5,5,9,5,2,1,6,6,16,7,6,3,1,7,7,25,9,
%T 12,7,1,1,8,8,36,11,20,13,8,4,1,9,9,49,13,30,21,27,9,3,1,10,10,64,15,
%U 42,31,64,16,10,5,1,11,11,81,17,56,43,125,25,21,11,2
%N Triangle read by rows, antidiagonals of an array (row r >= 1, column n >= 1) generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).
%C Companion to A178239 (the latter generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1)).
%C Row sums of the triangle = A169826: (1, 2, 5, 8, 16, 27, 45, 69, 109, ...).
%C Polcoeff row r of the array as f(x) satisfies f(x)/f(x^2) = (1 + r*x + x^2).
%C Let q(x) = (1 + r*x + x^2). Then polcoeff row r = q(x)*q(x^2)*q(x^4)*q(x^8)*...
%C Right border of the triangle = A002487: (1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, ...).
%C Terms in r-th row mod r (for r > 1): (1, 0, 1, 0, 1, 0, ...).
%F a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).
%F Given (1, r, 1, 0, 0, 0, ...) in each column of an infinite lower triangular matrix M; shifted down twice from the previous column. r-th row of the array = lim_{n->inf} M^n.
%F For the r-th row, a(2^k+n) = r*a(n) + a(2^k-n). - _Andrey Zabolotskiy_, Oct 21 2021
%e First few rows of the array:
%e 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, ..
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ..
%e 1, 3, 4, 9, 7, 12, 13, 27, 16, 21, 19, 36, 25, 39, ..
%e 1, 4, 5, 16, 9, 20, 21, 64, 25, 36, 29, 80, 41, 84, ..
%e 1, 5, 6, 25, 11, 30, 31, 125, 36, 55, 41, 150, 61, 155, ..
%e 1, 6, 7, 36, 13, 42, 43, 216, 49, 78, 55, 252, 85, 258, ..
%e 1, 7, 8, 49, 15, 56, 57, 343, 64, 105, 71, 392, 113, 399, ..
%e 1, 8, 9, 64, 17, 72, 73, 512, 81, 136, 89, 576, 145, 584, ..
%e ...
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 1, 2, 2;
%e 1, 3, 3, 1;
%e 1, 4, 4, 4, 3;
%e 1, 5, 5, 9, 5, 2;
%e 1, 6, 6, 16, 7, 6, 3;
%e 1, 7, 7, 25, 9, 12, 7, 1;
%e 1, 8, 8, 36, 11, 20, 13, 8, 4;
%e 1, 9, 9, 49, 13, 30, 21, 27, 9, 3;
%e 1, 10, 10, 64, 15, 42, 31, 64, 16, 10, 5;
%e 1, 11, 11, 81, 17, 56, 43, 125, 25, 21, 11, 2;
%e 1, 12, 12, 100, 19, 72, 57, 216, 36, 36, 19, 12, 5;
%e 1, 13, 13, 121, 21, 90, 73, 343, 49, 55, 29, 36, 13, 3;
%e 1, 14, 14, 144, 23, 110, 91, 512, 64, 78, 41, 80, 25, 14, 4;
%e ...
%o (PARI) A(r,n) = my(x=0,y=1); forstep(i=if(n,logint(n,2)),0,-1, if(bittest(n,i), x+=y;y*=r, y+=x;x*=r)); x;
%o T(r,n) = A(r-n+1,n); \\ _Kevin Ryde_, Mar 18 2021
%Y Array rows r=1 to r=10: A002487, A000027, A178590, A244643, A342610, A237711, A342611, A342614, A342615, A178569.
%Y Cf. A178239.
%K nonn,tabl
%O 1,5
%A _Gary W. Adamson_, May 29 2010
|
