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 A178239 Triangle read by rows, antidiagonals of an array generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1). 5
 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 5, 2, 1, 1, 1, 6, 1, 7, 3, 3, 1, 1, 1, 7, 1, 9, 4, 7, 1, 1, 1, 1, 8, 1, 11, 5, 13, 1, 4, 1, 1, 1, 9, 1, 13, 6, 21, 1, 7, 3, 1, 1, 1, 10, 1, 15, 7, 31, 1, 10, 5, 5, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Partial sums of array terms in groups of 1, next 2, next 4,...8 = powers of (r+2). Row sums = A178240: (1, 2, 3, 5, 7, 11, 16, 23,...) Row 1 of the array = A002487. Row 2 = .............A116528. Row 3 = .............A178241. Row 4 = .............A178242. ... Row 10 = ............A178243. Polcoeff row r of the array as f(x) satisfies f(x)/f(x^2) = (1 + x + r*x^2). Let q(x) = (1 + x + r*x^2). Then polcoeff row 4 = q(x) * q(x^2) * q(x^4) * q(x^8) * ... LINKS FORMULA Antidiagonals of an array generated from a(n) = a(2n); a(2n+1) = r*a(n) + a(n+1). Given a triangle M with columns stepped down twice from the previous column, for columns >0, with (1, 1, r, 0, 0, 0,...) in each column, r-th row of the array = Lim_{n->inf} M^n. EXAMPLE First few rows of the array = 1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,...1,... 1,...1,...2,...1,...3,...2,...3,...1,...4,...3,...5,...2,...5,...3,...4,... 1,...1,...3,...1,...5,...3,...7,...1,...7,...5,..13,...3,..13,...7,..15,... 1,...1,...4,...1,...7,...4,..13,...1,..10,...7,..25,...4,..25,..13,..40,... 1,...1,...5,...1,...9,...5,..21,...1,..13,...9,..41,...5,..41,..21,..85,... 1,...1,...6,...1,..11,...6,..31,...1,..16,..11,..61,...6,..61,..31,.156,... ... Example: In row 3: (1, 1, 4, 1, 7, 4, 13,...) = A178241, r = 3. A178241(7) = 13 = 3*4 + 1. In blocks of 1, 2, 4, 8,...terms, partial sums are powers of (r+2) = 5: (1, 5, 25,...). First few rows of the triangle = 1; 1, 1; 1, 1, 1; 1, 1, 2, 1; 1, 1, 3, 1, 1; 1, 1, 4, 1, 3, 1; 1, 1, 5, 1, 5, 2, 1; 1, 1, 6, 1, 7, 3, 3, 1; 1, 1, 7, 1, 9, 4, 7, 1, 1; 1, 1, 8, 1, 11, 5, 13, 1, 4, 1; 1, 1, 9, 1, 13, 6, 21, 1, 7, 3, 1; 1, 1, 10, 1, 15, 7, 31, 1, 10, 5, 5, 1; 1, 1, 11, 1, 17, 8, 43, 1, 13, 7, 13, 2, 1; 1, 1, 12, 1, 19, 9, 57, 1, 16, 9, 21, 3, 5, 1; 1, 1, 13, 1, 21, 11, 73, 1, 19, 11, 31, 4, 13, 2, 1; ... CROSSREFS Cf. A002487, A116528, A178240, A178241, A178241, A178243 Sequence in context: A294306 A181348 A107682 * A260534 A085476 A124944 Adjacent sequences:  A178236 A178237 A178238 * A178240 A178241 A178242 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, May 23 2010 STATUS approved

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Last modified August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)