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 A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0. 2
 1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 20, 5, 1, 3447, 1297, 461, 120, 28, 6, 1, 29093, 10063, 3447, 836, 181, 38, 7, 1, 273343, 87669, 29093, 6616, 1333, 270, 49, 8, 1, 2829325, 847015, 273343, 58576, 11029, 2150, 375, 61, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..54. FORMULA Column 0 = Column 0 of triangle A104980 = A003319. Column 1 = Column 1 of triangle A104980 = A104981. Column 3 = column 1 of A104988 (matrix square of A104980). Column 5 = column 2 of A104988 (matrix square of A104980). EXAMPLE To generate the array, start with all 1's in row 0; from then on, obtain row n+1 from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2,m>0} and then taking partial sums. This square array A begins: (1), (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, 1, 1, 1, 1, ...; (1), (2), 3, 4, (5), 6, 7, 8, (9), 10, 11, 12, 13, (14), 15, 16, ...; (3), (7), 13, 20, (28), 38, 49, 61, (74), 89, 105, 122, 140, (159),...; (13), (33), 71, 120, (181), 270, 375, 497, (637), 817, 1019, 1244, ...; (71), (191), 461, 836, (1333), 2150, 3169, 4413, (5906), 8001, ...; (461), (1297), 3447, 6616, (11029), 19030, 29483, 42775, (59324),...; (3447), (10063), 29093, 58576, (101351), 185674, 300329, 451277, ...; (29093), (87669), 273343, 573672, (1024949), 1982310, 3330651, ...; (273343), (847015), 2829325, 6159976, (11320359), 23009602, 39998897, ...; where terms in parenthesis at positions {0,1,4,8,13,..} in a row are removed before taking partial sums to obtain the next row. ... RELATION TO SPECIAL TRIANGLE. Triangle A104980 begins: 1; 1, 1; 3, 2, 1; 13, 7, 3, 1; 71, 33, 13, 4, 1; 461, 191, 71, 21, 5, 1; 3447, 1297, 461, 133, 31, 6, 1; 29093, 10063, 3447, 977, 225, 43, 7, 1; ... in which column 0 and column 1 are found in square array A. ... Matrix square of A104980 = triangle A104988 which begins: 1; 2, 1; 8, 4, 1; 42, 20, 6, 1; 266, 120, 38, 8, 1; 1954, 836, 270, 62, 10, 1; 16270, 6616, 2150, 516, 92, 12, 1; 151218, 58576, 19030, 4688, 882, 128, 14, 1; ... where column 1 and column 2 are also found in square array A. PROG (PARI) {T (n, k)=local (A=0, b=2, c=1, d=0); if (n==0, A=1, until (d>k, if (c==b* (b+1)/2-2, b+=1, A+=T (n-1, c); d+=1); c+=1)); A} CROSSREFS Cf. columns: A003319, A104981, A156629, related triangles: A104980, A104988. Cf. related tables: A136212, A136213, A125714, A135876, A127054, A125781, A136217. Sequence in context: A358310 A180190 A059438 * A104980 A316566 A134090 Adjacent sequences: A156625 A156626 A156627 * A156629 A156630 A156631 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Feb 17 2009 STATUS approved

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Last modified July 23 04:43 EDT 2024. Contains 374544 sequences. (Running on oeis4.)