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 A153123 Slowest-increasing monotonic sequence of positive integers such that for all n >= 1 the n-th term of the n-th differences is n. 1
 1, 2, 3, 6, 7, 9, 10, 12, 14, 17, 20, 22, 23, 31, 53, 56, 57, 683, 2983, 2984, 2985, 149411, 811160, 811162, 811163, 75426842, 476823319, 476823329, 476823330, 73727027930, 530824147003, 530824147008, 530824147009, 126824134344110 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS PROG (PARI) {calc_two_diags() = /* M, n, n2 inherited; down first diag., up next */ for(r=2, n, M[r, n2-r] = M[r-1, n2-r+1] - M[r-1, n2-r]); forstep(r=n, 1, -1, M[r, n2-r+1] = M[r, n2-r] + M[r+1, n2-r])} /* Find pair-by-pair first 2k terms of A153123 and put into row 1 of M. */ /* Row n+1 of M contains 2k-2n+1 terms of A153123's n-th differences -- */ /* for efficiency not starting until the n-th term in the n-th column. */ /* (Minor changes to program would calculate all n-th differences.) */ /* For larger k the PARI stack size may need to be increased first. */ {k = 450; M = matrix(k+1, 2*k); M[1, 1] = 1; M[1, 2] = 2; M[2, 1] = 1; for(n=2, k, M[n+1, n] = n; n2 = 2*n; M[1, n2-1] = M[1, n2-2] + 1; calc_two_diags(); if(M[1, n2] <= M[1, n2-1], /* If first try doesn't find increasing terms, */ M[1, n2-1] = M[1, n2-1] + 1 + floor(-M[2, n2-1]/(n-1)); calc_two_diags())); /* the above adjusts both terms just enough.*/ M[1, ] /* Print only first row of M */} /* The above program uses these general facts: */ /* Increasing (Decreasing) one term, a(j), of a sequence by a constant c */ /* i) increases (decreases) each element of the leftmost changing */ /* diagonal of its difference array by c and */ /* ii) increases (decreases) the elements of the diagonal immediately */ /* to its right -- including a(j+1) -- by c, 2c, 3c, ..., dc, respectively, */ /* for the d elements above a given element on this second diagonal*/ /* which is being held fixed, where dc is the change in a(j+1). */ /* (Incidentally, a similar pattern occurs for the elements below */ /* the fixed one but the sign of the changes is reversed.) */ CROSSREFS Sequence in context: A188094 A047561 A285567 * A100913 A047285 A191215 Adjacent sequences:  A153120 A153121 A153122 * A153124 A153125 A153126 KEYWORD nonn AUTHOR Rick L. Shepherd, Dec 18 2008 STATUS approved

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Last modified November 27 07:28 EST 2021. Contains 349365 sequences. (Running on oeis4.)