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, 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

Offset

1,2

Links

Table of n, a(n) for n=1..34.

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