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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

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

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)