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 A217626 First differences of A215940, or first differences of permutations of (0,1,2,...,m-1) reading them as decimal numbers, divided by 9 (with 10>=m, and m! > n). 10
 1, 9, 2, 9, 1, 78, 1, 19, 3, 8, 2, 77, 2, 8, 3, 19, 1, 78, 1, 9, 2, 9, 1, 657, 1, 9, 2, 9, 1, 178, 1, 29, 4, 7, 3, 66, 2, 18, 4, 18, 2, 67, 1, 19, 3, 8, 2, 646, 1, 19, 3, 8, 2, 67, 1, 29, 4, 7, 3, 176, 3, 7, 4, 29, 1, 67, 2, 8, 3, 19, 1, 646, 2, 8, 3, 19, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Terms do not depend on the choice of m, provided that m!>n (the index of the considered term), and the numbers associated to a permutation s of {0,...,m-1} are N(s)=sum_{i=1..m} s(i)*10^(m-i). This defines the present sequence for any arbitrarily large index, not limited to n <= 10!, for example. Similar sequences might be built in another base b, they would always start (1, b-1, 2, b-1, 1, ...). The partial sums of this kind of sequence would yield the analog of A215940 in the corresponding base. There are at least two palindromic patterns which are repeated throughout this sequence: one of them is "1,b-1,2,b-1,1" (It is optional here whether or not to include the 1's), another is built from the first 4!-1 terms (See the corresponding link for details). Also, for 1<=n<=(9!)-1: The repeating parts in the first differences of A030299 divided by nine, i.e. a(n) = A219664(n)/9. - Antti Karttunen, Dec 18 2012. Edited by: R. J. Cano, May 09 2017 There are more palindromic patterns than those mentioned above: Similar to the first 3!-1 and the first 4!-1 terms, the first k!-1 terms are repeated for all other k>4. Frequent are also multiples of these, e.g., k*[1,9,2,9,1] = [2,18,4,18,2], [3,27,6,27,3], ...), [1, 19, 3, 8, 2, 67, 1, 29, 4, 7, 3, 176, 3, 7, 4, 29, 1, 67, 2, 8, 3, 19, 1], and others. The "middle part" of roughly half the length (e.g., [9,2,9] or [67,...,67] in the last example), is repeated even more frequently. - M. F. Hasler, Jan 14 2013 From R. J. Cano, Apr 04 2016: (Start) Conjecture 1: Given 12, given P the set of permutations in increasing sequence for the letters 0..n-1, there are distributed with a symmetric pattern among its (n!)! permutations all those A000165(n!\2) of them such that their 1st differences are symmetric. Moreover by setting to zero the other elements whose 1st differences are not symmetric, we obtain an antisymmetric sequence. (End) Conjecture 4: If 2<=mA215940(n+1)-A215940(n); MATHEMATICA maxm = 5; Table[dd = FromDigits /@ Permutations[Range[m]]; (Drop[dd, If[m == 1, 0, (m - 1)!]] - First[dd])/9, {m, 1, maxm}] // Flatten // Differences (* Jean-François Alcover, Apr 25 2013 *) PROG (C) See LINKS. (Scheme): (define (A217626 n) (/ (A219664 n) 9)) ;; - Antti Karttunen, Dec 18 2012 (PARI) first_terms(n)={n=max(3, n); my(m:small=n!); my(a:vec=vector(m-1), i:small=0, x:vec=numtoperm(n, 0), y:vec, z:vec, u:small, B:small=11); m\=2; m--; while(i++<=m, u=!(i%6); y=numtoperm(n, i); z=(y-x)[1..n-1]; if(u, z=vector(#z, j, vecsum(z[1..j]))); a[i]=fromdigits(z, B-u); a[#a-i+1]=a[i]; x=y; ); z=(numtoperm(n, m+1)-y)[1..n-1]; a[m+1]=fromdigits(vector(#z, j, vecsum(z[1..j])), B--); return(a)} \\ Computes the first either 5 or n!-1 terms. - R. J. Cano, May 28 2017 CROSSREFS Cf. A215940, A207324, A219664, A220664, A030299. Cf. A219995 [ On the summation of 1/a(n) ]. Sequence in context: A157215 A021919 A078127 * A275362 A217174 A230157 Adjacent sequences:  A217623 A217624 A217625 * A217627 A217628 A217629 KEYWORD nonn,base,easy AUTHOR R. J. Cano, Oct 04 2012 EXTENSIONS Definition simplified by M. F. Hasler, Jan 12 2013 STATUS approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)