A038722 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,2,3,4,... .
1, 3, 2, 6, 5, 4, 10, 9, 8, 7, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 28, 27, 26, 25, 24, 23, 22, 36, 35, 34, 33, 32, 31, 30, 29, 45, 44, 43, 42, 41, 40, 39, 38, 37, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 78, 77, 76
Offset: 1
Examples
The rectangular array view is 1 2 4 7 11 16 22 29 37 46 3 5 8 12 17 23 30 38 47 57 6 9 13 18 24 31 39 48 58 69 10 14 19 25 32 40 49 59 70 82 15 20 26 33 41 50 60 71 83 96 21 27 34 42 51 61 72 84 97 111 28 35 43 52 62 73 85 98 112 127 36 44 53 63 74 86 99 113 128 144 45 54 64 75 87 100 114 129 145 162 55 65 76 88 101 115 130 146 163 181
References
- Suggested by correspondence with Michael Somos.
- R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.
Links
- Hieronymus Fischer, Table of n, a(n) for n = 1..11401
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
- Index entries for sequences that are permutations of the natural numbers
Crossrefs
Programs
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Haskell
a038722 n = a038722_list !! (n-1) a038722_list = concat a038722_tabl a038722_tabl = map reverse a000027_tabl a038722_row n = a038722_tabl !! (n-1) -- Reinhard Zumkeller, Nov 08 2013
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Mathematica
(* Program generates dispersion array T of the increasing sequence f[n] *) r=40; r1=12; c=40; c1=12; f[n_] := Floor[n+1/2+Sqrt[2n]] (* complement of column 1 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]] rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]] (* A038722 array *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A038722 sequence *) (* Clark Kimberling, Jun 06 2011, corrected Jan 26 2025 *) Table[ n, {m, 12}, {n, m (m + 1)/2, m (m - 1)/2 + 1, -1}] // Flatten (* or *) Table[ Ceiling[(Sqrt[8 n + 1] - 1)/2]^2 - n + 1, {n, 78}] (* Robert G. Wilson v, Jun 27 2014 *) With[{nn=20},Reverse/@TakeList[Range[(nn(1+nn))/2],Range[nn]]//Flatten] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Dec 14 2017 *) -
PARI
a(n)=local(t=floor(1/2+sqrt(2*n))); if(n<1, 0, t^2-n+1) /* Paul D. Hanna */
Formula
a(n) = (sqrt(2n-1) - 1/2)*(sqrt(2n-1) + 3/2) - n + 2 = A061579(n-1) + 1. Seen as a square table by antidiagonals, T(n, k) = k + (n+k-1)*(n+k-2)/2, i.e., the transpose of A000027 as a square table.
G.f.: g(x) = (x/(1-x))*(psi(x) - x/(1-x) + 2*Sum_{k>=0} k*x^(k*(k+1)/2)) where psi(x) = Sum_{k>=0} x^(k*(k+1)/2) = (1/2)*x^(-1/8)*theta_2(0,x^(1/2)) is a Ramanujan theta function. - Hieronymus Fischer, Aug 08 2007
a(n) = floor(sqrt(2*n) + 1/2)^2 - n + 1. - Clark Kimberling, Jun 05 2011; corrected by Paul D. Hanna, Jun 27 2011
From Hieronymus Fischer, Apr 30 2012: (Start)
a(n) = a(n-1)-1, if a(n-1)-1 > 0 is not in the set {a(k)| 1<=k
a(n) = n for n = 2k(k+1)+1, k >= 0.
a(n+1) = (m+2)(m+3)/2, if 8a(n)-7 is a square of an odd number, otherwise a(n+1) = a(n)-1, where m = (sqrt(8a(n)-7)-1)/2.
a(n) = ceiling((sqrt(8n+1)-1)/2)^2 - n + 1. (End)
G.f. as rectangular array: x*y*(1 - (1 + x)*y + (1 - x + x^2)*y^2)/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Dec 25 2022
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