Walt Rorie-Baety - cp's OEIS frontend

A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 2, 2, 12, 12, 3, 108, 120, 24, 4, 1280, 1520, 280, 40, 5, 18750, 23400, 3930, 510, 60, 6, 326592, 423360, 65016, 7644, 840, 84, 7, 6588344, 8800008, 1241464, 132552, 13440, 1288, 112, 8, 150994944, 206622720, 26911296, 2622528, 244944, 22032, 1872, 144, 9

Offset: 1

Walt Rorie-Baety, Aug 15 2013

			T(1,1) =  1: [1].
T(2,1) =  2: [1,2], [2,1].
T(2,2) =  2: [1,1], [2,2].
T(3,1) = 12: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3].
T(3,2) = 12: [1,1,2], [1,1,3], [1,2,2], [1,3,3], [2,1,1], [2,2,1], [2,2,3], [2,3,3], [3,1,1], [3,2,2], [3,3,1], [3,3,2].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
.       1;
.       2,       2;
.      12,      12,       3;
.     108,     120,      24,      4;
.    1280,    1520,     280,     40,     5;
.   18750,   23400,    3930,    510,    60,    6;
.  326592,  423360,   65016,   7644,   840,   84,   7;
. 6588344, 8800008, 1241464, 132552, 13440, 1288, 112,  8;

Alois P. Heinz, Rows n = 1..141, flattened
Project Euler, Problem 427: n-sequences

Row sums give: A000312.
Column k=1 gives: A055897.
Main diagonal gives: A000027.
Lower diagonal gives: 2*A180291.
Cf. A228194, A228273, A228617.

T:= proc(n) option remember; local b; b:=
      proc(m, s, i) option remember; `if`(m>i or s>m, 0,
        `if`(i=1, n, `if`(s=1, (n-1)*add(b(m, h, i-1), h=1..m),
         b(m, s-1, i-1) +`if`(s=m, b(m-1, s-1, i-1), 0))))
      end; forget(b);
      seq(add(b(k, s, n), s=1..k), k=1..n)
    end:
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 18 2013

T[n_] := T[n] = Module[{b}, b[m_, s_, i_] := b[m, s, i] = If[m>i || s>m, 0, If[i == 1, n, If[s == 1, (n-1)*Sum[b[m, h, i-1], {h, 1, m}], b[m, s-1, i-1] + If[s == m, b[m-1, s-1, i-1], 0]]]]; Table[Sum[b[k, s, n], {s, 1, k}], {k, 1, n}]]; Table[ T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A228194(n). - Alois P. Heinz, Dec 23 2020

A182082 Number of pairs, (x,y), with x >= y, whose LCM does not exceed n.

Original entry on oeis.org

1, 3, 5, 8, 10, 15, 17, 21, 24, 29, 31, 39, 41, 46, 51, 56, 58, 66, 68, 76, 81, 86, 88, 99, 102, 107, 111, 119, 121, 135, 137, 143, 148, 153, 158, 171, 173, 178, 183, 194, 196, 210, 212, 220, 228, 233, 235, 249, 252, 260, 265, 273, 275, 286, 291, 302, 307, 312

Offset: 1

Walt Rorie-Baety, Apr 10 2012

nonn

Note that this is the asymmetric count. If all pairs (x,y) are counted, A061503 is obtained. - T. D. Noe, Apr 10 2012

			a(1000000) = 37429395, according to Project Euler problem #379.

Walt Rorie-Baety, Table of n, a(n) for n = 1..1000
Project Euler, Problem 379: Least common multiple count.

Partial sums of A018892.

Cf. A000005, A007875, A013661, A061503 (symmetric case).

a n = length [(x,y)| x <- [1..n], y <- [x..n], lcm x y <= n]

Table[Count[Flatten[Table[LCM[i, j], {i, n}, {j, i, n}]], ?(# <= n &)], {n, 60}] (* _T. D. Noe, Apr 10 2012 *)
nn = 100; (Accumulate[Table[DivisorSigma[0, n^2], {n, nn}]] + Range[nn])/2 (* T. D. Noe, Apr 10 2012 *)

a(n)=(sum(k=1,n,numdiv(k^2))+n)/2 \\ Charles R Greathouse IV, Apr 10 2012

a(n) = Sum_{k=1..n} (d(k^2)+1)/2, where d is the number of divisors function (A000005). - Charles R Greathouse IV, Apr 10 2012
a(n) = Sum_{k=1..n} A007875(k) * floor(n/k). - Daniel Suteu, Jan 08 2021
a(n) ~ n * log(n)^2 /(4*zeta(2)) (see A018892 for a more accurate asymptotic formula). - Amiram Eldar, Feb 01 2025

A201992 Numbers whose binary representations are found in the Thue-Morse sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 22, 25, 26, 37, 38, 41, 44, 45, 50, 51, 52, 75, 76, 77, 82, 83, 89, 90, 101, 102, 105, 150, 153, 154, 165, 166, 179, 180, 203, 205, 210, 211, 300, 301, 306, 308, 331, 332, 358, 361, 406, 410, 421, 422, 601

Offset: 0

Walt Rorie-Baety, Dec 07 2011

Interpreting A010060 as a bit string, this sequence contains the decimal equivalents of the subsequences, in order.

			The binary representation of 21 (10101) has an overlapping square sequence (1X1X1, where X is any binary sequence, in this case, X = 0), and so is not in the sequence. Compare to A063037.

Walt Rorie-Baety, Table of n, a(n) for n = 0..2500
Project Euler, Problem 361: Subsequence of Thue-Morse sequence

Cf. A010060, A063037.

a201992 = 0: concatMap (\n -> Set.toList . Set.fromList . map binRep . filter ((==[1]).take 1) . window n . take (n*2^n) $ a010060) [1..] where
  {window n = takeWhile (full . drop (n-1)) . map (take n) .  tails; binRep = foldl' (\a b -> 2*a+b) 0}; full = not . null

Module[{nn=10000,tm},tm=Table[ThueMorse[n],{n,0,nn}];Join[{0},Position[ Table[ If[SequenceCount[tm,IntegerDigits[k,2]]>0,1,0],{k,1000}], 1]]]// Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 03 2018 *)

Helper function added and name of value in program changed for better understanding by Walt Rorie-Baety, Mar 25 2012

A179180 Partial sums of A007895.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 11, 13, 15, 17, 20, 21, 23, 25, 27, 30, 32, 35, 38, 39, 41, 43, 45, 48, 50, 53, 56, 58, 61, 64, 67, 71, 72, 74, 76, 78, 81, 83, 86, 89, 91, 94, 97, 100, 104, 106, 109, 112, 115, 119, 122, 126, 130, 131, 133, 135, 137, 140, 142, 145

Offset: 0

Walt Rorie-Baety, Jun 30 2010

nonn

Total number of summands in Zeckendorf representations of all the numbers 1,2,...,n (for n>0); see the conjecture at A214979. - Clark Kimberling, Oct 23 2012

			For n = 6, a(n) = 1+1+1+2+1+2 = 8.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Christian Ballot, On Zeckendorf and Base b Digit Sums, Fibonacci Quarterly, Vol. 51, No. 4 (2013), pp. 319-325.
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A001622, A007895, A214979.

s = Reverse[Table[Fibonacci[n + 1], {n, 1, 70}]];
t2 = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2, 1]], # > 0 &]] &, Range[z]]; v[n_] := Sum[t2[[k]], {k, 1, n}];
v1 = Table[v[n], {n, 1, z}]
(* Peter J. C. Moses, Oct 18 2012 *)
DigitCount[Select[Range[0, 500], BitAnd[#, 2*#] == 0&], 2, 1] // Accumulate (* Jean-François Alcover, Jan 25 2018 *)

a(n) ~ c * n * log(n), where c = (phi-1)/(sqrt(5)*log(phi)) = 0.574369... and phi is the golden ratio (A001622) (Ballot, 2013). - Amiram Eldar, Dec 09 2021

Corrected term a(17); the working list of the terms were not in order. Walt Rorie-Baety, Jun 30 2010

Extended by Clark Kimberling, Oct 23 2012

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User: Walt Rorie-Baety

A228154 T(n,k) is the number of s in {1,...,n}^n having longest contiguous subsequence with the same value of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A182082 Number of pairs, (x,y), with x >= y, whose LCM does not exceed n.

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A201992 Numbers whose binary representations are found in the Thue-Morse sequence.

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A179180 Partial sums of A007895.

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