Search: seq:40,40,40
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A009004
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Ordered short legs of Pythagorean triangles.
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+20
18
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3, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 15, 16, 16, 17, 18, 18, 19, 20, 20, 20, 21, 21, 21, 22, 23, 24, 24, 24, 24, 25, 25, 26, 27, 27, 27, 28, 28, 28, 29, 30, 30, 30, 31, 32, 32, 32, 33, 33, 33, 33, 34, 35, 35, 35, 36, 36, 36, 36, 36, 37, 38, 39, 39, 39, 39, 40, 40, 40, 40
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OFFSET
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1,1
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Robert Recorde, The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers, London, 1557. See p. 57.
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MATHEMATICA
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maxLeg = 40; r[a_] := a /. {ToRules[ Reduce[a <= b < c && a^2 + b^2 == c^2, {b, c}, Integers]]}; Flatten[r /@ Complement[ Range[maxLeg], {1, 2, 4}]] (* Jean-François Alcover, Jun 13 2012 *)
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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STATUS
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approved
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A062051
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Number of partitions of n into parts that are powers of 3.
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+20
14
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1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 23, 23, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, 72, 81, 81, 81, 93, 93, 93, 105, 105, 105, 117, 117, 117, 132, 132, 132, 147, 147, 147, 162
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OFFSET
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0,4
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COMMENTS
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Number of different partial sums of 1+[1,*3]+[1,*3]+..., where [1,*3] means we can either add 1 or multiply by 3. E.g., a(6)=3 because we have 6=1+1+1+1+1+1=(1+1)*3=1*3+1+1+1. - Jon Perry, Jan 01 2004
Also number of partitions of n into distinct 3-smooth parts. E.g., a(10) = #{9+1, 8+2, 6+4, 6+3+1, 4+3+2+1} = #{9+1, 3+3+3+1, 3+3+1+1+1+1, 3+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 5. - Reinhard Zumkeller, Apr 07 2005
Starts to differ from A008650 at a(81). - R. J. Mathar, Jul 31 2010
If m=ceiling(log_3(2k)) and n=(3^m+1)/2-k for k in the range (3^(m-1)+1)/2+(3^(m-2))<=k<=(3^m-1)/2, this sequence gives the number of "feasible" partitions described in the sequence A254296. For instance, the terms starting at 121st term of A254296 backwards to 68th term of A254296 provide the first 54 terms of this sequence. - Md. Towhidul Islam, Mar 01 2015
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LINKS
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Table of n, a(n) for n=0..69.
Vassil Dimitrov, Laurent Imbert, and Andrew Zakaluzny, Multiplication by a Constant is Sublinear, 18th IEEE Symposium on Computer Arithmetic (2007). See Theorem 1.
Md Towhidul Islam & Md Shahidul Islam, Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance, arXiv:1502.07730 [math.CO], 2015.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
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FORMULA
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a(n) = A005704([n/3]).
G.f.: product_{k>=0} 1/(1-x^(3^k)). - R. J. Mathar, Jul 31 2010
If m = ceiling(log_3(2k)), define n = (3^m + 1)/2 - k for k in the range (3^(m-1)+1)/2 + (3^(m-2)) <= k <= (3^m-1)/2. Then, a(n) = Sum_{s=ceiling((k-1)/3)..(3^(m-1)-1)/2} a(s). This gives the first 2(3^(m-1))/3 terms. - Md. Towhidul Islam, Mar 01 2015
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EXAMPLE
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a(4) = 2 and the partitions are 3+1, 1+1+1+1; a(9) = 5 and the partitions are 9; 3+3+3; 3+3+1+1+1; 3+1+1+1+1+1+1; 1+1+1+1+1+1+1+1+1.
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MATHEMATICA
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nn=70; a=Product[1/(1-x^(3^i)), {i, 0, 4}]; CoefficientList[Series[a, {x, 0, nn}], x] (* Geoffrey Critzer, Oct 30 2012 *)
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PROG
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(PARI) { n=15; v=vector(n); for (i=1, n, v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*3)); c=vector(n); for (i=1, n, for (j=1, 2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry
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CROSSREFS
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A005704 with terms repeated 3 times.
Cf. A000123, A018819, A000009, A003586, A105420, A039966, A018819, A023893, A105420, A106244, A131995, A179051, A254296.
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy, Jun 06 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001
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STATUS
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approved
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A254296
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The number of partitions of n having the minimum number of summands such that all integers from 1 to n can be represented as the sum of the summands times one of {-1, 0, 1}.
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+20
13
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1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 1, 1, 1, 10, 11, 12, 11, 12, 12, 11, 11, 12, 9, 9, 9, 7, 7, 7, 5, 5, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1, 131, 136, 140, 133, 137, 140, 133, 136, 138, 129, 131, 134, 125, 126, 128, 117, 119, 120, 109, 110, 111, 101, 102, 102, 92, 92, 93, 81, 81, 81, 72, 72, 72, 63, 63, 63, 54, 54, 54, 47, 47, 47, 40, 40, 40, 33, 33, 33
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OFFSET
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1,5
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COMMENTS
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Define a feasible partition of an n-kilogram stone as an ordered partition of minimum possible m parts W_1 <= W_2 <= ... <= W_m broken from the stone such that all integral weights from 1 to n can be weighed in one weighing using the parts/weights on a two pan balance. The minimum m for any n is m=ceiling(log_3(2n)). This sequence gives the number of feasible partitions of n.
From Robert G. Wilson v, Feb 04 2015: (Start)
Records: 1, 2, 3, 10, 11, 12, 131, 136, 140, 3887, 3921, 3950, 262555, 263112, 263707, 42240104, 42262878, 42285095, 16821037273, 16823225535, 16825391023, ..., .
Possible values: 1, 2, 3, 5, 7, 9, 10, 11, 12, 15, 18, 23, 28, 33, 40, 47, 54, 63, 72, 81, 92, 93, 101, 102, 105, ..., .
First occurrence on k, or 0 if not present: 1, 5, 7, 0 29, 0, 26, 0, 23, 14, 15, 16, 0, 0, 98, 0, 0, 95, 0, 0, 0, 0, 92, ..., .
1 occurs at: 1, 2, 3, 4, 11, 12, 13, 38, 39, 40, 119, 120, 121, 362, 363, 364, 1091, 1092, 1093, 3278, 3279, 3280, 9839, 9840, 9841, ..., .
2 occurs at: 5, 6, 8, 9, 10, 35, 36, 37, 116, 117, 118, 359, 360, 361, 1088, 1089, 1090, 3275, 3276, 3277, 9836, 9837, 9838, ..., .
3 occurs at: 7, 32, 33, 34, 113, 114, 115, 356, 357, 358, 1085, 1086, 1087, 3272, 3273, 3274, 9833, 9834, 9835, ..., .
5 occurs at: 29, 30, 31, 110, 111, 112, 353, 354, 355, 1082, 1083, 1084, 3269, 3270, 3271, 9830, 9831, 9832, ..., . (End)
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LINKS
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Md. Towhidul Islam, Table of n, a(n) for n = 1..88573
Md. Towhidul Islam, Formula for number of feasible partitions of n
Md Towhidul Islam & Md Shahidul Islam, Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance, arXiv:1502.07730 [math.CO], 2015.
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FORMULA
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Let us suppose, a(0)=1 and for (3^(m-1)+1)/2<=n<=(3^m-1)/2, m=ceiling(log_3(2n)).
Then for (3^(m-1)+1)/2<=n<=(3^(m-1)+1)/2+(3^(m-2)),a(n)=Sum{s=ceiling((n-1)/3..floor((2n+3^(m-2)-1)/4)}a(s)-Sum{d=ceiling((3n+2)/5)..(3^(m-1)-1)/2}Sum{p=ceiling((d-1)/3..2d-n-1}a(p)
and for (3^(m-1)+1)/2+3^(m-2)+1<=n<=(3^m-1)/2, a(n)=Sum_{s=ceiling((n-1)/3)..(3^(m-1)-1)/2}a(s).
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EXAMPLE
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For n=3, minimum number of weights m is 2. The only "feasible" set of weights is [1,2]. So, a(3)=1.
For n=7, m is 3. The "feasible" sets of weights are [1,1,5], [1,2,4], [1,3,3]. So, a(7)=3.
For n=19, m is 4. The "feasible" sets of weights are [1,1,4,13], [1,1,5,12], [1,2,3,13], [1,2,4,12], [1,2,5,11], [1,2,6,10], [1,2,7,9], [1,3,3,12], [1,3,4,11], [1,3,5,10], [1,3,6,9], [1,3,7,8]. There are no other "feasible" sets. So, a(19)=12.
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MATHEMATICA
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okQ[v_] := Module[{s=0}, For[i=1, i <= Length[v], i++, If[v[[i]] > 2*s+1, Return[ False], s += v[[i]] ] ]; Return[True]]; a[n_] := With[{k = Ceiling[Log[3, 2n]]}, Select[Reverse /@ IntegerPartitions[n, {k}], okQ] // Length]; Table[a[n], {n, 1, 88}] (* Jean-François Alcover, Feb 03 2015, after Charles R Greathouse IV *)
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PROG
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(PARI) ok(v)=my(s); for(i=1, #v, if(v[i]>2*s+1, return(0), s+=v[i])); 1
a(n)=my(k=ceil(log(2*n)/log(3))); #select(ok, partitions(n, , k)) \\ Charles R Greathouse IV, Feb 02 2015
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CROSSREFS
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Cf. A254296, A254430, A254431, A254432, A254433, A254435, A254436, A254437, A254438, A254439, A254440, A254441, A254442.
When we calculate a(n) for (3^(m-1)+1)/2+3^(m-2)+1 <= n <= (3^m-1)/2 starting from n=(3^m-1)/2 backwards, we get the sequence A062051 which is also the triplication of the terms of sequence A005704.
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KEYWORD
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nonn,look
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AUTHOR
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Md. Towhidul Islam, Jan 27 2015
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STATUS
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approved
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A007843
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Least positive integer k for which 2^n divides k!.
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+20
12
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1, 2, 4, 4, 6, 8, 8, 8, 10, 12, 12, 14, 16, 16, 16, 16, 18, 20, 20, 22, 24, 24, 24, 26, 28, 28, 30, 32, 32, 32, 32, 32, 34, 36, 36, 38, 40, 40, 40, 42, 44, 44, 46, 48, 48, 48, 48, 50, 52, 52, 54, 56, 56, 56, 58, 60, 60, 62, 64, 64, 64, 64, 64, 64, 66, 68, 68, 70, 72, 72, 72, 74, 76, 76, 78
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OFFSET
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0,2
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COMMENTS
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Obtained by writing every natural number n k times where 2^k divides n but 2^(k+1) does not divide n. - Amarnath Murthy, Aug 22 2002
Interval of the form (A007814(k!)-A007814(k), A007814(k!)] contains n >= 1 iff k = a(n). - Vladimir Shevelev, Mar 19 2012
It appears than for n>0, a(n) is divisible by 2, and that the resulting sequence a(n)/2 is A046699 (ignoring first term, this is the Meta-Fibonacci sequence for s=0). - Michel Marcus, Aug 19 2013
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REFERENCES
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H. Ibstedt, Smarandache Primitive Numbers, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 216-229.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
Oliver Kullmann and Xishun Zhao, Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses, arXiv preprint, 2015.
F. Smarandache, Only Problems, Not Solutions!.
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FORMULA
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a(n)=A002034(2^n). For n>1, it appears that a(n+1)=a(n)+2 if n is in A005187. - Benoit Cloitre, Sep 01 2002
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MAPLE
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with(numtheory): ans := [ ]: p := ithprime(1): t0 := 1/p: for n from 0 to 50 do t0 := t0*p: t1 := 1: i := 1: while t1 mod t0 <> 0 do i := i+1: t1 := t1*i: od: ans := [ op(ans), i ]: od: ans;
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MATHEMATICA
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a[n_] := (k=0; While[Mod[++k!, 2^n] > 0]; k); Table[a[n], {n, 0, 74}] (* Jean-François Alcover, Dec 08 2011 *)
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PROG
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(PARI) a(n)=if(n<0, 0, s=1; while(s!%(2^n)>0, s++); s)
(PARI) a(n) = {k = 1; while (valuation(k!, 2) < n, k++); k; } \\ Michel Marcus, Aug 19 2013
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CROSSREFS
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Cf. A007814, A007844, A007845, A020646, A048841-A048846.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Bruce Dearden and Jerry Metzger; R. Muller
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STATUS
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approved
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A201629
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a(n) = n if n is even and otherwise its nearest multiple of 4.
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+20
12
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0, 0, 2, 4, 4, 4, 6, 8, 8, 8, 10, 12, 12, 12, 14, 16, 16, 16, 18, 20, 20, 20, 22, 24, 24, 24, 26, 28, 28, 28, 30, 32, 32, 32, 34, 36, 36, 36, 38, 40, 40, 40, 42, 44, 44, 44, 46, 48, 48, 48, 50, 52, 52, 52, 54, 56, 56, 56, 58, 60, 60, 60, 62, 64, 64, 64, 66, 68, 68
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OFFSET
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0,3
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COMMENTS
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For n > 1, the maximal number of nonattacking knights on a 2 x (n-1) chessboard.
Compare this with the binary triangle construction of A240828.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Craig Knecht, Row sums of superimposed and added binary filled triangles.
V. Kotesovec, Non-attacking chess pieces
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FORMULA
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a(n) = n - sin(n*Pi/2).
G.f.: 2x^2/((-1+x)^2*(1+x^2)).
a(n) = 2*A004524(n+1). - R. J. Mathar, Feb 02 2012
a(n) = n+(1-(-1)^n)*(-1)^((n+1)/2)/2. [Bruno Berselli, Aug 06 2014]
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EXAMPLE
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G.f. = 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + 8*x^9 + ...
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MAPLE
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seq(n-sin(Pi*n/2), n=0..30); # Robert Israel, Jul 14 2015
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MATHEMATICA
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Table[2*(Floor[(Floor[(n + 1)/2] + 1)/2] + Floor[(Floor[n/2] + 1)/2]), {n, 1, 100}]
Table[If[EvenQ[n], n, 4*Round[n/4]], {n, 0, 68}] (* Alonso del Arte, Jan 27 2012 *)
CoefficientList[Series[2 x^2/((-1 + x)^2 (1 + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 06 2014 *)
a[ n_] := n - KroneckerSymbol[ -4, n]; (* Michael Somos, Jul 18 2015 *)
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PROG
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(PARI) a(n)=n\4*4+[0, 0, 2, 4][n%4+1] \\ Charles R Greathouse IV, Jan 27 2012
(PARI) {a(n) = n - kroncker( -4, n)}; /* Michael Somos, Jul 18 2015 */
(Haskell)
a201629 = (* 2) . a004524 . (+ 1) -- Reinhard Zumkeller, Aug 05 2014
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CROSSREFS
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Cf. A004524, A085801, A189889, A190394.
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KEYWORD
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nonn,easy
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AUTHOR
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Vaclav Kotesovec, Dec 03 2011
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EXTENSIONS
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Formula corrected by Robert Israel, Jul 14 2015
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STATUS
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approved
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2, 4, 4, 6, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 10, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 24, 24, 30, 30, 30, 30, 30, 30, 38, 38, 40, 40, 40, 40, 40, 40, 40, 46, 46, 46, 46, 46, 46, 46, 46, 54, 54, 54
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OFFSET
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1,1
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COMMENTS
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A251416(n) = Min{a(n), A251549(n)}. - Reinhard Zumkeller, Dec 19 2014
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LINKS
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Chai Wah Wu, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015.
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PROG
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(Haskell)
import Data.List ((\\))
a251546 n = head $ [2, 4 ..] \\ filter even (take n a098550_list)
-- Reinhard Zumkeller, Dec 19 2014
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CROSSREFS
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Cf. A098550, A247253, A251416, A251417, A251547-A251552.
See also A251557, A251558, A251559.
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Dec 18 2014
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STATUS
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approved
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A240542
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Numbers a(n) that are the coordinates of the midpoints of the (rotated) Dyck paths from (0, n) to (n, 0) defined by A237593. Also the alternating row sums of A235791.
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+20
10
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1, 2, 2, 3, 3, 5, 5, 6, 7, 7, 7, 9, 9, 9, 11, 12, 12, 13, 13, 15, 15, 15, 15, 17, 18, 18, 18, 20, 20, 22, 22, 23, 23, 23, 25, 26, 26, 26, 26, 28, 28, 30, 30, 30, 32, 32, 32, 34, 35, 36, 36, 36, 36, 38, 38, 40, 40, 40, 40, 42, 42, 42, 44, 45, 45, 47, 47, 47, 47, 49, 49, 52, 52
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OFFSET
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1,2
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COMMENTS
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The sequence is closely related to the alternating harmonic series.
Its asymptotic behavior is lim( k -> infinity : a(k)/k ) = log 2. The relative error is abs(a(k) - k*log2)/(k*log2) <= 2/sqrt(k).
Conjecture: the sequence of first positions of the alternating runs of odd and even numbers in a(k) equals A028982. Example: the positions in (1),(2),2,(3),3,5,5,(6),(7),7,7,9,9,9,11,(12),12,(13),13,15,... are 1,2,4,8,9,16,18,... Checked with a Mathematica function through a(1000000).
See A235791, A237591 and A237593 for additional formulas and properties.
The sequence of differences a(n) - a(n-1), for n>=1, appears to be equal to A067742(n), the sequence of middle divisors of n; as an empty sum, a(0) = 0, (which was conjectured by Michel Marcus in the entry A237593). Checked with the two respective Mathematica functions up to n=100000. - Hartmut F. W. Hoft, Jul 17 2014
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LINKS
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Table of n, a(n) for n=1..73.
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FORMULA
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a(n) = sum( k = 1 ... A003056(n) : (-1)^(k+1) A235791(n,k) ).
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MATHEMATICA
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a[n_] := Sum[(-1)^(k + 1) Ceiling[(n + 1)/k - (k + 1)/2], {k, 1, Floor[-1/2 + 1/2 Sqrt[8 n + 1]]}]; Table[a[n], {n, 40}]
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CROSSREFS
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Cf. A028982, A067742, A235791, A237270, A237271, A237591, A237593, A241562.
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KEYWORD
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nonn
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AUTHOR
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Hartmut F. W. Hoft, Apr 07 2014
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EXTENSIONS
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More terms from Omar E. Pol, Apr 16 2014
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STATUS
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approved
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A079438
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Number of rooted general plane trees which are symmetric and will stay symmetric also after the underlying plane binary tree has been reflected, i.e. number of integers i in range [A014137(n-1)..A014138(n-1)] such that A057164(i)=i and A057164(A057163(i)) = A057163(i).
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+20
9
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1, 1, 2, 2, 2, 4, 4, 4, 6, 6, 6, 8, 8, 8, 12, 12, 12, 14, 16, 16, 18, 18, 22, 24, 24, 24, 28, 28, 28, 30, 34, 34, 36, 36, 38, 40, 40, 40, 46, 46, 46, 48, 50, 50, 52, 52, 56, 58, 58, 58, 62, 62, 62, 64, 68, 68, 70, 70, 72, 74, 74, 74, 80, 80, 80, 82, 84, 84, 86, 86, 90, 92, 92, 92
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OFFSET
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0,3
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COMMENTS
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Also number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A071661 (= Donaghey's automorphism M "squared"), which is equal to condition A057164(i)=A069787(i)=i, i.e. the size of the intersect of fixed points of permutations A057164 and A069787 in the same range.
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REFERENCES
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R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
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LINKS
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Table of n, a(n) for n=0..73.
A. Karttunen, C-program for counting the initial terms of this sequence (empirically)
A. Karttunen, Illustration of initial terms for trees of sizes n=2..18
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FORMULA
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a(0)=a(1)=1, a(n) = 2*(floor((n+1)/3) + (if n>=14) (floor((n-10)/4)+floor((n-14)/8))) [This is the correct formula if the conjecture given in A080070 is true, otherwise it is only a lower bound, although known to be exact for up to very high values of n.]
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MAPLE
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A079438 := n -> `if`((n<2), 1, 2*(floor((n+1)/3) + `if`((n>=14), floor((n-10)/4)+floor((n-14)/8), 0)));
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CROSSREFS
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From n>= 2 onward A079440(n) = a(n)/2.
Occurs in A073202 as row 13373289. Cf. A079437, A079439, A079442, A080070.
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen Jan 27 2003
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STATUS
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approved
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A055592
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Longest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.
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+20
8
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5, 6, 8, 10, 12, 13, 13, 15, 15, 15, 16, 17, 17, 17, 18, 20, 20, 20, 21, 21, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 28, 29, 29, 30, 30, 30, 30, 30, 30, 30, 32, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 37, 37, 39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 40, 40, 41, 41
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OFFSET
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0,1
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LINKS
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Table of n, a(n) for n=0..69.
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MATHEMATICA
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max = 41; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055592 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 3]](* Jean-François Alcover, Jun 12 2012 *)
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CROSSREFS
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Cf. A055592, A055593, A055594, A055595.
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley, May 26 2000
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STATUS
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approved
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A120131
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Longest side of primitive Heronian triangles, sorted.
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+20
8
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5, 6, 8, 13, 13, 15, 15, 17, 17, 17, 20, 20, 21, 21, 24, 25, 25, 25, 26, 26, 28, 29, 29, 30, 30, 30, 35, 35, 36, 37, 37, 37, 37, 37, 39, 39, 39, 39, 40, 40, 40, 41, 41, 41, 41, 42, 44, 44, 45, 48, 48, 50, 50, 51, 51, 51, 51, 52, 52, 52, 52, 52, 52, 53, 53, 53, 53, 55, 55, 56
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