Search: seq:30,40,40
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A062857
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Size of smallest square multiplication table which contains some number n times.
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+20
6
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1, 2, 4, 6, 12, 12, 18, 20, 30, 30, 40, 40, 60, 60, 72, 72, 90, 90, 120, 120, 140, 140, 168, 168, 180, 180, 210, 210, 252, 252, 280, 280, 315, 315, 336, 336, 360, 360, 420, 420, 504, 504, 560, 560, 630, 630, 672, 672, 720, 720, 792, 792, 840, 840, 924, 924, 990
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OFFSET
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1,2
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LINKS
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Table of n, a(n) for n=1..57.
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EXAMPLE
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a(7)=18 because the 18 X 18 multiplication table is the smallest to contain a product of frequency 7 (namely the number A062856[7]=36).
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CROSSREFS
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The least such number is A062856[n].
Cf. A027424, A062851, A062854, A062855, A062856, A062858, A062859.
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KEYWORD
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nonn
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AUTHOR
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Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001
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EXTENSIONS
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More terms from Don Reble (djr(AT)nk.ca), Nov 08 2001
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STATUS
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approved
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1, 2, 4, 8, 6, 8, 6, 14, 14, 16, 10, 14, 12, 26, 26, 20, 18, 12, 26, 37, 31, 27, 50, 42, 38, 36, 32, 30, 41, 27, 23, 27, 25, 15, 16, 22, 16, 26, 20, 14, 29, 19, 34, 30, 40, 40, 28, 24, 22, 18, 12, 10, 20, 20, 14, 8, 16, 10, 26, 41, 31, 17, 13, 11, 45, 31, 47
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OFFSET
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2,2
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COMMENTS
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Conjecture: a(n) = o(prime(n)), as n goes to infinity.
If the conjecture is true, then A242720(n) ~ prime(n)^2. Indeed, A242720(n) >= prime(n)^2 + 2*prime(n) + 3; on the other hand, by the conjecture, we have A242720(n)/prime(n) <= a(n) + 1 + prime(n) = prime(n)*(1+o(1)).
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LINKS
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Peter J. C. Moses, Table of n, a(n) for n = 2..2501
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CROSSREFS
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Cf. A242719, A242720, A246748, A246819.
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KEYWORD
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nonn
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AUTHOR
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Vladimir Shevelev, Sep 04 2014
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EXTENSIONS
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More terms from Peter J. C. Moses, Sep 04 2014
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STATUS
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approved
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A078565
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Number of zeros in the binary expansion of n!.
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+20
3
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0, 0, 1, 1, 3, 3, 6, 7, 10, 13, 11, 19, 17, 21, 25, 23, 27, 27, 30, 40, 40, 41, 42, 44, 51, 54, 54, 56, 56, 63, 60, 71, 76, 77, 77, 77, 88, 86, 90, 90, 97, 99, 106, 105, 107, 117, 115, 117, 114, 122, 126, 130, 138, 138, 151, 144, 146, 157, 160, 158, 160, 176
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OFFSET
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0,5
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = Sum_COMPONENTS(COMPLEMENT(BINARY_VECTOR(n!)))
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EXAMPLE
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a(4) = 3 because 4!=24)D=11000)B and it has 3 zero digits.
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MATHEMATICA
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Table[Count[IntegerDigits[n!, 2], 0], {n, 0, 100}] (* T. D. Noe, Apr 10 2012 *)
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PROG
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(PARI) for(n=1, 300, b=binary(n!); print1(sum(k=1, length(b), if(b[k], 0, 1))", "))
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KEYWORD
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nonn
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AUTHOR
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Jose R. Brox (tautocrona(AT)terra.es), Jan 26 2003
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STATUS
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approved
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A222293
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Conjectured total number of times that k+n appears in the Collatz (3x+1) sequence of k for k = 1, 2, 3,...
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+20
2
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37, 30, 34, 30, 31, 29, 28, 38, 42, 32, 40, 40, 49, 30, 40, 40, 54, 45, 46, 40, 49, 44, 41, 48, 47, 54, 48, 41, 50, 44, 54, 45, 49, 60, 53, 47, 54, 50, 56, 44, 48, 50, 54, 47, 54, 38, 56, 47, 60, 48, 63, 48, 47, 45, 56, 53, 49, 49, 62, 52, 50, 54, 53, 52, 49, 46
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OFFSET
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1,1
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
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EXAMPLE
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a(1) = 37 because k+1 occurs in the Collatz sequence of k for the 37 values in A070993.
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MATHEMATICA
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Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 75; t = Table[0, {nn}]; lastChange = 10; k = 0; While[k < 2*lastChange, k++; c = Collatz[k]; d = Intersection[Range[nn], c - k]; If[Length[d] > 0, lastChange = k; t[[d]]++]]
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CROSSREFS
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Cf. A070993, A221213 (k-n).
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe, Feb 22 2013
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STATUS
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approved
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A029256
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Expansion of 1/((1-x^3)*(1-x^4)*(1-x^6)*(1-x^12)).
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+20
1
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1, 0, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 5, 2, 2, 5, 5, 2, 8, 5, 5, 8, 8, 5, 14, 8, 8, 14, 14, 8, 20, 14, 14, 20, 20, 14, 30, 20, 20, 30, 30, 20, 40, 30, 30, 40, 40, 30, 55, 40, 40, 55, 55, 40, 70, 55, 55, 70, 70, 55, 91, 70, 70, 91
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OFFSET
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0,7
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COMMENTS
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Number of partitions of n into parts 3, 4, 6, and 12. - Vincenzo Librandi, Jun 03 2014
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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MATHEMATICA
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CoefficientList[Series[1/((1 - x^3) (1 - x^4) (1 - x^6) (1 - x^12)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 03 2014 *)
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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A091657
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First occurrence of 1..n as terms in the continued fraction for Pi.
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+20
1
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4, 9, 9, 30, 40, 40, 40, 44, 130, 130, 276, 276, 276, 276, 276, 276, 647, 647, 647, 647, 647, 647, 647, 647, 791, 791, 791, 791, 791, 791, 878, 878, 878, 878, 1008, 1008, 1008, 3041, 3041, 3041, 3041, 3041, 3041, 3041, 3041, 3200, 3200, 3200, 3200, 3200, 3200
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OFFSET
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1,1
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LINKS
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Table of n, a(n) for n=1..51.
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MATHEMATICA
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f[n_] := Block[{k = 1}, While[ StringPosition[ ToString[ Union[ ContinuedFraction[Pi, k]]], StringDrop[ ToString[ Table[i, {i, n}]], -1]] == {}, k++ ]; k]; Table[ f[n], {n, 1, 51}]
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CROSSREFS
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Cf. A001203, A032523.
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KEYWORD
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cofr,nonn
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AUTHOR
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Robert G. Wilson v, Jan 26 2004
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STATUS
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approved
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0, 10, 10, 20, 20, 30, 30, 40, 40, 50, 50, 60, 60, 70, 70, 80, 80, 90, 90, 100, 100, 110, 110, 120, 120, 130, 130, 140, 140, 150, 150, 160, 160, 170, 170, 180, 180, 190, 190, 200, 200, 210, 210, 220, 220, 230, 230, 240, 240, 250, 250, 260, 260, 270, 270, 280
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OFFSET
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1,2
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
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FORMULA
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a(n) = 10*n-a(n-1)-10, with n>1, a(1)=0.
a(n) = 10*floor(n/2) = A168437(n) - 3. - Rick L. Shepherd, Jun 17 2010
G.f.: 10*x^2/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 19 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 19 2013
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MAPLE
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A168461:=n->10*floor(n/2); seq(A168461(n), n=1..100); # Wesley Ivan Hurt, Nov 25 2013
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MATHEMATICA
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Table[10 Floor[n/2], {n, 70}] (* or *) CoefficientList[Series[10 x/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
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PROG
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(MAGMA) [10*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013
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CROSSREFS
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Bisections are A008592 and (A008592 MINUS {0}). - Rick L. Shepherd, Jun 17 2010
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KEYWORD
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nonn,easy
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AUTHOR
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Vincenzo Librandi, Nov 26 2009
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EXTENSIONS
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Edited by Rick L. Shepherd, Jun 17 2010
Definition rewritten, using Shepherd's formula, by Vincenzo Librandi, Sep 19 2013
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STATUS
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approved
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 70, 70, 70, 70, 70
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OFFSET
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0,11
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COMMENTS
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Equivalently, set to zero the last (decimal) digit of n, i.e., subtract (n mod 10). The digit-wise addition defined in A169918 consists in multiplying the digits and taking this product modulo 10 for each digit, and "Blanks are ignored". Since 0 has only one digit, only the last digit of n is set to zero in that way. - M. F. Hasler, Mar 25 2015
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LINKS
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Table of n, a(n) for n=0..74.
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FORMULA
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a(n) = [n/10]*10. - M. F. Hasler, Mar 25 2015
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PROG
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(PARI) A169932(n) = n\10*10 \\ M. F. Hasler, Mar 25 2015
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CROSSREFS
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Cf. A169918.
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KEYWORD
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nonn,base
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AUTHOR
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David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010
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STATUS
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approved
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A180572
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Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the circular ladder P_2 X C_n (called also prism), where P_2 is the path graph on 2 nodes and C_n is the cycle graph on n nodes.
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+20
1
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9, 6, 12, 12, 4, 15, 20, 10, 18, 24, 18, 6, 21, 28, 28, 14, 24, 32, 32, 24, 8, 27, 36, 36, 36, 18, 30, 40, 40, 40, 30, 10, 33, 44, 44, 44, 44, 22, 36, 48, 48, 48, 48, 36, 12, 39, 52, 52, 52, 52, 52, 26, 42, 56, 56, 56, 56, 56, 42, 14, 45, 60, 60, 60, 60, 60, 60, 30, 48, 64, 64
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OFFSET
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3,1
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COMMENTS
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Row n contains 1+floor(n/2) entries.
Sum of entries in row n = n(2n-1)=A000384(n).
T(n,1)=3n = number of edges in the corresponding graph.
Sum(k*T(n,k),k>=1) = A138179(n).
The generating polynomial of row n (i.e. the Wiener polynomial of the circular ladder of order n) has been obtained from the Wiener polynomial of the cycle C_n (see the Sagan et al. paper) and by determining the distribution of the distances from the nodes of one cycle to the nodes of the other cycle. They can be derived also from the Doslic paper (Corollary 11 and Lemma 1).
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REFERENCES
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J. Gross and J. Yellen, Graph Theory and its Applications, CRC, Boca Raton, 1999 (p. 14).
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LINKS
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Table of n, a(n) for n=3..73.
T. Doslic, Vertex-weighted Wiener polynomials for composite graphs, Ars Mathematica Contemporanea, 1, 2008, 66-80.
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
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FORMULA
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The generating polynomial for row 2n+1 is (2n+1)(3t+t^2-2t^{n+1}-2t^{n+2})/(1-t) and for row 2n it is 2n(3t+t^2-t^n-2t^{n+1}-t^{n+2})/(1-t) (these are also the Wiener polynomials of the corresponding circular ladders).
The bivariate g.f. G=G(t,z) appears in the Maple program.
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EXAMPLE
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T(3,2)=6 because in P_2 X C_3 there are six unordered pairs of nodes at distance 2 (from the vertices of the outer triangle to the "opposite" vertices of the inner triangle).
Triangle starts:
9,6;
12,12,4;
15,20,10;
18,24,18,6;
21,28,28,14;
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MAPLE
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G := t*z^3*(9+6*t-6*z+4*t^2*z-16*t*z^2-10*t^2*z^2+8*t*z^3 +2*t^2*z^3 -2*t^3*z^3 +7*t^2*z^4+4*t^3*z^4-4*t^2*z^5-2*t^3*z^5) / ((1-z)^2*(1-t*z^2)^2): Gser := simplify(series(G, z = 0, 19)): for n from 3 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 3 to 16 do seq(coeff(P[n], t, j), j = 1 .. 1+floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A000384, A138179.
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Sep 16 2010
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STATUS
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approved
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A180573
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Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the sun graph on 2n nodes. The sun graph on 2n nodes is obtained by attaching n pendant edges to the cycle graph on n nodes.
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+20
1
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6, 6, 3, 8, 10, 8, 2, 10, 15, 15, 5, 12, 18, 21, 12, 3, 14, 21, 28, 21, 7, 16, 24, 32, 28, 16, 4, 18, 27, 36, 36, 27, 9, 20, 30, 40, 40, 35, 20, 5, 22, 33, 44, 44, 44, 33, 11, 24, 36, 48, 48, 48, 42, 24, 6, 26, 39, 52, 52, 52, 52, 39, 13, 28, 42, 56, 56, 56, 56, 49, 28, 7, 30, 45, 60
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OFFSET
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3,1
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COMMENTS
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Number of entries in row n = 2 + floor(n/2).
Sum of entries in row n = n(2n-1)=A000384(n).
Sum(k*T(n,k),k>=1) = A180574(n).
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REFERENCES
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B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
D. Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., 235 (2001), 237-244.
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LINKS
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Table of n, a(n) for n=3..77.
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FORMULA
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The generating polynomial of row 2n is 2nt-nt^n*(1+t)^2+2nt(1+t)^2*sum(t^j, j=0..n-1); the generating polynomial of row 2n+1 is (2n+1)t[1+(1+t)^2*sum(t^j,j=0..n-1)]; these are the Wiener polynomials of the corresponding graphs.
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EXAMPLE
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Triangle starts:
6,6,3;
8,10,8,2;
10,15,15,5;
12,18,21,12,3;
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MAPLE
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P := proc (n) if `mod`(n, 2) = 0 then sort(expand(n*t*(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-1))+n*t-(1/2)*n*t^((1/2)*n)*(1+t)^2)) else sort(expand(n*t*(1+(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-3/2))))) end if end proc; for n from 3 to 15 do P(n) end do: for n from 3 to 15 do seq(coeff(P(n), t, i), i = 1 .. 2+floor((1/2)*n)) end do; # yields sequence in trianguklar form
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CROSSREFS
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Cf. A000384, A180574
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Sep 19 2010
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STATUS
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approved
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