A001609 -id:A001609 - cp's OEIS frontend

A058365 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 8 sites wide.

1, 1, 1, 1, 1, 1, 1, 9, 10, 11, 12, 13, 14, 15, 16, 25, 35, 46, 58, 71, 85, 100, 116, 141, 176, 222, 280, 351, 436, 536, 652, 793, 969, 1191, 1471, 1822, 2258, 2794, 3446, 4239, 5208, 6399, 7870, 9692, 11950, 14744, 18190, 22429, 27637, 34036, 41906

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(8) = 9 because there is one way to put zero molecule to the necklace and 8 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-7*i, i-1)/i, i=1..n/8). a(n) = a(n-1) + a(n-8), a(n) = 1 for n = 1..7, a(8) = 9. generating function = (x+8*x^8)/(1-x-x^8).

A058366 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 7 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 9, 10, 11, 12, 13, 14, 22, 31, 41, 52, 64, 77, 91, 113, 144, 185, 237, 301, 378, 469, 582, 726, 911, 1148, 1449, 1827, 2296, 2878, 3604, 4515, 5663, 7112, 8939, 11235, 14113, 17717, 22232, 27895, 35007, 43946, 55181, 69294, 87011

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(7) = 8 because there is one way to put zero molecule to the necklace and 7 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-6*i, i-1)/i, i=1..n/7). a(n) = a(n-1) + a(n-7), a(n) = 1 for n = 1..6, a(7) = 8. generating function = (x+7*x^7)/(1-x-x^7).

A058367 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 6 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 8, 9, 10, 11, 12, 19, 27, 36, 46, 57, 69, 88, 115, 151, 197, 254, 323, 411, 526, 677, 874, 1128, 1451, 1862, 2388, 3065, 3939, 5067, 6518, 8380, 10768, 13833, 17772, 22839, 29357, 37737, 48505, 62338, 80110, 102949, 132306, 170043, 218548

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

			a(6) = 7 because there is one way to put zero molecule to the necklace and 6 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-5*i, i-1)/i, i=1..n/6). a(n) = a(n-1) + a(n-6), a(n) = 1 for n = 1..5, a(6) = 7. generating function = (x+6*x^6)/(1-x-x^6).

A049194 Number of digits in n-th term of A001387.

Original entry on oeis.org

1, 2, 3, 6, 8, 12, 18, 27, 39, 58, 85, 125, 183, 269, 394, 578, 847, 1242, 1820, 2668, 3910, 5731, 8399, 12310, 18041, 26441, 38751, 56793, 83234, 121986, 178779, 262014, 384000, 562780, 824794, 1208795, 1771575, 2596370, 3805165, 5576741

Offset: 1

Olivier Gérard

Peter A. Hendriks, "A binary variant of Conway's audioactive sequence", lecture at 1192nd meeting of WWWW, Groningen, The Netherlands (Jul 15 1999).

Edward Krogius, Table of n, a(n) for n = 1..1000
T. Sillke, The binary form of Conway's sequence
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1).

Cf. A001387, A001609.

CoefficientList[Series[(1+x+x^3-x^4-x^5)/(1-x-x^2+x^5),{x,0,50}],x] (* Peter J. C. Moses, Jun 21 2013 *)

a(n) = if (n==3, 3, if (n==4, 6, floor((8/9 + (1/18)*(748 - 36*sqrt(93))^(1/3) + (1/18)*(748 + 36*sqrt(93))^(1/3)) * (1/3 + (1/6)*(116 - 12*sqrt(93))^(1/3) + (1/6)*(116 + 12*sqrt(93))^(1/3))^(n-1)))) \\ Michel Marcus, Mar 04 2013

a(n) = my(v=vector(n), u=[1,2,3,6]); if(n<=4, u[n], for(i=1, 4, v[i]=u[i]); for(i=5, n, v[i]=v[i-1]+v[i-3]+!(i%2)); v[n]) \\ Jianing Song, Apr 28 2019

a(n) = (8/9 + (1/18)*(748 - 36*sqrt(93))^(1/3) + (1/18)*(748 + 36*sqrt(93))^(1/3)) * (1/3 + (1/6)*(116 - 12*sqrt(93))^(1/3) + (1/6)*(116 + 12*sqrt(93))^(1/3))^(n-1).
The number of digits is equal to c*l^n rounded down to the nearest integer, where c and l are the real roots of 3x^3 - 8x^2 + 5x - 1 and x^3 - x^2 - 1 respectively, for all n except n = 2 and n = 3.
From Jianing Song, Apr 28 2019: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) for n >= 7. [Derived from the T. Sillke link above.]
a(n) = a(n-1) + a(n-3) if n is odd, a(n-1) + a(n-3) + 1 if n is even, n >= 5 (this does not hold for n = 4).
Limit_{n->oo} a(n)/A001609(n) = c, where c = 1.276742... is the unique real root of 3x^3 - 4x^2 + x - 1. (End)

More terms and formulas supplied by Gerton Lunter (gerton(AT)math.rug.nl)

A112455 a(n) = -a(n-2) - a(n-3).

Original entry on oeis.org

-3, 0, 2, 3, -2, -5, -1, 7, 6, -6, -13, 0, 19, 13, -19, -32, 6, 51, 26, -57, -77, 31, 134, 46, -165, -180, 119, 345, 61, -464, -406, 403, 870, 3, -1273, -873, 1270, 2146, -397, -3416, -1749, 3813, 5165, -2064, -8978, -3101, 11042, 12079, -7941

Offset: 0

Anthony C Robin, Dec 13 2005

This sequence resembles the Perrin sequence, A001608. Like many such sequences with a(1)=0, any prime p divides a(p). The first pseudoprime (composite n divides a(n)) is 121.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,-1,-1).

Cf. A001608, A112458.

a:=[-3,0,2];; for n in [4..60] do a[n]:=-a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 19 2019

I:=[-3,0,2]; [n le 3 select I[n] else -Self(n-2) -Self(n-3): n in [1..60]]; // G. C. Greubel, May 19 2019

A112455 := proc(n)
    option remember ;
    if n <= 2 then
        op(n+1,[-3,0,2]) ;
    else
        -procname(n-2)-procname(n-3) ;
    end if;
end proc: # R. J. Mathar, Feb 18 2024

Table[ -Tr[MatrixPower[{{0, 0, -1}, {1, 0, -1}, {0, 1, 0}}, n]], {n, 1, 60}] (* Artur Jasinski, Jan 10 2007 *)
LinearRecurrence[{0,-1,-1}, {-3,0,2}, 60] (* G. C. Greubel, May 19 2019 *)

Vec(-(3+x^2)/(1+x^2+x^3)+O(x^60)) \\ Charles R Greathouse IV, May 15 2013

(-(3+x^2)/(1+x^2+x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, May 19 2019

a(n) = - trace({{0, 0, -1}, {1, 0, -1}, {0, 1, 0}})^n. - Artur Jasinski, Jan 10 2007
From R. J. Mathar, Oct 24 2009: (Start)
G.f.: -(3+x^2)/(1+x^2+x^3).
a(n) = -3*A077962(n) - A077962(n-2). (End)
a(n) = (-1)^(n+1)*(A001609(n)^2 - A001609(2*n))/2. - Greg Dresden, Apr 14 2023

Edited by Don Reble, Jan 25 2006

A001645 A Fielder sequence.

Original entry on oeis.org

1, 3, 7, 11, 26, 45, 85, 163, 304, 578, 1090, 2057, 3888, 7339, 13862, 26179, 49437, 93366, 176321, 332986, 628852, 1187596, 2242800, 4235569, 7998951, 15106172, 28528288, 53876211, 101746240, 192149690, 362878313, 685302531, 1294206745, 2444133829

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Wikipedia, Companion matrix.
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,1).

Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649.

I:=[1,3,7,11,26]; [n le 5 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017

A001645:=-(1+2*z+3*z**2+5*z**4)/(-1+z+z**2+z**3+z**5); [Conjectured by Simon Plouffe in his 1992 dissertation.]

LinearRecurrence[{1, 1, 1, 0, 1}, {1, 3, 7, 11, 26}, 50] (* T. D. Noe, Aug 09 2012 *)
CoefficientList[Series[x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

a(n)=if(n<0,0,polcoeff(x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5)+x*O(x^n),n))

G.f.: x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5).

a(n) = trace(M^n), where M = [0, 0, 0, 0, 1; 1, 0, 0, 0, 0; 0, 1, 0, 0, 1; 0, 0, 1, 0, 1; 0, 0, 0, 1, 1] is the 5 x 5 companion matrix to the monic polynomial x^5 - x^4 - x^3 - x^2 - 1. It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Jan 09 2023

A080839 Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.

Original entry on oeis.org

1, 1, 1, 2, 6, 27, 180, 1786, 26094, 559127, 17535396, 804131875, 53833201737

Offset: 1

John W. Layman, Mar 28 2003

From T. D. Noe, Feb 05 2007: (Start)
The slowest-growing sequence of length n is 1,2,4,6,...,2(n-1). The fastest-growing sequence is 1,2,4,8,...,2^(n-1).
The ratio a(n+1)a(n-1)/a(n)^2 appears to converge to a constant near 1.46, which is the approximate growth rate of A001609. Are the sequences related?
(End)
Also, a(n) is the number of (not necessarily increasing) positive integer sequences of length n-1 with Gilbreath transform (1, ..., 1). - Pontus von Brömssen, May 13 2023

			The table below shows that {1,2,4,6,10} is one of the 6 sequences of length 5 that satisfy the stated condition:
   1
   2 1
   4 2 1
   6 2 0 1
  10 4 2 2 1

Cf. A001609, A036262, A363002, A363003, A363004, A363005.

Cf. also A136465, the total number of increasing sequences with the same maximum length. [From Charles R Greathouse IV, Aug 08 2010]

More terms from T. D. Noe, Feb 05 2007

Added "positive" to definition. - N. J. A. Sloane, May 13 2023

A049064 Describe the previous term in binary (method A - initial term is 0).

Original entry on oeis.org

0, 10, 1110, 11110, 100110, 1110010110, 111100111010110, 100110011110111010110, 1110010110010011011110111010110, 1111001110101100111001011010011011110111010110, 1001100111101110101100111100111010110111001011010011011110111010110

Offset: 1

Olivier de Mouzon

Method A = 'frequency' (in binary mode) followed by 'digit'-indication.

The number of digits of a(n) is A001609(n) except for n = 2. See the link from T. Sillke below. - Jianing Song, Mar 16 2019

			E.g., the term after 11110 is obtained by saying "four (i.e., 100 in binary mode) 1, one 0", which gives 100110.

Kade Robertson, Table of n, a(n) for n = 1..18
Kade Robertson, Table of n, a(n) for n = 1..31
T. Sillke, The binary form of Conway's sequence

Cf. A001387 (initial term is 1), A001391, A001609 (number of digits), A259710 (written in decimal).

Decimal look-and-say sequences: A005150, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154.

a(n) = A001391(n-1), n > 1. - R. J. Mathar, Oct 15 2008

Edited by Charles R Greathouse IV, Apr 06 2010
a(11) from Kade Robertson, Jun 24 2015
Offset corrected by Jianing Song, Mar 16 2019

A001641 A Fielder sequence: a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

1, 3, 4, 11, 16, 30, 50, 91, 157, 278, 485, 854, 1496, 2628, 4609, 8091, 14196, 24915, 43720, 76726, 134642, 236283, 414645, 727654, 1276941, 2240878, 3932464, 6900996, 12110401, 21252275, 37295140, 65448411, 114853952, 201554638, 353703730, 620706779

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Wikipedia, Companion matrix.
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no.
6, 2006, pp. 840-855.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1).

Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649, A060945.

I:=[1,3,4,11]; [n le 4 select I[n] else Self(n-1) + Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 09 2018

A001641:=-(1+2*z+4*z**3)/(z+1)/(z**3-z**2+2*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{1, 1, 0, 1}, {1, 3, 4, 11}, 50] (* T. D. Noe, Aug 09 2012 *)

a(n):=(sum(sum(binomial(j,n-4*k+3*j)*binomial(k,j),j,floor((4*k-n)/3),floor((4*k-n)/2))/k,k,1,n))*n; /* Vladimir Kruchinin, May 25 2011 */

a(n)=if(n<0,0,polcoeff(x*(1+2*x+4*x^3)/(1-x-x^2-x^4)+x*O(x^n),n))

G.f.: x*(1+2*x+4*x^3)/(1-x-x^2-x^4).
a(n) = n*Sum_{k=1..n} Sum_{j=floor((4*k-n)/3)..floor((4*k-n)/2)} binomial(j,n-4*k+3*j)*binomial(k,j)/k. - Vladimir Kruchinin, May 25 2011
a(n) = Trace(M^n), where M = [0, 0, 0, 1; 1, 0, 0, 0; 0, 1, 0, 1; 0, 0, 1, 1] is the companion matrix to the monic polynomial x^4 - x^3 - x^2 - 1. It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Dec 31 2022

A293343 Dimensions of the squares of a certain family of error-correcting codes.

Original entry on oeis.org

7, 11, 16, 37, 71, 123, 232, 441, 804, 1475

Offset: 3

Eric M. Schmidt, Oct 07 2017

The codes themselves have dimensions A001609(n). (Take s=3 in Theorem 6.1 of [Cascudo].) - Eric M. Schmidt, Oct 12 2017

Ignacio Cascudo, On squares of cyclic codes, arXiv:1703.01267 [cs.IT], 2017. See Table 1.

Cf. A293344, A293345.

cp's OEIS Frontend

A058365 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 8 sites wide.

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A058366 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 7 sites wide.

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A058367 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 6 sites wide.

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A049194 Number of digits in n-th term of A001387.

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A112455 a(n) = -a(n-2) - a(n-3).

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GAP

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A001645 A Fielder sequence.

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Magma

Maple

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A080839 Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.

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A049064 Describe the previous term in binary (method A - initial term is 0).

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