A004001 -id:A004001 - cp's OEIS frontend

A005708 a(n) = a(n-1) + a(n-6), with a(i) = 1 for i = 0..5.

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 43, 55, 71, 92, 119, 153, 196, 251, 322, 414, 533, 686, 882, 1133, 1455, 1869, 2402, 3088, 3970, 5103, 6558, 8427, 10829, 13917, 17887, 22990, 29548, 37975, 48804, 62721, 80608, 103598, 133146, 171121, 219925, 282646

Offset: 0

N. J. A. Sloane

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 = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = sum_{i=0..n/m} binomial(n-(m-1)*i, i). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
For n>=6, a(n-6) = number of compositions of n in which each part is >=6. - Milan Janjic, Jun 28 2010
Number of compositions of n into parts 1 and 6. - Joerg Arndt, Jun 24 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=6, 2*a(n-6) equals the number of 2-colored compositions of n with all parts >=6, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+5) equals the number of binary words of length n having at least 5 zeros between every two successive ones. - Milan Janjic, Feb 07 2015
Number of tilings of a 6 X n rectangle with 6 X 1 hexominoes. - M. Poyraz Torcuk, Mar 26 2022

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=0..500
Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See pp. 18, 22.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 10.
Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,5,1).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
S. Kitaev, Independent sets on path-schemes, JIS 9 (2006) # 06.2.2 G(x) for M={1,2,3,4,5} gives seq. shifted 5 places left
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO] (2016), Section 4.5
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 379
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1).

Cf. A000045, A000079, A000930, A003269, A003520, A005709, A005710, A005711.

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 5)}, unlabeled]: seq(count(SeqSetU, size=j), j=6..59); # Zerinvary Lajos, Oct 10 2006
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 5)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=5..58); # Zerinvary Lajos, Mar 26 2008
M := Matrix(6, (i,j)-> if j=1 and member(i,[1,6]) then 1 elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1,1]; seq(a(n), n=0..60); # Alois P. Heinz, Jul 27 2008

LinearRecurrence[{1, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

x='x+O('x^66); Vec(x/(1-(x+x^6))) /* Joerg Arndt, Jun 25 2011 */

G.f.: 1/(1-x-x^6). - Simon Plouffe in his 1992 dissertation
a(n) = term (1,1) in the 6 X 6 matrix [1,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,0,0,0,0,0]^n. - Alois P. Heinz, Jul 27 2008
For positive integers n and k such that k <= n <= 6*k and 5 divides n-k, define c(n,k) = binomial(k,(n-k)/5), and c(n,k)=0, otherwise. Then, for n>= 1, a(n) = sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([1/6-n/6, 1/3-n/6, 1/2-n/6, 2/3-n/6, 5/6-n/6, -n/6], [1/5-n/5, 2/5-n/5, 3/5- n/5, 4/5-n/5, -n/5], -6^6/5^5) for n>=25. - Peter Luschny, Sep 19 2014

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

A005710 a(n) = a(n-1) + a(n-8), with a(i) = 1 for i = 0..7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078, 22268, 27428, 33786, 41623

Offset: 0

N. J. A. Sloane

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 = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = Sum_{i=0..n/m} binomial(n-(m-1)*i, i). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
For n >= 8, a(n-8) = number of compositions of n in which each part is >= 8. - Milan Janjic, Jun 28 2010
Number of compositions of n into parts 1 and 8. - Joerg Arndt, Jun 24 2011
a(n+7) equals the number of binary words of length n having at least 7 zeros between every two successive ones. - Milan Janjic, Feb 09 2015

P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194.
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 = 0..500
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 18.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 10.
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
A. O. Munagi, Euler-type identities for integer compositions via zig-zag graphs, Integers 12 (2012), Paper No. A60, 10 pp.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
Djamila Oudrar and Maurice Pouzet, Ordered structures with no finite monomorphic decomposition. Application to the profile of hereditary classes, arXiv:2312.05913 [math.CO], 2023. See p. 18.
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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 381
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1).

Cf. A000045, A000079, A000930, A003269, A003520, A005708, A005709, A005711.

A005710:=-1/(-1+z+z**8); # Simon Plouffe in his 1992 dissertation.
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 7)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=7..62); # Zerinvary Lajos, Mar 26 2008
M := Matrix(8, (i,j)-> if j=1 and member(i,[1,8]) then 1 elif (i=j-1) then 1 else 0 fi); a := n -> (M^(n))[1,1]; seq(a(n), n=0..55); # Alois P. Heinz, Jul 27 2008

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
CoefficientList[Series[1/(1-x-x^8),{x,0,60}],x] (* Harvey P. Dale, Jun 14 2016 *)

x='x+O('x^66); Vec(x/(1-(x+x^8))) /* Joerg Arndt, Jun 25 2011 */

G.f.: 1/(1-x-x^8).
For positive integers n and k such that k <= n <= 8*k, and 7 divides n-k, define c(n,k) = binomial(k,(n-k)/7), and c(n,k) = 0, otherwise. Then, for n >= 1, a(n-1) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([1/8-n/8, 1/4-n/8, 3/8-n/8, 1/2-n/8, 5/8-n/8, 3/4-n/8, 7/8-n/8, -n/8], [1/7-n/7, 2/7-n/7, 3/7-n/7, 4/7-n/7, 5/7-n/7, 6/7-n/7, -n/7], -8^8/7^7) for n >= 49. - Peter Luschny, Sep 19 2014

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

A002421 Expansion of (1-4*x)^(3/2) in powers of x.

Original entry on oeis.org

1, -6, 6, 4, 6, 12, 28, 72, 198, 572, 1716, 5304, 16796, 54264, 178296, 594320, 2005830, 6843420, 23571780, 81880920, 286583220, 1009864680, 3580429320, 12765008880, 45741281820, 164668614552, 595340375688, 2160865067312, 7871722745208, 28772503827312

Offset: 0

N. J. A. Sloane

Terms that are not divisible by 12 have indices in A019469. - Ralf Stephan, Aug 26 2004
From Ralf Steiner, Apr 06 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/8^k = 2F1(-3/2,-3/2,1,2).
Sum_{k>=0} a(k) / 2^k = -i. (End)

			G.f. = 1 - 6*x + 6*x^2 + 4*x^3 + 6*x^4 + 12*x^5 + 28*x^6 + 72*x^7 + 198*x^8 + 572*x^9 + ...

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
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. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Notes on A984 and A2420-A2424

Cf. A007054, A004001, A002420, A002422, A002423, A002424.

Cf. A000257, A001622, A071721, A071724, A085687.

Concatenation([1], List([1..40], n-> 12*Factorial(2*n-4) /( Factorial(n)*Factorial(n-2)) )) # G. C. Greubel, Jul 03 2019

[1,-6] cat [12*Catalan(n-2)/n: n in [2..30]]; // Vincenzo Librandi, Jun 11 2012

A002421 := n -> 3*4^(n-1)*GAMMA(-3/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002421(n), n=0..29); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^(3/2),{x,0,40}],x] (* Vincenzo Librandi, Jun 11 2012 *)
a[n_]:= Binomial[ 3/2, n] (-4)^n; (* Michael Somos, Dec 04 2013 *)
a[n_]:= SeriesCoefficient[(1-4x)^(3/2), {x, 0, n}]; (* Michael Somos, Dec 04 2013 *)

{a(n) = binomial( 3/2, n) * (-4)^n}; /* Michael Somos, Dec 04 2013 */

{a(n) = if( n<0, 0, polcoeff( (1 - 4*x + x * O(x^n))^(3/2), n))}; /* Michael Somos, Dec 04 2013 */

((1-4*x)^(3/2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m)*K_m(4), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
a(n) ~ (3/4)*Pi^(-1/2)*n^(-5/2)*2^(2*n)*(1 + 15/8*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
From Ralf Stephan, Mar 11 2004: (Start)
a(n) = 12*(2*n-4)! /(n!*(n-2)!), n > 1.
a(n) = 12*Cat(n-2)/n = 2(Cat(n-1) - 4*Cat(n-2)), in terms of Catalan numbers (A000108).
Terms that are not divisible by 12 have indices in A019469. (End)
Let rho(x)=(1/Pi)*(x*(4-x))^(3/2), then for n >= 4, a(n) = Integral_{x=0..4} (x^(n-4) *rho(x)) dx. - Groux Roland, Mar 16 2011
G.f.: (1-4*x)^(3/2) = 1 - 6*x + 12*x^2/(G(0) + 2*x); G(k) = (4*x+1)*k-2*x+2-2*x*(k+2)*(2*k+1)/G(k+1); for -1/4 <= x < 1/4, otherwise G(0) = 2*x; (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
G.f.: 1/G(0) where G(k) = 1 + 4*x*(2*k+1)/(1 - 1/(1 + (2*k+2)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
G.f.: G(0)/2, where G(k) = 2 + 2*x*(2*k-3)*G(k+1)/(k+1). - Sergei N. Gladkovskii, Jun 06 2013 [Edited by Michael Somos, Dec 04 2013]
0 = a(n+2) * (a(n+1) - 14*a(n)) + a(n+1) * (6*a(n+1) + 16*a(n)) for all n in Z. - Michael Somos, Dec 04 2013
A232546(n) = 3^n * a(n). - Michael Somos, Dec 04 2013
G.f.: hypergeometric1F0(-3/2;;4*x). - R. J. Mathar, Aug 09 2015
a(n) = 3*4^(n-1)*Gamma(-3/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
From Ralf Steiner, Apr 06 2017: (Start)
Sum_{k>=0} a(k)/4^k = 0.
Sum_{k>=0} a(k)^2/16^k = 32/(3*Pi).
Sum_{k>=0} a(k)^2*(k/8)/16^k = 1/Pi.
Sum_{k>=0} a(k)^2*(-k/24+1/8)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(k-1/4)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(2k-2)/16^k = 1/Pi.(End)
D-finite with recurrence: n*a(n) +2*(-2*n+5)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/3 + 10*Pi/(81*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 92/75 - 4*sqrt(5)*log(phi)/125, where phi is the golden ratio (A001622). (End)

A265903 Square array read by descending antidiagonals: A(1,k) = A188163(k), and for n > 1, A(n,k) = A087686(1+A(n-1,k)).

Original entry on oeis.org

1, 3, 2, 5, 7, 4, 6, 12, 15, 8, 9, 14, 27, 31, 16, 10, 21, 30, 58, 63, 32, 11, 24, 48, 62, 121, 127, 64, 13, 26, 54, 106, 126, 248, 255, 128, 17, 29, 57, 116, 227, 254, 503, 511, 256, 18, 38, 61, 120, 242, 475, 510, 1014, 1023, 512, 19, 42, 86, 125, 247, 496, 978, 1022, 2037, 2047, 1024, 20, 45, 96, 192, 253, 502, 1006, 1992, 2046, 4084, 4095, 2048

Offset: 1

Antti Karttunen, Dec 18 2015

Square array A(n,k) [where n is row and k is column] is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

For n >= 3, each row n lists the numbers that appear n times in A004001. See also A051135.

			The top left corner of the array:
     1,    3,    5,    6,     9,    10,    11,    13,    17,    18,    19
     2,    7,   12,   14,    21,    24,    26,    29,    38,    42,    45
     4,   15,   27,   30,    48,    54,    57,    61,    86,    96,   102
     8,   31,   58,   62,   106,   116,   120,   125,   192,   212,   222
    16,   63,  121,  126,   227,   242,   247,   253,   419,   454,   469
    32,  127,  248,  254,   475,   496,   502,   509,   894,   950,   971
    64,  255,  503,  510,   978,  1006,  1013,  1021,  1872,  1956,  1984
   128,  511, 1014, 1022,  1992,  2028,  2036,  2045,  3864,  3984,  4020
   256, 1023, 2037, 2046,  4029,  4074,  4083,  4093,  7893,  8058,  8103
   512, 2047, 4084, 4094,  8113,  8168,  8178,  8189, 16006, 16226, 16281
  1024, 4095, 8179, 8190, 16292, 16358, 16369, 16381, 32298, 32584, 32650
  ...

Inverse permutation: A267104.
Transpose: A265901.
Row 1: A188163.
Row 2: A266109.
Row 3: A267103.
For the known and suspected columns, see the rows listed for transposed array A265901.
Cf. A004001, A051135, A087686.
Cf. A265900 (main diagonal), A265909 (submain diagonal).
Cf. A162598 (column index of n in array), A265332 (row index of n in array).
Cf. also permutations A267111, A267112.

(define (A265903 n) (A265903bi (A002260 n) (A004736 n)))
(define (A265903bi row col) (if (= 1 row) (A188163 col) (A087686 (+ 1 (A265903bi (- row 1) col)))))

For the first row n=1, A(1,k) = A188163(k), for rows n > 1, A(n,k) = A087686(1+A(n-1,k)).

A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 7, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4

Offset: 1

Antti Karttunen, Jan 09 2016

nonn

If all 1's are deleted, the remaining terms are the sequence incremented. - after Franklin T. Adams-Watters Oct 05 2006 comment in A051135.

Ordinal transform of A162598.

			Illustration how the sequence can be constructed by concatenating the frequency counts Q_n of each successive level n of A004001-tree:
--
             1                                      Q_0 = (1)
             |
            _2__                                    Q_1 = (2)
           /    \
         _3    __4_____                             Q_2 = (1,3)
        /     /  |     \
      _5    _6  _7    __8___________                Q_3 = (1,1,2,4)
     /     /   / |   /  |  \        \
   _9    10  11 12  13  14  15___    16_________    Q_4 = (1,1,1,2,1,2,3,5)
  /     /   /  / |  /  / |   |\  \   | \  \  \  \
17    18  19 20 21 22 23 24 25 26 27 28 29 30 31 32
--
The above illustration copied from the page 229 of Kubo and Vakil paper (page 5 in PDF).

Antti Karttunen, Table of n, a(n) for n = 1..8192
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Essentially same as A051135 apart from the initial term, which here is set as a(1)=1.
Cf. A004001, A265901, A265903.
Cf. A162598 (corresponding other index).
Cf. A265754.
Cf. also A267108, A267109, A267110.

terms = 120;
h[1] = 1; h[2] = 1;
h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
seq[nmax_] := seq[nmax] = (Length /@ Split[Sort @ Array[h, nmax, 2]])[[;; terms]];
seq[nmax = 2 terms];
seq[nmax += terms];
While[seq[nmax] != seq[nmax - terms], nmax += terms];
seq[nmax] (* Jean-François Alcover, Dec 19 2021 *)

(define (A265332 n) (if (= 1 n) 1 (A051135 n)))

a(1) = 1; for n > 1, a(n) = A051135(n).

A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 8, 15, 12, 6, 16, 31, 27, 14, 9, 32, 63, 58, 30, 21, 10, 64, 127, 121, 62, 48, 24, 11, 128, 255, 248, 126, 106, 54, 26, 13, 256, 511, 503, 254, 227, 116, 57, 29, 17, 512, 1023, 1014, 510, 475, 242, 120, 61, 38, 18, 1024, 2047, 2037, 1022, 978, 496, 247, 125, 86, 42, 19, 2048, 4095, 4084, 2046, 1992, 1006, 502, 253, 192, 96, 45, 20

Offset: 1

Antti Karttunen, Dec 18 2015

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).
Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

			The top left corner of the array:
   1,  2,   4,   8,  16,   32,   64,  128,  256,   512,  1024, ...
   3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095, ...
   5, 12,  27,  58, 121,  248,  503, 1014, 2037,  4084,  8179, ...
   6, 14,  30,  62, 126,  254,  510, 1022, 2046,  4094,  8190, ...
   9, 21,  48, 106, 227,  475,  978, 1992, 4029,  8113, 16292, ...
  10, 24,  54, 116, 242,  496, 1006, 2028, 4074,  8168, 16358, ...
  11, 26,  57, 120, 247,  502, 1013, 2036, 4083,  8178, 16369, ...
  13, 29,  61, 125, 253,  509, 1021, 2045, 4093,  8189, 16381, ...
  17, 38,  86, 192, 419,  894, 1872, 3864, 7893, 16006, 32298, ...
  18, 42,  96, 212, 454,  950, 1956, 3984, 8058, 16226, 32584, ...
  19, 45, 102, 222, 469,  971, 1984, 4020, 8103, 16281, 32650, ...
  20, 47, 105, 226, 474,  977, 1991, 4028, 8112, 16291, 32661, ...
  22, 51, 112, 237, 490,  999, 2020, 4065, 8158, 16347, 32728, ...
  23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, ...
  25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, ...
  28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, ...
  ...

Antti Karttunen, Table of n, a(n) for n = 1..210; the first 20 antidiagonals of array
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A267102.
Transpose: A265903.
Cf. A265900 (main diagonal).
Cf. A162598 (row index of n in array), A265332 (column index of n in array).
Cf. A004001, A051135, A088359, A087686.
Column 1: A188163.
Column 2: A266109.
Row 1: A000079 (2^n).
Row 2: A000225 (2^n - 1, from 3 onward).
Row 3: A000325 (2^n - n, from 5 onward).
Row 4: A000918 (2^n - 2, from 6 onward).
Row 5: A084634 (?, from 9 onward).
Row 6: A132732 (2^n - 2n + 2, from 10 onward).
Row 7: A000295 (2^n - n - 1, from 11 onward).
Row 8: A036563 (2^n - 3).
Row 9: A084635 (?, from 17 onward).
Row 12: A048492 (?, from 20 onward).
Row 13: A249453 (?, from 22 onward).
Row 14: A183155 (2^n - 2n + 1, from 23 onward. Cf. also A244331).
Row 15: A000247 (2^n - n - 2, from 25 onward).
Row 16: A028399 (2^n - 4).
Cf. also permutations A267111, A267112.

(define (A265901 n) (A265901bi (A002260 n) (A004736 n)))
(define (A265901bi row col) (if (= 1 col) (A188163 row) (A087686 (+ 1 (A265901bi row (- col 1))))))

For the first column k=1, A(n,1) = A188163(n), for columns k > 1, A(n,k) = A087686(1+A(n,k-1)).

A002422 Expansion of (1-4*x)^(5/2).

Original entry on oeis.org

1, -10, 30, -20, -10, -12, -20, -40, -90, -220, -572, -1560, -4420, -12920, -38760, -118864, -371450, -1179900, -3801900, -12406200, -40940460, -136468200, -459029400, -1556708400, -5318753700, -18296512728, -63334082520

Offset: 0

N. J. A. Sloane

sign

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
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. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
N. J. A. Sloane, Notes on A984 and A2420-A2424

Cf. A007054, A004001, A002420, A002421, A002423, A002424, A007272, A001622.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(5/2) )); // G. C. Greubel, Jul 03 2019

A002422 := n -> -(15/8)*4^n*GAMMA(n-5/2)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002422(n), n=0..26); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^{5/2},{x,0,30}],x] (* Vincenzo Librandi, Jun 11 2012 *)

vector(30, n, n--; (-4)^n*binomial(5/2, n)) \\ G. C. Greubel, Jul 03 2019

[(-4)^n*binomial(5/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

a(n+3) = -2 * A007272(n).
a(n) = Sum_{m=0..n} binomial(n, m) * K_m(6), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com).
a(n) ~ -15/8*Pi^(-1/2)*n^(-7/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = -(15/8)*4^n*Gamma(n-5/2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^n*binomial(5/2, n). - Peter Luschny, Oct 22 2018
D-finite with recurrence: n*a(n) +2*(-2*n+7)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 32/45 - 14*Pi/(3^5*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 2144/1875 - 28*log(phi)/(5^4*sqrt(5)), where phi is the golden ratio (A001622). (End)

A002423 Expansion of (1-4*x)^(7/2).

Original entry on oeis.org

1, -14, 70, -140, 70, 28, 28, 40, 70, 140, 308, 728, 1820, 4760, 12920, 36176, 104006, 305900, 917700, 2801400, 8684340, 27293640, 86843400, 279409200, 908079900, 2978502072, 9851968392, 32839894640

Offset: 0

N. J. A. Sloane

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
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. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Notes on A984 and A2420-A2424

Cf. A007054, A004001, A002420, A002421, A002422, A002424, A007272, A001622.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(7/2) )); // G. C. Greubel, Jul 03 2019

A002423 := n -> (105/16)*4^n*GAMMA(-7/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002423(n), n=0..27); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4*x)^(7/2),{x,0,30}],x] (* Jean-François Alcover, Mar 21 2011 *)
Table[(4^(-1+x) Pochhammer[-(7/2),-1+x])/Pochhammer[1,-1+x],{x,30}] (* Harvey P. Dale, Jul 13 2011 *)

vector(30, n, n--; (-4)^n*binomial(7/2, n)) \\ G. C. Greubel, Jul 03 2019

[(-4)^n*binomial(7/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m) * K_m(8), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
a(n) ~ 105*4^(n-2)/(sqrt(Pi)*n^(9/2)). - Vaclav Kotesovec, Jul 28 2013
a(n) = (105/16)*4^n*Gamma(-7/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^n * binomial(7/2, n). - G. C. Greubel, Jul 03 2019
D-finite with recurrence: n*a(n) +2*(-2*n+9)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 36/35 + 2*Pi/(3^4*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 23932/21875 - 36*log(phi)/(5^5*sqrt(5)), where phi is the golden ratio (A001622). (End)

A005711 a(n) = a(n-1) + a(n-9) for n >= 9; a(n) = 1 for n=0..7; a(8) = 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 19, 24, 30, 37, 45, 54, 64, 76, 91, 110, 134, 164, 201, 246, 300, 364, 440, 531, 641, 775, 939, 1140, 1386, 1686, 2050, 2490, 3021, 3662, 4437, 5376, 6516, 7902, 9588, 11638, 14128, 17149, 20811, 25248

Offset: 0

N. J. A. Sloane

a(n+7) equals the number of binary words of length n having at least 8 zeros between every two successive ones. - Milan Janjic, Feb 09 2015

a(n) is the number of compositions of n+1 into parts 1 and 9. - Joerg Arndt, May 19 2018

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 382
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A005710.

A005711:=-(1+z**8)/(-1+z+z**9); # Simon Plouffe in his 1992 dissertation
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 8)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=9..65); # Zerinvary Lajos, Mar 26 2008
M:= Matrix(9, (i,j)-> if j=1 and member(i,[1,9]) then 1 elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n+1))[1,1]; seq(a(n), n=0..60); # Alois P. Heinz, Jul 27 2008

CoefficientList[Series[(1+x^8)/(1-x-x^9), {x, 0, 57}], x] (* Michael De Vlieger, May 20 2018 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,1},{1,1,1,1,1,1,1,1,2},60] (* Harvey P. Dale, Jul 30 2022 *)

x='x+O('x^66); Vec((1+x^8)/(1-x-x^9)) /* Joerg Arndt, Jun 25 2011 */

G.f.: (1+x^8)/(1-x-x^9).

For positive integers n and k such that k <= n <= 9*k, and 8 divides n-k, define c(n,k) = binomial(k,(n-k)/8), and c(n,k) = 0, otherwise. Then, for n>= 1, a(n-1) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011

A002424 Expansion of (1-4*x)^(9/2).

Original entry on oeis.org

1, -18, 126, -420, 630, -252, -84, -72, -90, -140, -252, -504, -1092, -2520, -6120, -15504, -40698, -110124, -305900, -869400, -2521260, -7443720, -22331160, -67964400, -209556900, -653817528, -2062039896, -6567978928, -21111360840

Offset: 0

N. J. A. Sloane

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
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. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020.
N. J. A. Sloane, Notes on A984 and A2420-A2424.

Cf. A001622, A002420, A002421, A002422, A002423, A004001, A007054, A007272.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(9/2) )); // G. C. Greubel, Jul 03 2019

A002424 := n -> -(945/32)*4^n*GAMMA(-9/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002424(n),n=0..28); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^(9/2),{x,0,30}],x] (* Harvey P. Dale, Dec 27 2011 *)

my(x='x+O('x^30)); Vec((1-4*x)^(9/2)) \\ Altug Alkan, Dec 14 2015

vector(30, n, n--; (-4)^n*binomial(9/2, n)) \\ G. C. Greubel, Jul 03 2019

[(-4)^n*binomial(9/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m) * K_m(10), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg, abarg(AT)research.bell-labs.com.
a(n) = -(945/32)*4^n*Gamma(-9/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^n*binomial(9/2, n). - G. C. Greubel, Jul 03 2019
D-finite with recurrence: n*a(n) +2*(-2*n+11)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 32/35 - 22*Pi/(3^7*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 1050752/984375 - 44*log(phi)/(5^6*sqrt(5)), where phi is the golden ratio (A001622). (End)

cp's OEIS Frontend

A005708 a(n) = a(n-1) + a(n-6), with a(i) = 1 for i = 0..5.

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A005710 a(n) = a(n-1) + a(n-8), with a(i) = 1 for i = 0..7.

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A002421 Expansion of (1-4*x)^(3/2) in powers of x.

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A265903 Square array read by descending antidiagonals: A(1,k) = A188163(k), and for n > 1, A(n,k) = A087686(1+A(n-1,k)).

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A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

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A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

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