A051502 -id:A051502 - cp's OEIS frontend

A090314 a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.

2, 23, 531, 12236, 281959, 6497293, 149719698, 3450050347, 79500877679, 1831970236964, 42214816327851, 972772745777537, 22415987969211202, 516540496037635183, 11902847396834820411, 274282030623238504636, 6320389551731320427039, 145643241720443608326533, 3356114949121934311937298

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Lim_{n -> infinity} a(n)/a(n+1) = 0.04339638... = 2/(23+sqrt(533)) = (sqrt(533)-23)/2.

Lim_{n -> infinity} a(n+1)/a(n) = 23.04339638... = (23+sqrt(533))/2 = 2/(sqrt(533) - 23).

			a(4) = 281959 = 23*a(3) + a(2) = 23*12236 + 531 = ((23 + sqrt(533))/2)^4 + ((23 - sqrt(533))/2)^4 = 281958.999996453 + 0.000003546 = 281959.

G. C. Greubel, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (23,1).

Cf. A089141, A051502.

Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), this sequence (m=23), A090316 (m=24), A330767 (m=25).

a:=[2,23];; for n in [3..20] do a[n]:=23*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 29 2019

I:=[2,23]; [n le 2 select I[n] else 23*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 29 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 23*I/2)), n = 0..20); # G. C. Greubel, Dec 29 2019

LinearRecurrence[{23,1},{2,23},20] (* Harvey P. Dale, Jul 11 2014 *)
LucasL[Range[20]-1,23] (* G. C. Greubel, Dec 29 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 23*I/2) ) \\ G. C. Greubel, Dec 29 2019

[2*(-I)^n*chebyshev_T(n, 23*I/2) for n in (0..20)] # G. C. Greubel, Dec 29 2019

a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.
a(n) = ((23 + sqrt(533))/2)^n + ((23 - sqrt(533))/2)^n.
(a(n))^2 = a(2n) - 2 if n=1, 3, 5....
(a(n))^2 = a(2n) + 2 if n=2, 4, 6....
G.f.: (2-23*x)/(1-23*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = Lucas(n, 23) = 2*(-i)^n * ChebyshevT(n, 23*i/2). - G. C. Greubel, Dec 29 2019

More terms from Ray Chandler, Feb 14 2004

Terms a(16) onward added by G. C. Greubel, Dec 29 2019

A000231 Number of inequivalent Boolean functions of n variables under action of complementing group.

Original entry on oeis.org

2, 3, 7, 46, 4336, 134281216, 288230380379570176, 2658455991569831764110243006194384896, 452312848583266388373324160190187140390789016525312000869601987902398529536

Offset: 0

N. J. A. Sloane

The next term has 152 digits. - Harvey P. Dale, Jun 21 2011

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 143.
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, correctly, although in the Preface on page viii 4336 is mis-typed as 4436).

Michael De Vlieger, Table of n, a(n) for n = 0..11
R. L. Ashenhurst, The application of counting techniques, Proc. ACM Nat. Mtg., Pittsburg, 1952, 293-305.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
M. A. Harrison, The number of transitivity sets of Boolean functions, J. Soc. Indust. Appl. Math., 11 (1963), 806-828.
Index entries for sequences related to Boolean functions

Cf. A051502.

Row sums of A054724.

a:= n-> (2^(2^n)+(2^n-1)*2^(2^(n-1)))/2^n:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 27 2023

Table[(2^(2^n)+(2^n-1)*2^(2^(n-1)))/2^n,{n,10}] (* Harvey P. Dale, Jun 21 2011 *)

a(n)=(2^(2^n-n)+(2^n-1)*2^(2^(n-1)-n)) \\ Charles R Greathouse IV, Jul 29 2016

a(n) = (2^(2^n)+(2^n-1)*2^(2^(n-1)))/2^n.

More terms from Vladeta Jovovic, Apr 20 2000

a(0)=2 prepended by Alois P. Heinz, Jan 27 2023

A090731 a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.

Original entry on oeis.org

2, 23, 527, 12098, 277727, 6375623, 146361602, 3359941223, 77132286527, 1770682648898, 40648568638127, 933146396028023, 21421718540006402, 491766380024119223, 11289205022014735727, 259159949126314802498

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 21*(5*b)^2 =+4 with companion sequence b(n)=A097778(n-1), n>=1; b(0):=0.

			(x;y) = (0;2), (23;1), (527;23), (12098;528), ... give the
nonnegative integer solutions to x^2 - 21*(5*y)^2 = 4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Indranil Ghosh, Table of n, a(n) for n = 0..733
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (23,-1).

Cf. A037088, A051502.
a(n)=sqrt(4 + 21*(5*A097778(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).

a[0] = 2; a[1] = 23; a[n_] := 23a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{23,-1},{2,23},30] (* Harvey P. Dale, Feb 20 2012 *)

[lucas_number2(n,23,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = S(n, 23) - S(n-2, 23) = 2*T(n, 23/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 23)=A097778(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (23+5*sqrt(21))/2 and am := (23-5*sqrt(21))/2.
G.f.: (2-23*x)/(1-23*x+x^2).

Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004

A022619 Triangle T(n,k)of numbers of asymmetric Boolean functions of n variables with exactly k = 0..2^n nonzero values (atoms) under action of complementing group C(n,2).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 7, 7, 7, 0, 1, 0, 0, 1, 0, 35, 105, 273, 448, 715, 750, 715, 448, 273, 105, 35, 0, 1, 0, 0, 1, 0, 155, 1085, 6293, 27776, 105183, 327050, 876525, 2011776, 4032015, 7048811, 10855425, 14721280, 17678835, 18771864

Offset: 1

Vladeta Jovovic, Jul 13 2000

			Triangle begins:
  [0,1,0],
  [0,1,0,1,0],
  [0,1,0,7,7,7,0,1,0],
  ...;
T(5,k) = coefficient of x^k in (1/32)*((1+x)^32-31*(1+x^2)^16+310*(1+x^4)^8-1240*(1+x^8)^4+1984*(1+x^16)^2-1024*(1+x^32)),k = 0..32.

Row sums give A051502.

Cf. A054724.

T[n_,0]:=0; T[n_, k_] := (1/2^n)*Coefficient[Sum[(-1)^j*2^(Binomial[j, 2])* QBinomial[n, j, 2]*(1 + x^(2^j))^(2^(n - j)), {j, 0, n}], x^k];
Table[T[n, k], {n, 1, 5}, {k, 0, 2^n}] // Flatten (* G. C. Greubel, Feb 15 2018 *)

T(n, k) = coefficient of x^k in (1/2^n)*Sum_{j = 0..n} (-1)^j*2^C(j, 2)*[n, j]*(1+x^(2^j))^(2^(n-j)), where [n, j] is Gaussian 2-binomial coefficient; k = 0..2^n.

A029941 Number of symmetric types of (4,2n)-hypergraphs under action of complementing group C(4,2).

Original entry on oeis.org

1, 15, 50, 225, 590, 1485, 3130, 6435, 11931, 21450, 36220, 59670, 94140, 145350, 217500, 319770, 458981, 648945, 900350, 1233375, 1663850, 2220075, 2924870, 3817125, 4928511, 6310980, 8007640, 10086780, 12605560, 15651900, 19300440, 23662980, 28835081

Offset: 0

Vladeta Jovovic, Jul 13 2000

The first g.f. gives a 0 between each two terms of the sequence - Colin Barker, Jul 12 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Cf. A051502.

CoefficientList[Series[(9 x^12 - 21 x^11 + 26 x^10 + 121 x^9 - 149 x^8 + 132 x^7 + 20 x^6 + 68 x^5 - 61 x^4 + 89 x^3 - 6 x^2 + 11 x + 1)/((x - 1)^8 (x + 1)^4 (x^2 + 1)^2 (x^4 + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1},{1,15,50,225,590,1485,3130,6435,11931,21450,36220,59670,94140,145350,217500,319770,458981,648945,900350,1233375},40] (* Harvey P. Dale, Aug 14 2021 *)

G.f.: (30/(1-x^2)^8-70/(1-x^4)^4+120/(1-x^8)^2-64/(1-x^16))/16.
G.f.: (9*x^12 -21*x^11 +26*x^10 +121*x^9 -149*x^8 +132*x^7 +20*x^6 +68*x^5 -61*x^4 +89*x^3 -6*x^2 +11*x +1) / ((x-1)^8 *(x+1)^4 *(x^2+1)^2 *(x^4+1)). - Colin Barker, Jul 12 2013
a(n) = 4*a(n-1)-4*a(n-2)-4*a(n-3)+11*a(n-4)-8*a(n-5)+8*a(n-7)-10*a(n-8)+8*a(n-10)-10*a(n-12)+8*a(n-13)-8*a(n-15)+11*a(n-16)-4*a(n-17)-4*a(n-18)+4*a(n-19)-a(n-20). - Wesley Ivan Hurt, May 24 2021

More terms from James Sellers, Aug 08 2000

More terms from Colin Barker, Jul 12 2013

A227725 T(n,k) = number of small equivalence classes of n-ary Boolean functions that contain 2^k functions.

Original entry on oeis.org

2, 2, 1, 2, 3, 2, 2, 7, 14, 23, 2, 15, 70, 345, 3904

Offset: 0

Tilman Piesk, Jul 22 2013

Left diagonal (k=0) has only 2s. Two functions (contradiction and tautology) are always alone in their respective sec, regardless of arity.
Second diagonal (k=1) is 2^n-1 (A000225). These are the n-ary linear Boolean functions. Each sec contains a row of a binary Walsh matrix and its complement.
Right diagonal (k=n) is A051502, the numbers of small equivalence classes of n-ary functions, that contain the highest possible number of 2^n functions.

			Triangle begins:              Row sums (A000231)
            2                         2
         2     1                      3
      2     3     2                   7
   2     7    14    23               46
2    15    70    345   3904        4336

Cf. A000231, A051502, A000225, A227722.

A055788 Number of symmetric types of (5,2n)-hypergraphs under action of complementing group C(5,2).

Original entry on oeis.org

1, 31, 186, 1581, 7316, 30039, 103974, 330429, 947050, 2533289, 6325550, 14969435, 33665380, 72544185, 150236850, 300540195, 582235491, 1096087770, 2009383420, 3595937070, 6292703640, 10787811210, 18142828740, 29975617710

Offset: 0

Vladeta Jovovic, Jul 13 2000

nonn

Cf. A051502.

G.f.: (62/(1-x^2)^16-310/(1-x^4)^8+1240/(1-x^8)^4-1984/(1-x^16)^2+1024/(1-x^32))/32.

More terms from James Sellers, Aug 22 2000

cp's OEIS Frontend

A090314 a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.

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A000231 Number of inequivalent Boolean functions of n variables under action of complementing group.

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A090731 a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.

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A022619 Triangle T(n,k)of numbers of asymmetric Boolean functions of n variables with exactly k = 0..2^n nonzero values (atoms) under action of complementing group C(n,2).

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A029941 Number of symmetric types of (4,2n)-hypergraphs under action of complementing group C(4,2).

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Mathematica

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A227725 T(n,k) = number of small equivalence classes of n-ary Boolean functions that contain 2^k functions.

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A055788 Number of symmetric types of (5,2n)-hypergraphs under action of complementing group C(5,2).

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