Search: a001025 -id:a001025
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A060219
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Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025).
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+20
2
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16, 120, 1360, 16320, 209712, 2795480, 38347920, 536862720, 7635496960, 109951057896, 1599289640400, 23456246655680, 346430740566960, 5146970983535160, 76861433640386288, 1152921504338411520, 17361641481138401520, 262353693488939386880, 3976729669784964390480
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OFFSET
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1,1
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COMMENTS
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Number of monic irreducible polynomials of degree n over GF(16). - Robert Israel, Jan 07 2015
Number of Lyndon words (aperiodic necklaces) with n beads of 16 colors. - Andrew Howroyd, Dec 10 2017
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LINKS
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FORMULA
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a(n) = (1/n)* Sum_{d|n} mu(d)*16^(n/d).
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 16*x^k))/k. - Ilya Gutkovskiy, May 19 2019
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EXAMPLE
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a(2)=120 since there are 256 points of period 2 in the full 16-shift and 16 fixed points, so there must be (256-16)/2 = 120 orbits of length 2.
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MAPLE
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f:= (n, p) -> add(numtheory:-mobius(d)*p^(n/d), d=numtheory:-divisors(n))/n:
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MATHEMATICA
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PROG
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(PARI) a(n) = sumdiv(n, d, moebius(d)*16^(n/d))/n; \\ Michel Marcus, Jan 07 2015
(Magma)A060219:= func< n | (&+[MoebiusMu(d)*16^Floor(n/d): d in Divisors(n)])/n >;
(SageMath)
def A060219(n): return sum(moebius(k)*16^(n//k) for k in (1..n) if (k).divides(n))/n
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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4, 9, 8, 0, 9, 8, 5, 0, 8, 3, 9, 8, 6, 3, 6, 0, 4, 3, 7, 3, 4, 2, 9, 2, 2, 3, 9, 3, 9, 7, 4, 6, 2, 7, 6, 1, 5, 6, 0, 4, 1, 5, 8, 6, 3, 2, 5, 0, 4, 2, 7, 7, 6, 5, 0, 5, 6, 5, 9, 2, 2, 4, 3, 0, 0, 1, 8, 1, 3, 4, 4, 8, 6, 0, 3, 9, 6, 5, 4, 1
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OFFSET
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-1,1
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COMMENTS
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This Ramanujan-like series may be evaluated in an elegant way in terms of 1/Pi and Catalan's constant, as indicated below in the Formula section.
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LINKS
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FORMULA
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Equals (24 + 16*log(2) - 16*Catalan)/Pi + 8*log(2) - 12, letting Catalan denote Catalan's constant (see A006752).
Equals Sum_{n>=0} H'(2n)*C(n)^2/16^n, letting H'(i) denote the i-th alternating harmonic number, and letting C(i) denote the i-th Catalan number.
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EXAMPLE
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Equals 0.04980985083986360437342922393974627615604158632504...
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MATHEMATICA
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First[RealDigits[(24 + 16 Log[2] - 16 Catalan)/\[Pi] + 8 Log[2] - 12,
10, 80]]
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PROG
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(PARI) default(realprecision, 100); (24 + 16*log(2) - 16*Catalan)/Pi + 8*log(2) - 12 \\ G. C. Greubel, Aug 25 2018
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); (24 + 16*Log(2) - 16*Catalan(R))/Pi(R) + 8*Log(2) - 12; // G. C. Greubel, Aug 25 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A049541
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Decimal expansion of 1/Pi.
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+10
87
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3, 1, 8, 3, 0, 9, 8, 8, 6, 1, 8, 3, 7, 9, 0, 6, 7, 1, 5, 3, 7, 7, 6, 7, 5, 2, 6, 7, 4, 5, 0, 2, 8, 7, 2, 4, 0, 6, 8, 9, 1, 9, 2, 9, 1, 4, 8, 0, 9, 1, 2, 8, 9, 7, 4, 9, 5, 3, 3, 4, 6, 8, 8, 1, 1, 7, 7, 9, 3, 5, 9, 5, 2, 6, 8, 4, 5, 3, 0, 7, 0, 1, 8, 0, 2, 2, 7, 6, 0, 5, 5, 3, 2, 5, 0, 6, 1, 7, 1
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OFFSET
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0,1
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COMMENTS
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The ratio of the volume of a regular octahedron to the volume of the circumscribed sphere (which has circumradius a*sqrt(2)/2 = a*A010503, where a is the octahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A165952, A165953 and A165954. - Rick L. Shepherd, Oct 01 2009
Corresponds to a gauge point marked "M" on slide rule calculating devices in the 20th century. The Pickworth reference notes its use in calculating the area of the curved surface of a cylinder. - Peter Munn, Aug 14 2020
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REFERENCES
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J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Baasel, p. 245. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London, 1945, p. 53, Gauge Points.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.
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LINKS
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Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
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FORMULA
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1/Pi = Sum_{m>=0} binomial(2*m, m)^3 * (42*m+5)/(2^(12*m+4)), Ramanujan, from the J.-P. Delahaye reference. - Wolfdieter Lang, Sep 18 2018; corrected by Bernard Schott, Mar 26 2020
1/Pi = 12*Sum_{n >= 0} (-1)^n*((6*n)!/(n!^3*(3*n)!))*(13591409 + 545140134*n)/640320^(3*n + 3/2) [Chudnovsky]. - Sanjar Abrarov, Mar 31 2020
1/Pi = (sqrt(8)/9801) * Sum_{n >= 0} ((4*n)!/((n!)^4)) * (26390*n + 1103)/(396^(4*n)) [Ramanujan, 1914]. - Bernard Schott, Mar 26 2020
Equal Sum_{k>=2} tan(Pi/2^k)/2^k. - Amiram Eldar, Aug 05 2020
Floor((3/8)*Sum_{n>=1} sigma[3](n)*n/exp(Pi*n/(10^((1/5)*k+(1/5))))) mod 10, will give the k-th digit of 1/Pi. - Simon Plouffe, Dec 19 2023
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EXAMPLE
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0.3183098861837906715377675267450287240689192914809128974953...
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MAPLE
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MATHEMATICA
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PROG
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(Magma) R:= RealField(100); 1/Pi(R); // G. C. Greubel, Aug 21 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A131865
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Partial sums of powers of 16.
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+10
52
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1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057, 314824432191309680913, 5037190915060954894609
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OFFSET
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0,2
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COMMENTS
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16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=16, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, Feb 21 2010
a(n) is the total number of holes in a certain box fractal (start with 16 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015
Except for 1 and 17, all terms are Brazilian repunits numbers in base 16, and so belong to A125134. All terms >= 273 are composite because a(n) = ((4^(n+1) + 1) * (4^(n+1) - 1))/15. - Bernard Schott, Jun 06 2017
The sequence in binary is 1, 10001, 100010001, 1000100010001, 10001000100010001, ... cf. Plouffe link, A330135. - Frank Ellermann, Mar 05 2020
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LINKS
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FORMULA
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a(n) = if n=0 then 1 else a(n-1) + A001025(n).
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EXAMPLE
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a(3) = 1 + 16 + 256 + 4096 = 4369 = in binary: 1000100010001.
a(4) = (16^5 - 1)/15 = (4^5 + 1) * (4^5 - 1)/15 = 1025 * 1023/15 = 205 * 341 = 69905 = 11111_16. - Bernard Schott, Jun 06 2017
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MAPLE
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MATHEMATICA
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Accumulate[16^Range[0, 20]] (* or *) LinearRecurrence[{17, -16}, {1, 17}, 20] (* Harvey P. Dale, Jul 19 2019 *)
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PROG
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(Sage) [gaussian_binomial(n, 1, 16) for n in range(1, 18)] # Zerinvary Lajos, May 28 2009
(Maxima)
a[0]:0$
a[n]:=16*a[n-1]+1$
(Python)
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CROSSREFS
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Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723. - M. F. Hasler, Nov 05 2012
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A003992
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Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.
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+10
24
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0
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OFFSET
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0,8
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COMMENTS
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If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018
T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020
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LINKS
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FORMULA
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E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004
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EXAMPLE
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Rows begin:
[1, 0, 0, 0, 0, 0, 0, 0, ...],
[1, 1, 1, 1, 1, 1, 1, 1, ...],
[1, 2, 4, 8, 16, 32, 64, 128, ...],
[1, 3, 9, 27, 81, 243, 729, 2187, ...],
[1, 4, 16, 64, 256, 1024, 4096, 16384, ...],
[1, 5, 25, 125, 625, 3125, 15625, 78125, ...],
[1, 6, 36, 216, 1296, 7776, 46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...
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MATHEMATICA
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Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten
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PROG
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(Magma) [[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018
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CROSSREFS
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Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A160700
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a(n) = if n<16 then n else a(floor(n/16)) XOR (n mod 16).
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+10
21
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13, 3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11, 5, 4, 7, 6, 1, 0, 3, 2, 13
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OFFSET
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0,3
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COMMENTS
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A very simple hash function for the nonnegative integers.
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LINKS
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MAPLE
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read("transforms") ;
if n < 16 then
n;
else
XORnos(procname(floor(n/16)), modp(n, 16))
end if;
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MATHEMATICA
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a[n_] := a[n] = If[n < 16, n, a[Floor[n/16]] ~BitXor~ Mod[n, 16]];
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PROG
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(Maxima)
load(functs)$
A160700(n):=if n<16 then n else logxor(floor(n/16), mod(n, 16))$
(Haskell)
import Data.Bits (xor)
a160700 n = a160700_list !! n
a160700_list = [0..15] ++ map f [16..] where
f x = a160700 x' `xor` m :: Int where (x', m) = divMod x 16
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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1, 16, 35, 50, 69, 84, 103, 118, 137, 152, 171, 186, 205, 220, 239, 254, 256, 273, 290, 307, 324, 341, 358, 375, 392, 409, 426, 443, 460, 477, 494, 511, 515, 530, 545, 560, 583, 598, 613, 628, 651, 666, 681, 696, 719, 734, 749, 764, 770, 787, 800, 817, 838
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OFFSET
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1,2
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LINKS
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FORMULA
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MATHEMATICA
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b[n_] := b[n] = If[n < 16, n, b[Floor[n/16]]~BitXor~Mod[n, 16]];
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PROG
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(PARI) A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
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CROSSREFS
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Cf. A161440, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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4, 64, 1024, 16384, 262144, 4194304, 67108864, 1073741824, 17179869184, 274877906944, 4398046511104, 70368744177664, 1125899906842624, 18014398509481984, 288230376151711744, 4611686018427387904, 73786976294838206464, 1180591620717411303424, 18889465931478580854784
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OFFSET
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0,1
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COMMENTS
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The partial sum of A000888(n) = Catalan(n)^2*(n + 1) resp. A267844(n) = Catalan(n)^2*(4n + 3) resp. A267987(n) = Catalan(n)^2*(4n + 4) divided by A013709(n) (this) a(n) = 2^(4n+2) absolutely converge to 1/Pi resp. 1 resp. 4/Pi. Thus this series is 1/Pi resp. 1 resp. 4/Pi. - Ralf Steiner, Jan 23 2016
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LINKS
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FORMULA
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a(n) = 16*a(n-1), n > 0; a(0) = 4. G.f.: 4/(1 - 16*x). [Philippe Deléham, Nov 23 2008]
E.g.f.: 4*exp(16*x).
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MAPLE
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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A223599
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T(n,k)=Petersen graph (8,2) coloring a rectangular array: number of nXk 0..15 arrays where 0..15 label nodes of a graph with edges 0,1 0,8 8,14 8,10 1,2 1,9 9,15 9,11 2,3 2,10 10,12 3,4 3,11 11,13 4,5 4,12 12,14 5,6 5,13 13,15 6,7 6,14 7,0 7,15 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph
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+10
13
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16, 48, 256, 144, 256, 4096, 432, 1504, 1376, 65536, 1296, 6736, 16192, 7424, 1048576, 3888, 32768, 122608, 176224, 40160, 16777216, 11664, 156592, 1124064, 2372080, 1931968, 217600, 268435456, 34992, 755200, 9902320, 43725920, 47659632
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OFFSET
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1,1
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COMMENTS
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Table starts
............16........48..........144.............432...............1296
...........256.......256.........1504............6736..............32768
..........4096......1376........16192..........122608............1124064
.........65536......7424.......176224.........2372080...........43725920
.......1048576.....40160......1931968........47659632.........1807461152
......16777216....217600.....21308000.......982848688........77164934624
.....268435456...1180256....236213312.....20631729648......3355919411936
....4294967296...6405888...2629972704....438231627440....147579242411936
...68719476736..34782688..29389265856...9379905920496...6534353238114336
.1099511627776.188912640.329426847840.201754894742320.290550417324168160
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LINKS
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FORMULA
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Empirical for column k:
k=1: a(n) = 16*a(n-1)
k=2: a(n) = 8*a(n-1) -11*a(n-2) -16*a(n-3)
k=3: a(n) = 23*a(n-1) -153*a(n-2) +217*a(n-3) +258*a(n-4) -456*a(n-5) -104*a(n-6) +192*a(n-7)
k=4: [order 9]
k=5: [order 29]
k=6: [order 55]
Empirical for row n:
n=1: a(n) = 3*a(n-1)
n=2: a(n) = 6*a(n-1) +3*a(n-2) -42*a(n-3) -8*a(n-4) +48*a(n-5) for n>6
n=3: [order 11] for n>12
n=4: [order 28] for n>29
n=5: [order 74] for n>75
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EXAMPLE
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Some solutions for n=3 k=4
.14..6..5.13...13.15..9.15...12..4.12.10....6..5.13.15....8.14..8.10
..7..6..5..6...13.15..9..1...12..4.12..4....6..5.13..5....8.14..8.14
..5..6.14..6....9.15..9.11....5..4.12.14...13..5..6..5....6.14..6.14
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A223692
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T(n,k)=Petersen graph (8,2) coloring a rectangular array: number of nXk 0..15 arrays where 0..15 label nodes of a graph with edges 0,1 0,8 8,14 8,10 1,2 1,9 9,15 9,11 2,3 2,10 10,12 3,4 3,11 11,13 4,5 4,12 12,14 5,6 5,13 13,15 6,7 6,14 7,0 7,15 and every array movement to a horizontal or antidiagonal neighbor moves along an edge of this graph
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+10
13
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16, 48, 256, 144, 432, 4096, 432, 2304, 3888, 65536, 1296, 12384, 37008, 34992, 1048576, 3888, 66816, 363600, 595584, 314928, 16777216, 11664, 361440, 3788640, 10817856, 9594000, 2834352, 268435456, 34992, 1958400, 40075632, 223096320
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OFFSET
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1,1
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COMMENTS
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Table starts
............16..........48............144..............432.................1296
...........256.........432...........2304............12384................66816
..........4096........3888..........37008...........363600..............3788640
.........65536.......34992.........595584.........10817856............223096320
.......1048576......314928........9594000........324280368..........13402129824
......16777216.....2834352......154616832.......9762152544.........814399853760
.....268435456....25509168.....2492365968.....294583794768.......49817845241568
....4294967296...229582512....40180445568....8901308553408.....3059068970173824
...68719476736..2066242608...647800215696..269168305340592...188252023352797728
.1099511627776.18596183472.10444288589568.8142829402619232.11599193857488796224
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LINKS
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FORMULA
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Empirical for column k:
k=1: a(n) = 16*a(n-1)
k=2: a(n) = 9*a(n-1)
k=3: a(n) = 24*a(n-1) -127*a(n-2)
k=4: a(n) = 59*a(n-1) -1103*a(n-2) +7621*a(n-3) -16900*a(n-4)
k=5: [order 7] for n>8
k=6: [order 17]) for n>18
k=7: [order 37] for n>39
Empirical for row n:
n=1: a(n) = 3*a(n-1)
n=2: a(n) = 8*a(n-1) -11*a(n-2) -16*a(n-3) for n>4
n=3: a(n) = [order 10]) for n>12
n=4: a(n) = [order 24] for n>27
n=5: a(n) = [order 56] for n>61
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EXAMPLE
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Some solutions for n=3 k=4
..2..1..9..1....6..5..4..5....6.14..6.14....4..3..2.10....2..3..4..3
..2..1..9.11....4..5..6.14...12.14..8.14....2.10..2.10....4..3.11.13
..9.11..9.15....6..7..6.14....8..0..8..0....8.10..8.10...11.13.11..9
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CROSSREFS
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Column 2 is 48*9^(n-1)
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KEYWORD
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AUTHOR
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STATUS
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approved
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