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Search: a001025 -id:a001025
Displaying 1-10 of 66 results found. page 1 2 3 4 5 6 7
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A060219 Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025). +20
2
16, 120, 1360, 16320, 209712, 2795480, 38347920, 536862720, 7635496960, 109951057896, 1599289640400, 23456246655680, 346430740566960, 5146970983535160, 76861433640386288, 1152921504338411520, 17361641481138401520, 262353693488939386880, 3976729669784964390480 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..825
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Exactly realizable sequences
FORMULA
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
EXAMPLE
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.
MAPLE
f:= (n, p) -> add(numtheory:-mobius(d)*p^(n/d), d=numtheory:-divisors(n))/n:
seq(f(n, 16), n=1..30); # Robert Israel, Jan 07 2015
MATHEMATICA
A060219[n_]:= DivisorSum[n, MoebiusMu[#]*16^(n/#) &]/n; Table[A060219[n], {n, 40}] (* G. C. Greubel, Aug 01 2024 *)
PROG
(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 >;
[A060219(n): n in [1..40]]; // G. C. Greubel, Aug 01 2024
(SageMath)
def A060219(n): return sum(moebius(k)*16^(n//k) for k in (1..n) if (k).divides(n))/n
[A060219(n) for n in range(1, 41)] # G. C. Greubel, Aug 01 2024
CROSSREFS
Column 16 of A074650.
Cf. A001025.
KEYWORD
easy,nonn
AUTHOR
Thomas Ward, Mar 21 2001
EXTENSIONS
Terms a(17) onward added by G. C. Greubel, Aug 01 2024
STATUS
approved
A280630 Decimal expansion of Sum_{n>=1} (A001246(n)*A201546(n)) / (A001025(n)*A010050(n)). +20
1
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
-1,1
COMMENTS
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..10000
J. M. Campbell, A. Sofo, An integral transform related to series involving alternating harmonic numbers, Integr. Transf. Spec. F., 28 (7) (2017), 547-559.
R. B. Paris, Review Zbl 1376.33023, zbMATH 2018.
FORMULA
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.
EXAMPLE
Equals 0.04980985083986360437342922393974627615604158632504...
MATHEMATICA
First[RealDigits[(24 + 16 Log[2] - 16 Catalan)/\[Pi] + 8 Log[2] - 12,
10, 80]]
PROG
(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
CROSSREFS
Cf. A001025, A001246, A006752, A010050, A201546.
KEYWORD
nonn,cons,base
AUTHOR
John M. Campbell, Jan 06 2017
STATUS
approved
A049541 Decimal expansion of 1/Pi. +10
87
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
REFERENCES
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
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.
J. Bohr, Ramanujan's Method of Approximating Pi.
J. Borwein, Ramanujan's Sum.
Heng Huat Chan, Shaun Cooper, and Wen-Chin Liaw, The Rogers-Ramanujan continued fraction and a quintic iteration for 1/Pi, Proc. Amer. Math. Soc. 135 (2007), 3417-3424.
D. V. Chudnovsky and G. V. Chudnovsky, The computation of classical constants, Proc. Nati. Acad. Sci. USA, Vol. 86, pp. 8178-8182, November 1989.
J. Guillera, A New Method to Obtain Series for 1/Pi and 1/Pi^2, Experimental Mathematics, Volume 15, Issue 1, 2006.
R. Matsumoto, Ramanujan Type Series. [Broken link]
A. S. Nimbran, Deriving Forsyth-Glaisher type series for 1/π and Catalan's constant by an elementary method, The Mathematics Student, Indian Math. Soc., Vol. 84, Nos. 1-2, Jan.-June (2015), 69-86. [Broken link]
Eric W. Weisstein, Octahedron.
Index entries for transcendental numbers.
FORMULA
Equals (1/(12-16*A002162))*Sum_{n>=0} A002894(n)*H(n)/(A001025(n) * A016754(n-1)), where H(n) denotes the n-th harmonic number. - John M. Campbell, Aug 28 2016
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
EXAMPLE
0.3183098861837906715377675267450287240689192914809128974953...
MAPLE
Digits:=100: evalf(1/Pi); # Wesley Ivan Hurt, Aug 29 2016
MATHEMATICA
RealDigits[N[1/Pi, 10, 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)
PROG
(PARI) 1/Pi \\ Charles R Greathouse IV, Jun 16 2011
(MATLAB) 1/pi \\ Altug Alkan, Apr 10 2016
(Magma) R:= RealField(100); 1/Pi(R); // G. C. Greubel, Aug 21 2018
CROSSREFS
Cf. A000796, A165922, A165952, A165953, A165954, A063723, A010503. - Rick L. Shepherd, Oct 01 2009
Cf. A088538 (4/Pi).
KEYWORD
nonn,cons,changed
AUTHOR
N. J. A. Sloane, Dec 11 1999
STATUS
approved
A131865 Partial sums of powers of 16. +10
52
1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057, 314824432191309680913, 5037190915060954894609 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008
Second quadrisection of A115451. - Paul Curtz, May 21 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
Partial sums are in A014899. Also, the sequence is related to A014931 by A014931(n+1) = (n+1)*a(n) - Sum_{i=0..n-1} a(i) for n>0. - Bruno Berselli, Nov 07 2012
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..800
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Kival Ngaokrajang, Illustration of initial terms
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Simon Plouffe, Identities and approximations inspired from Ramanujan notebooks, III, 2009.
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (17,-16).
FORMULA
a(n) = if n=0 then 1 else a(n-1) + A001025(n).
for n > 0: A131851(a(n)) = n and abs(A131851(m)) < n for m < a(n).
a(n) = A098704(n+2)/2.
a(n) = (16^(n+1) - 1)/15. - Bernard Schott, Jun 06 2017
a(n) = (A001025(n+1) - 1)/15.
a(n) = 16*a(n-1) + 1. - Paul Curtz, May 20 2008
G.f.: 1 / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011
E.g.f.: exp(x)*(16*exp(15*x) - 1)/15. - Stefano Spezia, Mar 06 2020
EXAMPLE
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
MAPLE
A131865:=n->(16^(n+1)-1)/15: seq(A131865(n), n=0..30); # Wesley Ivan Hurt, Apr 29 2017
MATHEMATICA
Table[(2^(4 n) - 1)/15, {n, 16}] (* Robert G. Wilson v, Aug 22 2007 *)
Accumulate[16^Range[0, 20]] (* or *) LinearRecurrence[{17, -16}, {1, 17}, 20] (* Harvey P. Dale, Jul 19 2019 *)
PROG
(Sage) [gaussian_binomial(n, 1, 16) for n in range(1, 18)] # Zerinvary Lajos, May 28 2009
(Magma) [(16^(n+1)-1)/15: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011
(Maxima)
a[0]:0$
a[n]:=16*a[n-1]+1$
A131865(n):=a[n]$
makelist(A131865(n), n, 1, 30); /* Martin Ettl, Nov 05 2012 */
(PARI) A131865(n)=16^n\15 \\ M. F. Hasler, Nov 05 2012
(Python)
def A131865(n): return (1<<(n+1<<2))//15 # Chai Wah Wu, Nov 10 2022
CROSSREFS
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
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Jul 22 2007
STATUS
approved
A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0. +10
24
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
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
LINKS
T. D. Noe, Rows n = 0..50 of triangle, flattened
FORMULA
E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004
E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006
EXAMPLE
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, ...], ...
MATHEMATICA
Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten
PROG
(PARI) T(n, k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017
(Magma) [[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018
CROSSREFS
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.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).
KEYWORD
easy,nice,nonn,tabl
AUTHOR
Marc LeBrun
EXTENSIONS
More terms from David W. Wilson
Edited by Paul D. Hanna, Oct 22 2004
STATUS
approved
A160700 a(n) = if n<16 then n else a(floor(n/16)) XOR (n mod 16). +10
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
A very simple hash function for the nonnegative integers.
a(A000079(n))=A133145(n); a(A000302(n))=A010685(n); a(A001025(n))=A161452(n); a(A161440(n))=0; a(A161441(n))=1; a(A161442(n))=2; a(A161443(n))=3; a(A161444(n))=4; a(A161445(n))=5; a(A161446(n))=6; a(A161447(n))=7; a(A161448(n))=8; a(A161449(n))=9; a(A161450(n))=10; a(A161451(n))=11; a(A161452(n))=12; a(A161453(n))=13; a(A161454(n))=14; a(A161455(n))=15. - Reinhard Zumkeller, Jun 10 2009
LINKS
R. Zumkeller, Table of n, a(n) for n = 0..10000
MAPLE
read("transforms") ;
A160700 := proc(n)
if n < 16 then
n;
else
XORnos(procname(floor(n/16)), modp(n, 16))
end if;
end proc: # R. J. Mathar, Jul 12 2016
MATHEMATICA
a[n_] := a[n] = If[n < 16, n, a[Floor[n/16]] ~BitXor~ Mod[n, 16]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 25 2018 *)
PROG
(Maxima)
load(functs)$
A160700(n):=if n<16 then n else logxor(floor(n/16), mod(n, 16))$
makelist(A160700(n), n, 0, 60); /* Martin Ettl, Nov 05 2012 */
(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
-- Reinhard Zumkeller, Nov 07 2012
(PARI) a(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t \\ Charles R Greathouse IV, Jan 25 2018
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Jun 01 2009
STATUS
approved
A161441 Numbers n such that A160700(n) = 1. +10
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
16n - 16 <= a(n) <= 16n - 1. - Charles R Greathouse IV, Jan 25 2018
MATHEMATICA
b[n_] := b[n] = If[n < 16, n, b[Floor[n/16]]~BitXor~Mod[n, 16]];
Select[Range[0, 1000], b[#] == 1&] (* Jean-François Alcover, Dec 01 2021 *)
PROG
(PARI) A160700(n)=my(t=n%16); while(n>15, n>>=4; t=bitxor(t, n%16)); t
a(n)=for(k=16*n-16, 16*n-1, if(a(k)==1, return(k))) \\ Charles R Greathouse IV, Jan 25 2018
CROSSREFS
A001025 is a subsequence.
Cf. A161440, A161442, A161443, A161444, A161445, A161446, A161447, A161448, A161449, A161450, A161451, A161452, A161453, A161454, A161455.
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Jun 10 2009
STATUS
approved
A013709 a(n) = 4^(2*n+1). +10
16
4, 64, 1024, 16384, 262144, 4194304, 67108864, 1073741824, 17179869184, 274877906944, 4398046511104, 70368744177664, 1125899906842624, 18014398509481984, 288230376151711744, 4611686018427387904, 73786976294838206464, 1180591620717411303424, 18889465931478580854784 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Also powers of 2 with singly even numbers (A016825) as exponents. - Alonso del Arte, Sep 03 2012
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (16).
FORMULA
a(n) = 16*a(n-1), n > 0; a(0) = 4. G.f.: 4/(1 - 16*x). [Philippe Deléham, Nov 23 2008]
a(n) = 4^(2*n + 1) = 2^(4*n + 2). - Alonso del Arte, Sep 03 2012
a(n) = 4*A001025(n). - Michel Marcus, Jan 30 2016
From Elmo R. Oliveira, Aug 26 2024: (Start)
E.g.f.: 4*exp(16*x).
a(n) = A000302(A005408(n)). (End)
MAPLE
A013709:=n->4^(2*n+1): seq(A013709(n), n=0..20); # Wesley Ivan Hurt, Jan 30 2016
MATHEMATICA
2^(4 Range[0, 15] + 2) (* Alonso del Arte, Sep 03 2012 *)
NestList[16#&, 4, 20] (* Harvey P. Dale, Jun 03 2013 *)
PROG
(Magma) [4^(2*n+1): n in [0..20]]; // Vincenzo Librandi, May 26 2011
(PARI) a(n)=4<<(4*n) \\ Charles R Greathouse IV, Apr 07 2012
CROSSREFS
Cf. A000888, A001025, A016825, A267844, A267987.
Cf. A000302, A005408.
KEYWORD
nonn,easy,changed
AUTHOR
N. J. A. Sloane
STATUS
approved
A223599 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 +10
13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..161
FORMULA
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
EXAMPLE
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
CROSSREFS
Column 1 is A001025
Column 2 is A223434
Row 1 is A188825(n+1)
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin Mar 23 2013
STATUS
approved
A223692 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 +10
13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..218
FORMULA
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
EXAMPLE
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
CROSSREFS
Column 1 is A001025
Column 2 is 48*9^(n-1)
Row 1 is A188825(n+1)
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin Mar 25 2013
STATUS
approved
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Last modified August 30 13:06 EDT 2024. Contains 375543 sequences. (Running on oeis4.)