A001350 -id:A001350 - cp's OEIS frontend

A001351 Associated Mersenne numbers.

0, 1, 3, 1, 3, 11, 9, 8, 27, 37, 33, 67, 117, 131, 192, 341, 459, 613, 999, 1483, 2013, 3032, 4623, 6533, 9477, 14311, 20829, 30007, 44544, 65657, 95139, 139625, 206091, 300763, 439521, 646888, 948051, 1385429, 2033193, 2983787, 4366197, 6397723, 9387072

Offset: 0

N. J. A. Sloane, R. K. Guy

From Peter Bala, Sep 15 2019: (Start)
This is a linear divisibility sequence of order 6 (Haselgrove, p. 21). It is a particular case of a family of divisibility sequences studied by Roettger et al. The o.g.f. has the form x*d/dx(f(x)/(x^3*f(1/x))) where f(x) = x^3 - x^2 - 1.
More generally, if f(x) = 1 + P*x + Q*x^2 + x^3 or f(x) = -1 + P*x + Q*x^2 + x^3, where P and Q are integers, then the rational function x*d/dx(f(x)/(x^3*f(1/x))) is the generating function for a linear divisibility sequence of order 6. Cf. A001945. There are corresponding results when f(x) is a monic quartic polynomial with constant term 1. (End)

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).

Danny Rorabaugh, Table of n, a(n) for n = 0..6000
Peter Bala, Some linear divisibility sequences of order 6
C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22. [Annotated and scanned copy]
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
E. L. Roettger, H. C. Williams, and R. K. Guy, Some extensions of the Lucas functions, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics, Vol. 43, J. Borwein, I. Shparlinski, W. Zudilin (Eds.) 2013.
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-1,1,-1).

Cf. A001350, A001945.

I:=[0,1,3,1,3,11]; [n le 6 select I[n] else Self(n-1) - Self(n-2) + 3*Self(n-3) - Self(n-4) + Self(n-5) - Self(n-6): n in [1..50]]; // Vincenzo Librandi, Sep 23 2015

A001351:=z*(z^2-z+1)*(z^2+3*z+1)/(z^3+z-1)/(z^3-z^2-1); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{1, -1, 3, -1, 1, -1}, {0, 1, 3, 1, 3, 11}, 50] (* Vincenzo Librandi, Sep 23 2015 *)

a(n) = a(n-1) - a(n-2) + 3*a(n-3) - a(n-4) + a(n-5) - a(n-6) for n >= 6. - Sean A. Irvine, Sep 23 2015

a(n) = (alpha^n - 1)*(beta^n - 1)*(gamma^n - 1) where alpha, beta and gamma are the zeros of x^3 - x^2 - 1. - Peter Bala, Sep 15 2019

More terms from Vincenzo Librandi, Sep 23 2015

A129744 a(n) = -(u^n-1)*(v^n-1) with u = 1+sqrt(2), v = 1-sqrt(2).

Original entry on oeis.org

2, 4, 14, 32, 82, 196, 478, 1152, 2786, 6724, 16238, 39200, 94642, 228484, 551614, 1331712, 3215042, 7761796, 18738638, 45239072, 109216786, 263672644, 636562078, 1536796800, 3710155682, 8957108164, 21624372014, 52205852192

Offset: 1

N. J. A. Sloane, May 13 2007

Seiichi Manyama, Table of n, a(n) for n = 1..1000
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A002203, A000129, A001350, A113224.

u:=1+sqrt(2): v:=1-sqrt(2): a:=n->expand(-(u^n-1)*(v^n-1)): seq(a(n),n=1..33); # Emeric Deutsch, May 13 2007

Table[Simplify[ -((1 + Sqrt[2])^n - 1)*((1 - Sqrt[2])^n - 1)], {n, 1, 30}] (* Stefan Steinerberger, May 15 2007 *)

w = quadgen(8); vector(30, n, -((1+w)^n-1)*((1-w)^n-1)) \\ Michel Marcus, Mar 21 2015

Vec(2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x))+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015

a(2n) = A002203(2n)-2. a(2n+1) = A002203(2n+1). - R. J. Mathar, corrected Dec 05 2007.
G.f.: 2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x)).
From Peter Bala, Mar 19 2015: (Start)
a(n) = -det(I - M^n) where I is the 2X2 identity matrix and M = [2, 1; 1, 0]. Cf. A001350.
a(n) = 2*A113224(n-1).
This is divisibility sequence, that is, if n | m then a(n) | a(m).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n > 4. - Seiichi Manyama, Jun 07 2018

More terms from Emeric Deutsch and Stefan Steinerberger, May 13 2007

A152152 a(n) = Product_{k=1..n} (1 + 4sin(2Pi*k/n)^2).

Original entry on oeis.org

0, 1, 1, 16, 25, 121, 256, 841, 2025, 5776, 14641, 39601, 102400, 271441, 707281, 1860496, 4862025, 12752041, 33362176, 87403801, 228765625, 599074576, 1568239201, 4106118241, 10749542400, 28143753121, 73680216481, 192900153616

Offset: 0

Roger L. Bagula and Gary W. Adamson, Nov 26 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2000
M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 4.
Kh. Bibak and M. H. Shirdareh Haghighi, Some Trigonometric Identities Involving Fibonacci and Lucas Numbers , Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.4
N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities, April 2006.
N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities, Fibonacci Quart. 46/47 (2008/2009), no. 1, 56-61.

Cf. A000032, A001350, A324487.

[(1-Lucas(n)+(-1)^n)^2: n in [0..30]]; // G. C. Greubel, Mar 13 2019

Table[(1 + Fibonacci[n] - 2*Fibonacci[n+1] + (-1)^n)^2, {n, 0, 30}]

{a(n) = (1-fibonacci(n-1)-fibonacci(n+1)+(-1)^n)^2}; \\ G. C. Greubel, Mar 13 2019

[(1-lucas_number2(n,1,-1)+(-1)^n)^2 for n in (0..30)] # G. C. Greubel, Mar 13 2019

a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).
a(n) = (1 + Fibonacci(n) - 2*Fibonacci(n + 1) + (-1)^n)^2.
G.f.: -x*(x^6 -2*x^5 +10*x^4 -14*x^3 +10*x^2 -2*x +1)/((x -1)*(x +1)*(x^2 -3*x +1)*(x^2 -x -1)*(x^2 +x -1)). - Colin Barker, Apr 13 2014
a(n) = A001350(n)^2. - Colin Barker, Apr 13 2014
a(n) = (1 + (-1)^n - Lucas(n))^2. - G. C. Greubel, Mar 13 2019

A288913 a(n) = Lucas(4*n + 3).

Original entry on oeis.org

4, 29, 199, 1364, 9349, 64079, 439204, 3010349, 20633239, 141422324, 969323029, 6643838879, 45537549124, 312119004989, 2139295485799, 14662949395604, 100501350283429, 688846502588399, 4721424167835364, 32361122672259149, 221806434537978679, 1520283919093591604

Offset: 0

Bruno Berselli, Jun 19 2017

a(n) mod 4 gives A101000.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7,-1).

Cf. A000032, A000045, A004187, A101000.
Cf. A033891: fourth quadrisection of A000045.
Partial sums are in A081007 (after 0).
Positive terms of A098149, and subsequence of A001350, A002878, A016897, A093960, A068397.
Quadrisection of A000032: A056854 (first), A056914 (second), A246453 (third, without 11), this sequence (fourth).

[Lucas(4*n + 3): n in [0..30]]; // G. C. Greubel, Dec 22 2017

LucasL[4 Range[0, 21] + 3]
LinearRecurrence[{7,-1}, {4,29}, 30] (* G. C. Greubel, Dec 22 2017 *)

Vec((4 + x)/(1 - 7*x + x^2) + O(x^30)) \\ Colin Barker, Jun 20 2017

from sympy import lucas
def a(n):  return lucas(4*n + 3)
print([a(n) for n in range(22)]) # Michael S. Branicky, Apr 29 2021

def L():
    x, y = -1, 4
    while True:
        yield y
        x, y = y, 7*y - x
r = L(); [next(r) for  in (0..21)] # _Peter Luschny, Jun 20 2017

G.f.: (4 + x)/(1 - 7*x + x^2).
a(n) = 7*a(n-1) - a(n-2) for n>1, with a(0)=4, a(1)=29.
a(n) = ((sqrt(5) + 1)^(4*n + 3) - (sqrt(5) - 1)^(4*n + 3))/(8*16^n).
a(n) = Fibonacci(4*n+4) + Fibonacci(4*n+2).
a(n) = 4*A004187(n+1) + A004187(n).
a(n) = 5*A003482(n) + 4 = 5*A081016(n) - 1.
a(n) = A002878(2*n+1) = A093960(2*n+3) = A001350(4*n+3) = A068397(4*n+3).
a(n+1)*a(n+k) - a(n)*a(n+k+1) = 15*Fibonacci(4*k). Example: for k=6, a(n+1)*a(n+6) - a(n)*a(n+7) = 15*Fibonacci(24) = 695520.

A324489 a(n) = A324488(n)/n.

Original entry on oeis.org

1, 0, 21, 31, 266, 672, 3484, 11375, 48768, 177023, 716418, 2730315, 10878520, 42485638, 169181010, 670042125, 2678678730, 10705526976, 43007270292, 173003915322, 698235680844, 2822901487191, 11439823946306, 46438021798875, 188856966693230, 769224288476860, 3137871076604544, 12817404260955810

Offset: 1

N. J. A. Sloane, Mar 12 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Tables 5 and 6.

Cf. A060280, A324485, A324486, A324487, A324488.

a001350(n) = fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n;
a(n) = sumdiv(n, d, moebius(n/d)*a001350(d)^3)/n; \\ Seiichi Manyama, Apr 29 2021

f(x) = ((1-3*x+x^2)*(1+3*x+x^2))^3*(1-x^2)^10/((1-4*x-x^2)*(1-x-x^2)^6*(1+x-x^2)^9);
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*log(f(x^k))/k)) \\ Seiichi Manyama, Apr 29 2021

From Seiichi Manyama, Apr 29 2021: (Start)
a(n) = (1/n) * Sum_{d|n} mu(n/d) * A001350(d)^3 = (1/n) * Sum_{d|n} mu(n/d) * A324487(d).
G.f.: Sum_{k>=1} mu(k) * log(f(x^k))/k , where f(x) = ((1-3*x+x^2) * (1+3*x+x^2))^3 * (1-x^2)^10/((1-4*x-x^2) * (1-x-x^2)^6 * (1+x-x^2)^9). (End)

More terms from Seiichi Manyama, Apr 29 2021

A032189 Number of ways to partition n elements into pie slices each with an odd number of elements.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 19, 30, 41, 63, 94, 142, 211, 328, 493, 765, 1170, 1810, 2787, 4340, 6713, 10461, 16274, 25414, 39651, 62074, 97109, 152287, 238838, 375166, 589527, 927554, 1459961, 2300347, 3626242, 5721044, 9030451, 14264308, 22542397, 35646311, 56393862, 89264834, 141358275

Offset: 1

Christian G. Bower

nonn

a(n) is also the total number of cyclic compositions of n into odd parts assuming that two compositions are equivalent if one can be obtained from the other by a cyclic shift. For example, a(5)=3 because 5 has the following three cyclic compositions into odd parts: 5, 1+3+1, 1+1+1+1+1. - Petros Hadjicostas, Dec 27 2016

C. G. Bower, Transforms (2)
P. Flajolet and M. Soria, The Cycle Construction In SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
Petros Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021.
Index entries for sequences related to necklaces

Cf. A000358, A001350, A008965.

a1350[n_] := Sum[Binomial[k - 1, 2k - n] n/(n - k), {k, 0, n - 1}];
a[n_] := 1/n Sum[EulerPhi[n/d] a1350[d], {d, Divisors[n]}];
Array[a, 50] (* Jean-François Alcover, Jul 29 2018, after Petros Hadjicostas *)

N=66;  x='x+O('x^N);
B(x)=x/(1-x^2);
A=sum(k=1,N,eulerphi(k)/k*log(1/(1-B(x^k))));
Vec(A)
/* Joerg Arndt, Aug 06 2012 */

from sympy import totient, lucas, divisors
def A032189(n): return sum(totient(n//k)*(lucas(k)-((k&1^1)<<1)) for k in divisors(n,generator=True))//n # Chai Wah Wu, Sep 23 2023

a(n) = A000358(n)-(1+(-1)^n)/2.
"CIK" (necklace, indistinct, unlabeled) transform of 1, 0, 1, 0...(odds)
G.f.: Sum_{k>=1} phi(k)/k * log( 1/(1-B(x^k)) ) where B(x) = x/(1-x^2). [Joerg Arndt, Aug 06 2012]
a(n) = (1/n)*Sum_{d divides n} phi(n/d)*A001350(d). - Petros Hadjicostas, Dec 27 2016

A324485 a(n) = A324484(n)/n.

Original entry on oeis.org

1, 0, 5, 6, 24, 40, 120, 250, 640, 1452, 3600, 8510, 20880, 50460, 124024, 303750, 750120, 1853120, 4600200, 11437548, 28527320, 71281800, 178526880, 447893250, 1125750120, 2833844040, 7144449920, 18036271740, 45591631800, 115381449692, 292329067800, 741410192250

Offset: 1

N. J. A. Sloane, Mar 12 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..2000
M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 4.

Cf. A001350, A060280, A152152, A324483, A324484, A324489.

a001350(n) = fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n;
a(n) = sumdiv(n, d, moebius(n/d)*a001350(d)^2)/n; \\ Seiichi Manyama, Apr 29 2021

f(x) = ((1-x-x^2)*(1+x-x^2))^2/((1-3*x+x^2)*(1-x)^2*(1+x)^4);
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*log(f(x^k))/k)) \\ Seiichi Manyama, Apr 29 2021

From Seiichi Manyama, Apr 29 2021: (Start)
a(n) = (1/n) * Sum_{d|n} mu(n/d) * A001350(d)^2 = (1/n) * Sum_{d|n} mu(n/d) * A152152(d).
G.f.: Sum_{k>=1} mu(k) * log(f(x^k))/k , where f(x) = ((1-x-x^2) * (1+x-x^2))^2/((1-3*x+x^2) * (1-x)^2 * (1+x)^4). (End)

More terms from Seiichi Manyama, Apr 29 2021

A294203 Number of partitions of n into distinct Lucas parts (A000204) greater than 1.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 3, 0, 0, 2, 2, 0, 0, 3, 0, 0, 1, 3, 0, 0, 3, 2, 0, 0, 4, 0, 0, 2, 3, 0, 0, 3, 1, 0, 0, 4, 0, 0, 3, 3, 0, 0, 5, 0, 0, 2, 4, 0, 0, 4, 2, 0, 0, 5, 0, 0, 3, 3, 0, 0, 4, 0, 0, 1, 4, 0, 0, 4, 3, 0, 0, 6, 0, 0, 3, 5, 0, 0, 5, 2, 0, 0, 6, 0, 0, 4, 4

Offset: 0

Ilya Gutkovskiy, Oct 24 2017

nonn

Convolution of the sequences A003263 and A033999.
Positions of 0: 1, 2, 5, 6, 8, 9, 12, 13, ... = A287775(n) - 1 (conjecture).
From Michel Dekking, Dec 30 2017: (Start)
Proof of the 'positions of 0' conjecture: let (z(n))=1,2,5,6,8,9,12,... be the positions of 0. The crucial observation is that if a number n is the sum of distinct Lucas parts greater than 1, then n+1 is a sum of Lucas parts. This implies that (z(2n))=2,6,9,13,... is the sequence of numbers A054770 that are not a sum of Lucas numbers. We see there that Ian Agol proved that b(n):=A054770(n)=floor(phi*n)+2n-1. But then the sequence of first differences (b(n+1)-b(n)) equals the Fibonacci word on the alphabet {4,3}, yielding that (z(2n)-z(2n-1)) equals the Fibonacci word on {3,2}, and we already know that z(2n+1)-z(2n)=1 for all n. On the other hand, A287775 has the same first difference sequence given by A108103. Since A287775(1)=2, the conjecture follows. (End)
Positions of 1: 0, 3, 4, 10, 15, 28, 44, 75, ... = A001350(n+1) - 1 (conjecture).

			a(7) = 2 because we have [7] and [4, 3].

Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000
Index entries for sequences related to partitions

Cf. A000204, A003263, A033999, A067592, A239002, A294204, A054770.

CoefficientList[Series[Product[1 + x^LucasL[k], {k, 2, 15}], {x, 0, 100}], x]

G.f.: Product_{k>=2} (1 + x^Lucas(k)).

A324488 Inflation orbit counts b^{(3)}_n for Danzer's F-type tiling and other 3D cut and project patterns with tau-inflation.

Original entry on oeis.org

1, 0, 63, 124, 1330, 4032, 24388, 91000, 438912, 1770230, 7880598, 32763780, 141420760, 594798932, 2537715150, 10720674000, 45537538410, 192699485568, 817138135548, 3460078306440, 14662949297724, 62103832718202, 263115950765038, 1114512523173000, 4721424167330750

Offset: 1

N. J. A. Sloane, Mar 12 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Tables 5 and 6.

Cf. A001350, A031367, A324487, A324489.

a001350(n) = fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n;
a(n) = sumdiv(n, d, moebius(n/d)*a001350(d)^3); \\ Seiichi Manyama, Apr 29 2021

a(n) = Sum_{d|n} mu(n/d) * A001350(d)^3 = Sum_{d|n} mu(n/d) * A324487(d). - Seiichi Manyama, Apr 29 2021

More terms from Seiichi Manyama, Apr 29 2021

A365857 Number of cyclic compositions of 2*n into odd parts.

Original entry on oeis.org

1, 2, 4, 7, 14, 30, 63, 142, 328, 765, 1810, 4340, 10461, 25414, 62074, 152287, 375166, 927554, 2300347, 5721044, 14264308, 35646311, 89264834, 223959710, 562878429, 1416953362, 3572233420, 9018211989, 22795835726, 57690911720, 146164582455, 370705552702, 941109975022, 2391391374017, 6081865318124

Offset: 1

Joshua P. Bowman, Sep 20 2023

nonn

Even bisection of A032189.
Also the number of cyclic compositions into an even number of odd parts; because such a sum must be even, alternating terms are zero and have been removed.
Also the number of dual classes of cyclic n-color compositions of n. A cyclic composition is a sum of positive integers in which the order of the parts is considered up to cyclic permutation. In other words, it is the collection of components remaining in the cycle graph C_n on n vertices when one or more edges are removed, and rotations are considered equivalent. In an n-color composition, each part of size k is assigned one of k "colors" which may be represented graphically by marking one vertex in the part. (See A032198 for the number of cyclic n-color compositions.) The dual of a cyclic n-color composition is obtained by switching the roles of edges and vertices in C_n, then removing each edge that came from a previously marked vertex while marking each vertex that came from a previously removed edge. Each cyclic n-color composition of n either belongs to a dual pair or is self-dual. (See A365859 for the number of self-dual cyclic n-color compositions.)

A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 25.
Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color cyclic compositions, Discrete Mathematics 341 (2018), 3209-3226.

Cf. A001350, A032189, A032198, A365858, A365859.

N=99;  x='x+O('x^N); B(x)=x/(1-x^2);
A=Vec(sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k)))));
vector(#A\2,n,A[2*n]) \\ Joerg Arndt, Sep 22 2023

from sympy import totient, lucas, divisors
def A365857(n): return sum(totient((n<<1)//k)*(lucas(k)-((k&1^1)<<1)) for k in divisors(n<<1,generator=True))//n>>1 # Chai Wah Wu, Sep 23 2023

G.f.: (1/2)*(Sum_{k>=1} phi(k)/k * log((1-2*x^k+x^(2*k))/(1-3*x^k+x^(2*k))) + Sum_{m>=1} phi(2*m)/(2*m) * log((1+x^m-x^(2*m))/(1-x^m-x^(2*m)))).
a(n) = (1/(2*n)) * Sum_{k divides 2*n} phi(k)*A001350((2*n)/k).
a(n) = (A032198(n) + A365859(n))/2.

cp's OEIS Frontend

A001351 Associated Mersenne numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A129744 a(n) = -(u^n-1)*(v^n-1) with u = 1+sqrt(2), v = 1-sqrt(2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A152152 a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A288913 a(n) = Lucas(4*n + 3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Sage

Formula

A324489 a(n) = A324488(n)/n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A032189 Number of ways to partition n elements into pie slices each with an odd number of elements.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A324485 a(n) = A324484(n)/n.

Views

Author

A152152 a(n) = Product_{k=1..n} (1 + 4sin(2Pi*k/n)^2).