A005059 -id:A005059 - cp's OEIS frontend

A006516 a(n) = 2^(n-1)*(2^n - 1), n >= 0.

0, 1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528, 140737479966720, 562949936644096

Offset: 0

N. J. A. Sloane

a(n) is also the number of different lines determined by pair of vertices in an n-dimensional hypercube. The number of these lines modulo being parallel is in A003462. - Ola Veshta (olaveshta(AT)my-deja.com), Feb 15 2001
Let G_n be the elementary Abelian group G_n = (C_2)^n for n >= 1: A006516 is the number of times the number -1 appears in the character table of G_n and A007582 is the number of times the number 1. Together the two sequences cover all the values in the table, i.e., A006516(n) + A007582(n) = 2^(2n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 01 2001
a(n) is the number of n-letter words formed using four distinct letters, one of which appears an odd number of times. - Lekraj Beedassy, Jul 22 2003 [See, e.g., the Balakrishnan reference, problems 2.67 and 2.68, p. 69. - Wolfdieter Lang, Jul 16 2017]
Number of 0's making up the central triangle in a Pascal's triangle mod 2 gasket. - Lekraj Beedassy, May 14 2004
m-th triangular number, where m is the n-th Mersenne number, i.e., a(n)=A000217(A000225(n)). - Lekraj Beedassy, May 25 2004
Number of walks of length 2n+1 between two nodes at distance 3 in the cycle graph C_8. - Herbert Kociemba, Jul 02 2004
The sequence of fractions a(n+1)/(n+1) is the 3rd binomial transform of (1, 0, 1/3, 0, 1/5, 0, 1/7, ...). - Paul Barry, Aug 05 2005
Number of monic irreducible polynomials of degree 2 in GF(2^n)[x]. - Max Alekseyev, Jan 23 2006
(A007582(n))^2 + a(n)^2 = A007582(2n). E.g., A007582(3) = 36, a(3) = 28; A007582(6) = 2080. 36^2 + 28^2 = 2080. - Gary W. Adamson, Jun 17 2006
The sequence 6*a(n), n>=1, gives the number of edges of the Hanoi graph H_4^{n} with 4 pegs and n>=1 discs. - Daniele Parisse, Jul 28 2006
8*a(n) is the total border length of the 4*n masks used when making an order n regular DNA chip, using the bidimensional Gray code suggested by Pevzner in the book "Computational Molecular Biology." - Bruno Petazzoni (bruno(AT)enix.org), Apr 05 2007
If we start with 1 in binary and at each step we prepend 1 and append 0, we construct this sequence: 1 110 11100 1111000 etc.; see A109241(n-1). - Artur Jasinski, Nov 26 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which x does not equal y. - Ross La Haye, Jan 02 2008
Wieder calls these "conjoint usual 2-combinations." The set of "conjoint strict k-combinations" is the subset of conjoint usual k-combinations where the empty set and the set itself are excluded from possible selection. These numbers C(2^n - 2,k), which for k = 2 (i.e., {x,y} of the power set of a set) give {1, 0, 1, 15, 91, 435, 1891, 7875, 32131, 129795, 521731, ...}. - Ross La Haye, Jan 15 2008
If n is a member of A000043 then a(n) is also a perfect number (A000396). - Omar E. Pol, Aug 30 2008
a(n) is also the number whose binary representation is A109241(n-1), for n>0. - Omar E. Pol, Aug 31 2008
From Daniel Forgues, Nov 10 2009: (Start)
If we define a spoof-perfect number as:
A spoof-perfect number is a number that would be perfect if some (one or more) of its odd composite factors were wrongly assumed to be prime, i.e., taken as a spoof prime.
And if we define a "strong" spoof-perfect number as:
A "strong" spoof-perfect number is a spoof-perfect number where sigma(n) does not reveal the compositeness of the odd composite factors of n which are wrongly assumed to be prime, i.e., taken as a spoof prime.
The odd composite factors of n which are wrongly assumed to be prime then have to be obtained additively in sigma(n) and not multiplicatively.
Then:
If 2^n-1 is odd composite but taken as a spoof prime then 2^(n-1)*(2^n - 1) is an even spoof perfect number (and moreover "strong" spoof-perfect).
For example:
a(8) = 2^(8-1)*(2^8 - 1) = 128*255 = 32640 (where 255 (with factors 3*5*17) is taken as a spoof prime);
sigma(a(8)) = (2^8 - 1)*(255 + 1) = 255*256 = 2*(128*255) = 2*32640 = 2n is spoof-perfect (and also "strong" spoof-perfect since 255 is obtained additively);
a(11) = 2^(11-1)*(2^11 - 1) = 1024*2047 = 2096128 (where 2047 (with factors 23*89) is taken as a spoof prime);
sigma(a(11)) = (2^11 - 1)*(2047 + 1) = 2047*2048 = 2*(1024*2047) = 2*2096128 = 2n is spoof-perfect (and also "strong" spoof-perfect since 2047 is obtained additively).
I did a Google search and didn't find anything about the distinction between "strong" versus "weak" spoof-perfect numbers. Maybe some other terminology is used.
An example of an even "weak" spoof-perfect number would be:
n = 90 = 2*5*9 (where 9 (with factors 3^2) is taken as a spoof prime);
sigma(n) = (1+2)*(1+5)*(1+9) = 3*(2*3)*(2*5) = 2*(2*5*(3^2)) = 2*90 = 2n is spoof-perfect (but is not "strong" spoof-perfect since 9 is obtained multiplicatively as 3^2 and is thus revealed composite).
Euler proved:
If 2^k - 1 is a prime number, then 2^(k-1)*(2^k - 1) is a perfect number and every even perfect number has this form.
The following seems to be true (is there a proof?):
If 2^k - 1 is an odd composite number taken as a spoof prime, then 2^(k-1)*(2^k - 1) is a "strong" spoof-perfect number and every even "strong" spoof-perfect number has this form?
There is only one known odd spoof-perfect number (found by Rene Descartes) but it is a "weak" spoof-perfect number (cf. 'Descartes numbers' and 'Unsolved problems in number theory' links below). (End)
a(n+1) = A173787(2*n+1,n); cf. A020522, A059153. - Reinhard Zumkeller, Feb 28 2010
Also, row sums of triangle A139251. - Omar E. Pol, May 25 2010
Starting with "1" = (1, 1, 2, 4, 8, ...) convolved with A002450: (1, 5, 21, 85, 341, ...); and (1, 3, 7, 15, 31, ...) convolved with A002001: (1, 3, 12, 48, 192, ...). - Gary W. Adamson, Oct 26 2010
a(n) is also the number of toothpicks in the corner toothpick structure of A153006 after 2^n - 1 stages. - Omar E. Pol, Nov 20 2010
The number of n-dimensional odd theta functions of half-integral characteristic. (Gunning, p.22) - Michael Somos, Jan 03 2014
a(n) = A000217((2^n)-1) = 2^(2n-1) - 2^(n-1) is the nearest triangular number below 2^(2n-1); cf. A007582, A233327. - Antti Karttunen, Feb 26 2014
a(n) is the sum of all the remainders when all the odd numbers < 2^n are divided by each of the powers 2,4,8,...,2^n. - J. M. Bergot, May 07 2014
Let b(m,k) = number of ways to form a sequence of m selections, without replacement, from a circular array of m labeled cells, such that the first selection of a cell whose adjacent cells have already been selected (a "first connect") occurs on the k-th selection. b(m,k) is defined for m >=3, and for 3 <= k <= m. Then b(m,k)/2m ignores rotations and reflection. Let m=n+2, then a(n) = b(m,m-1)/2m. Reiterated, a(n) is the (m-1)th column of the triangle b(m,k)/2m, whose initial rows are (1), (1 2), (2 6 4), (6 18 28 8), (24 72 128 120 16), (120 360 672 840 496 32), (720 2160 4128 5760 5312 2016 64); see A249796. Note also that b(m,3)/2m = n!, and b(m,m)/2m = 2^n. Proofs are easy. - Tony Bartoletti, Oct 30 2014
Beginning at a(1) = 1, this sequence is the sum of the first 2^(n-1) numbers of the form 4*k + 1 = A016813(k). For example, a(4) = 120 = 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29. - J. M. Bergot, Dec 07 2014
a(n) is the number of edges in the (2^n - 1)-dimensional simplex. - Dimitri Boscainos, Oct 05 2015
a(n) is the number of linear elements in a complete plane graph in 2^n points. - Dimitri Boscainos, Oct 05 2015
a(n) is the number of linear elements in a complete parallelotope graph in n dimensions. - Dimitri Boscainos, Oct 05 2015
a(n) is the number of lattices L in Z^n such that the quotient group Z^n / L is C_4. - Álvar Ibeas, Nov 26 2015
a(n) gives the quadratic coefficient of the polynomial ((x + 1)^(2^n) + (x - 1)^(2^n))/2, cf. A201461. - Martin Renner, Jan 14 2017
Let f(x)=x+2*sqrt(x) and g(x)=x-2*sqrt(x). Then f(4^n*x)=b(n)*f(x)+a(n)*g(x) and g(4^n*x)=a(n)*f(x)+b(n)*g(x), where b is A007582. - Luc Rousseau, Dec 06 2018
For n>=1, a(n) is the covering radius of the first order Reed-Muller code RM(1,2n). - Christof Beierle, Dec 22 2021
a(n) =

			G.f. = x + 6*x^2 + 28*x^3 + 120*x^4 + 496*x^5 + 2016*x^6 + 8128*x^7 + 32640*x^8 + ...

V. K. Balakrishnan, Theory and problems of Combinatorics, "Schaum's Outline Series", McGraw-Hill, 1995, p. 69.
Martin Gardner, Mathematical Carnival, "Pascal's Triangle", p. 201, Alfred A. Knopf NY, 1975.
Richard K. Guy, Unsolved problems in number theory, (p. 72).
Ross Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
Clifford A. Pickover, Wonders of Numbers, Chap. 55, Oxford Univ. Press NY 2000.
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..200
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
M. Archibald, A. Blecher, A. Knopfmacher and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
William Banks, Ahmet Güloğlu, Wesley Nevans and Filip Saidak, Descartes numbers, Anatomy of integers, 167-173, CRM Proc. Lecture Notes, 46, Amer. Math. Soc., Providence, RI, 2008. MathSciNet review (subscription required).
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
John Elias, Illustration of initial terms: Mersenne-Sierpinski Triangles
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
R. C. Gunning, Riemann Surfaces and Second-Order Theta Functions, Springer-Verlag, 1976. See page 22.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
T. Helleseth, T. Klove and J.Mykkeltveit, On the covering radius of binary codes (Corresp.), IEEE Transactions on Information Theory, Vol. 24 (1978).
V. Meally, Letter to N. J. A. Sloane, May 1975
Axel Muller, Metod Saniga, Alain Giorgetti, Henri de Boutray, and Frédéric Holweck, New and improved bounds on the contextuality degree of multi-qubit configurations, arXiv:2305.10225 [quant-ph], 2023.
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.
O. S. Rothaus, On "bent" functions, Journal of Combinatorial Theory, Series A, Vol. 20 (1976).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Equals A006095(n+1) - A006095(n). In other words, A006095 gives the partial sums.
Cf. A000108, A007582, A010036, A016290, A003462, A079598, A134346, A048473, A151821, A010684.
Cf. A000043, A000396. - Omar E. Pol, Aug 30 2008
Cf. A109241, A139251, A153006. - Omar E. Pol, Aug 31 2008, May 25 2010, Nov 20 2010
Cf. A002450, A002001. - Gary W. Adamson, Oct 26 2010
Cf. A049072, A000384, A201461, A005059 (binomial transform, and special 5-letter words), A065442, A211705.
Cf. A171476.

List([0..25],n->2^(n-1)*(2^n-1)); # Muniru A Asiru, Dec 06 2018

a006516 n = a006516_list !! n
a006516_list = 0 : 1 :
    zipWith (-) (map (* 6) $ tail a006516_list) (map (* 8) a006516_list)
-- Reinhard Zumkeller, Oct 25 2013

[2^(n-1)*(2^n - 1): n in [0..30]]; // Vincenzo Librandi, Oct 31 2014

GBC := proc(n,k,q) local i; mul( (q^(n-i)-1)/(q^(k-i)-1),i=0..k-1); end; # define q-ary Gaussian binomial coefficient [ n,k ]_q
[ seq(GBC(n+1,2,2)-GBC(n,2,2), n=0..30) ]; # produces A006516
A006516:=1/(4*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation
seq(binomial(2^n, 2), n=0..19); # Zerinvary Lajos, Feb 22 2008

Table[2^(n - 1)(2^n - 1), {n, 0, 30}] (* or *) LinearRecurrence[{6, -8}, {0, 1}, 30] (* Harvey P. Dale, Jul 15 2011 *)

A006516(n):=2^(n-1)*(2^n - 1)$ makelist(A006516(n),n,0,30); /* Martin Ettl, Nov 15 2012 */

a(n)=(1<Charles R Greathouse IV, Jun 10 2011

vector(100, n, n--; 2^(n-1)*(2^n-1)) \\ Altug Alkan, Oct 06 2015

for n in range(0, 30): print(2**(n-1)*(2**n - 1), end=', ') # Stefano Spezia, Dec 06 2018

[lucas_number1(n,6,8) for n in range(24)]  # Zerinvary Lajos, Apr 22 2009

[(4**n - 2**n) / 2 for n in range(24)]  # Zerinvary Lajos, Jun 05 2009

G.f.: x/((1 - 2*x)*(1 - 4*x)).
E.g.f. for a(n+1), n>=0: 2*exp(4*x) - exp(2*x).
a(n) = 2^(n-1)*Stirling2(n+1,2), n>=0, with Stirling2(n,m)=A008277(n,m).
Second column of triangle A075497.
a(n) = Stirling2(2^n,2^n-1) = binomial(2^n,2). - Ross La Haye, Jan 12 2008
a(n+1) = 4*a(n) + 2^n. - Philippe Deléham, Feb 20 2004
Convolution of 4^n and 2^n. - Ross La Haye, Oct 29 2004
a(n+1) = Sum_{k=0..n} Sum_{j=0..n} 4^(n-j)*binomial(j,k). - Paul Barry, Aug 05 2005
a(n+2) = 6*a(n+1) - 8*a(n), a(1) = 1, a(2) = 6. - Daniele Parisse, Jul 28 2006 [Typo corrected by Yosu Yurramendi, Aug 06 2008]
Row sums of triangle A134346. Also, binomial transform of A048473: (1, 5, 17, 53, 161, ...); double bt of A151821: (1, 4, 8, 16, 32, 64, ...) and triple bt of A010684: (1, 3, 1, 3, 1, 3, ...). - Gary W. Adamson, Oct 21 2007
a(n) = 3*Stirling2(n+1,4) + Stirling2(n+2,3). - Ross La Haye, Jun 01 2008
a(n) = (4^n - 2^n)/2.
a(n) = A153006(2^n-1). - Omar E. Pol, Nov 20 2010
Sum_{n>=1} 1/a(n) = 2 * (A065442 - 1) = A211705 - 2. - Amiram Eldar, Dec 24 2020
a(n) = binomial(2*n+2, n+1) - Catalan(n+2). - N. J. A. Sloane, Apr 01 2021
a(n) = A171476(n-1), for n >= 1, and a(0) = 0. - Wolfdieter Lang, Jul 27 2022

A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

Original entry on oeis.org

3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789

Offset: 1

Alexander Adamchuk, Sep 14 2006

(3^k + 5^k)/8 = A074606(k)/8 = A081186(k)/4.
Corresponding primes of the form (3^k + 5^k)/2^3 are listed in {A121938(n)} = {A079773(a(n))} = {19, 421, 10039, 95383574161, 2384331073699, ...}.
No other terms less than 100000. - Robert Price, Apr 28 2012

Cf. A074606, A081186, A121824, A121877, A005058, A005059, A121938, A109347, A079773.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}]

select(n->isprime((3^n + 5^n)/8), vector(2000,i,i)) \\ Charles R Greathouse IV, Feb 13 2011

a(11)-a(15) from Robert Price, Apr 28 2012

A005058 a(n) = 5^n - 3^n.

Original entry on oeis.org

0, 2, 16, 98, 544, 2882, 14896, 75938, 384064, 1933442, 9706576, 48650978, 243609184, 1219108802, 6098732656, 30503229218, 152544843904, 762810312962, 3814309845136, 19072324066658, 95363944856224

Offset: 0

N. J. A. Sloane

The resistance distance between two apex nodes of the n-Hanoi graph is given by a(n)/3^n. - Pontus von Brömssen, Nov 01 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..300
M. A. Alekseyev and T. Berger, Solving the Tower of Hanoi with random moves. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
Index entries for linear recurrences with constant coefficients, signature (8, -15).

Cf. A005059.

[5^n - 3^n: n in [0..30]]; // Vincenzo Librandi, Jun 08 2013

Join[{a = 0, b = 1}, Table[c = 8*b - 15 * a; a = b; b = c, {n, 60}]] * 2 (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
5^Range[0, 20] - 3^Range[0, 20] (* Harvey P. Dale, Jan 23 2011 *)
CoefficientList[ Series[1/(1 - 5 x) - 1/(1 - 3 x), {x, 0, 20}], x] (* Robert G. Wilson v *)
Range[0, 20]! CoefficientList[Series[Exp[5 x] - Exp[3 x], {x, 0, 20}], x] (* Robert G. Wilson v *)
a[0] := 0; a[1] := 2; a[n_] := 8 a[n - 1] - 15 a[n - 2]; Table[a[n], {n, 0, 24}] (* Alonso del Arte, Jan 23 2011 *)

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-5*x) - 1/(1-3*x).
E.g.f.: e^(5*x) - e^(3*x). (End)
a(n) = 8*a(n - 1) - 15*a(n - 2) for n>1, a(0)=0, a(1)=2. - Vincenzo Librandi, Jan 24 2011

A081186 4th binomial transform of (1,0,1,0,1,...), A059841.

Original entry on oeis.org

1, 4, 17, 76, 353, 1684, 8177, 40156, 198593, 986404, 4912337, 24502636, 122336033, 611148724, 3054149297, 15265963516, 76315468673, 381534296644, 1907542343057, 9537324294796, 47685459212513, 238423809278164, 1192108586037617, 5960511549128476

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A007582.
a(n) is a companion to A005059(n): a(n) + A005059(n) = 5^n; e.g. a(4) = A005059(4) = 353 + 272 = 625 = 5^4. - Gary W. Adamson, Jun 30 2006
Number of words of length n from an alphabet of 5 letters in which a chosen letter appears an even number of times. - James Mahoney, Feb 03 2012 [See a comment in A007582, also for crossrefs. for the 1- to 11-letter word cases. - Wolfdieter Lang, Jul 17 2017]
The sequence of fractions x(n) = a(n+1)/a(n) satisfies a simple recurrence x(n+1) = 108 - (815 - 1500 / x(n-1)) / x(n) known as Muller's recurrence. It is used for the demonstration of an unexpected failure of floating-point computations. - Andrey Zabolotskiy, Sep 17 2019

			Say the alphabet is {a,b,c,d,e} and we want to know how many words of length one and two contain c an even number of times. a(1) = 4, which we can see by the four words {(a),(b),(d),(e)} and a(2) = 17, which we can see by the seventeen words {(a,a), (a,b), (a,d), (a,e), (b,a), (b,b), (b,d), (b,e), (c,c), (d,a), (d,b), (d,d), (d,e), (e,a), (e,b), (e,d), (e,e)}. - _James Mahoney_, Feb 03 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Christian Hill, Muller's Recurrence, 2017.
Index entries for linear recurrences with constant coefficients, signature (8,-15).

Cf. A005059, A007582.

List([0..25], n-> (3^n + 5^n)/2); # G. C. Greubel, Dec 26 2019

[3^n/2+5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

seq( (3^n + 5^n)/2, n=0..30); # G. C. Greubel, Dec 26 2019

CoefficientList[Series[(1-4x)/((1-3x)(1-5x)), {x,0,25}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{8,-15},{1,4},30] (* Harvey P. Dale, Apr 13 2019 *)

vector(31, n,(3^(n-1) + 5^(n-1))/2 ) \\ G. C. Greubel, Dec 26 2019

[(3^n + 5^n)/2 for n in (0..25)] # G. C. Greubel, Dec 26 2019

a(n) = 8*a(n-1) - 15*a(n-2) with n>1, a(0)=1, a(1)=4.
G.f.: (1-4*x)/((1-3*x)*(1-5*x)).
a(n) = (3^n + 5^n)/2.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*4^(n-2*k).
E.g.f.: exp(4*x) * cosh(x). - Paul Barry, Oct 06 2004

A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

Original entry on oeis.org

0, 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509, 39848449432985688, 279034513462540441, 1953718431395986212

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A081199.
Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X ... n-dimensional cubic grid. - Ron Hardin, May 24 2010
a(n) is also the total number of words of length n, over an alphabet of seven letters, of which one of them appears an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5-, 6- and 8-letter case analogs see A131577, A003462, A005059, A081199, A081201 respectively. - Wolfdieter Lang, Jul 17 2017

			The a(2) = 12 words of length 2 over {A, B, C, D, E, F, G} with say, A, appearing an odd number of times (that is once) are: AB, AC, AD, AE, AF, AG; BA, CA, DA, EA, FA, GA. - _Wolfdieter Lang_, Jul 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Index entries for linear recurrences with constant coefficients, signature (12,-35).

Cf. A000035, A003462, A005059, A006516, A081199, A081201 (binomial transform, and 8-letter analog), A121213, A131577.

Apart from offset same as A016161.

[7^n/2-5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12,-35},{0,1},30] (* Harvey P. Dale, Feb 07 2014 *)

[lucas_number1(n,12,35) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009

a(n) = 12*a(n-1) - 35*a(n-2), a(0) = 0, a(1) = 1.
G.f.: x/((1-5*x)*(1-7*x)).
a(n) = 7^n/2 - 5^n/2.
a(n) = Sum_{k=0..n-1} 7^k * 5^(n-k-1), with a(0)=0. - Reinhard Zumkeller, Aug 01 2010
a(n) = A121213(n)/2. - Reinhard Zumkeller, Aug 01 2010
E.g.f.: exp(5*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jun 19 2021

A120612 For n>1, a(n) = 2a(n-1) + 15a(n-2); a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 17, 49, 353, 1441, 8177, 37969, 198593, 966721, 4912337, 24325489, 122336033, 609554401, 3054149297, 15251614609, 76315468673, 381405156481, 1907542343057, 9536162033329, 47685459212513, 238413348924961, 1192108586037617, 5960417405949649

Offset: 0

Gary W. Adamson, Jun 17 2006

Characteristic polynomial of matrix M = x^2 - 2x - 15. a(n)/a(n-1) tends to 5, largest eigenvalue of M and a root of the characteristic polynomial.
Binomial transform of [1, 0, 16, 0, 256, 0, 4096, 0, 65536, 0, ...]=: powers of 16 (A001025) with interpolated zeros. - Philippe Deléham, Dec 02 2008
a(n) is the number of compositions of n when there are 1 type of 1 and 16 types of other natural numbers. - Milan Janjic, Aug 13 2010

			a(4) = 353 = 2*49 + 15*17 = 2*a(3) + 15*a(2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,15).

Cf. A005059, A081186, A059841.

Table[(5^n+(-1)^n*3^n)/2,{n,1,30}] (* Alexander Adamchuk, Aug 31 2006 *)
a[n_] := (5^n + (-3)^n)/2; Array[a, 24, 0] (* Or *)
CoefficientList[Series[(1 + 15 x)/(1 - 2 x - 15 x^2), {x, 0, 23}], x] (* Or *)
LinearRecurrence[{2, 15}, {1, 1}, 25] (* Or *)
Table[ MatrixPower[{{1, 2}, {8, 1}}, n][[1, 1]], {n, 0, 30}]  (* Robert G. Wilson v, Sep 18 2013 *)

a(n)=([1,4; 4,1]^n)[1,1] \\ Charles R Greathouse IV, Oct 16 2013

concat(1, Vec((15*x+1)/(-15*x^2-2*x+1) + O(x^100))) \\ Colin Barker, Mar 12 2014

a(n) = ( 5^n + (-1)^n * 3^n ) / 2 \\ Charles R Greathouse IV, May 18 2015

Let M = the 2 X 2 matrix [1,4; 4,1], then a(n) = M^n * [1,0], left term.
From Alexander Adamchuk, Aug 31 2006: (Start)
a(n) = ( 5^n + (-1)^n * 3^n ) / 2.
a(2n+1) = A005059(2n+1).
a(2n) = A081186(2n). (End)
a(n) = Sum_{k=0..n} A098158(n,k)*16^(n-k). - Philippe Deléham, Dec 26 2007
If p(1)=1, and p(i)=16, (i > 1), and if A is Hessenberg matrix of order n defined by: A(i,j) = p(j-i+1), (i <= j), A(i,j)=-1, (i = j+1), and A(i,j)=0 otherwise. Then, for n >= 1, a(n)=det A. - Milan Janjic, Apr 29 2010

More terms from Alexander Adamchuk, Aug 31 2006
Entry revised by Philippe Deléham, Dec 02 2008
More terms from Colin Barker, Mar 12 2014

A016209 Expansion of 1/((1-x)(1-3x)(1-5x)).

Original entry on oeis.org

1, 9, 58, 330, 1771, 9219, 47188, 239220, 1205941, 6059229, 30384718, 152189310, 761743711, 3811110039, 19062724648, 95335146600, 476740303081, 2383895225649, 11920057258978, 59602029687090

Offset: 0

N. J. A. Sloane

For a combinatorial interpretation following from a(n) = A039755(n+2,2) = h^{(3)}A039755.%20-%20_Wolfdieter%20Lang">n, the complete homogeneous symmetric function of degree n in the symbols {1, 3, 5} see A039755. - _Wolfdieter Lang, May 26 2017

			a(2) = h^{(3)}_2 = 1^2 + 3^2 + 5^2 + 1^1*(3^1 + 5^1) + 3^1*5^1 = 58. - _Wolfdieter Lang_, May 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A016218, A016208, A000392, A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256, A039755, A021424.

[(5^(n+2)-2*3^(n+2)+1)/8: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

A016209 := proc(n) (5^(n+2)-2*3^(n+2)+1)/8; end proc: # R. J. Mathar, Mar 22 2011

Join[{a=1,b=9},Table[c=8*b-15*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2011 *)
CoefficientList[Series[1/((1-x)(1-3x)(1-5x)),{x,0,30}],x] (* or *) LinearRecurrence[ {9,-23,15},{1,9,58},30] (* Harvey P. Dale, Feb 20 2020 *)

PARI
```
a(n)=if(n<0,0,n+=2; (5^n-2*3^n+1)/8)
```

a(n) = A039755(n+2, 2).
a(n) = (5^(n+2) - 2*3^(n+2)+1)/8 = a(n-1) + A005059(n+1) = 8*a(n-1) - 15*a(n-2) + 1 = (A003463(n+2) - A003462(n+2))/2. - Henry Bottomley, Jun 06 2000
G.f.: 1/((1-x)(1-3*x)(1-5*x)). See the name.
E.g.f.: (25*exp(5*x) - 18*exp(3*x) + exp(x))/8, from the e.g.f. of the third column (k=2) of A039755. - Wolfdieter Lang, May 26 2017

A081199 5th binomial transform of (0,1,0,1,...), A000035.

Original entry on oeis.org

0, 1, 10, 76, 520, 3376, 21280, 131776, 807040, 4907776, 29708800, 179301376, 1080002560, 6496792576, 39047864320, 234555621376, 1408407470080, 8454739787776, 50745618595840, 304542431051776, 1827529464217600, 10966276296933376, 65802055828111360, 394829927154712576

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A005059.
Conjecture (verified up to a(9)): Number of collinear 4-tuples of points in a 4 X 4 X 4 X ... n-dimensional cubic grid. - R. H. Hardin, May 24 2010
a(n) is also the total number of words of length n, over an alphabet of six letters, of which one of them appears an odd number of times. See a Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5- and 7-letter case analogs see A131577, A003462, A005059 and A081200, respectively. - Wolfdieter Lang, Jul 16 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-24).

Cf. A000035, A003462, A005059, A006516, A081200 (binomial transform of a(n), and 7-letter case), A131577.

Apart from offset the same as A016149.

[6^n/2-4^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

seq(add(2^(2*n-k)*binomial(n,k)/2,k=1..n),n=0..20); # Zerinvary Lajos, Apr 18 2009

CoefficientList[Series[x / ((1 - 4 x) (1 - 6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{10, -24}, {0, 1}, 21] (* Michael De Vlieger, Jul 16 2017 *)

a(n) = 10*a(n-1) - 24*a(n-2) with n>1, a(0)=0, a(1)=1.
G.f.: x/((1-4*x)*(1-6*x)).
a(n) = 6^n/2 - 4^n/2.
E.g.f.: exp(4*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jul 23 2024

A162590 Polynomials with e.g.f. exp(x*t)/csch(t), triangle of coefficients read by rows.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 0, 4, 0, 4, 0, 1, 0, 10, 0, 5, 0, 0, 6, 0, 20, 0, 6, 0, 1, 0, 21, 0, 35, 0, 7, 0, 0, 8, 0, 56, 0, 56, 0, 8, 0, 1, 0, 36, 0, 126, 0, 84, 0, 9, 0, 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 1, 0, 55, 0, 330, 0, 462, 0, 165, 0, 11, 0, 0, 12, 0, 220, 0, 792, 0, 792, 0

Offset: 0

Peter Luschny, Jul 07 2009

Comment from Peter Bala (Dec 06 2011): "Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of (I-t*M)^-1 (apart from the initial 1) lists the row polynomials for" A196776(n,k), which gives the number of ordered partitions of an n set into k odd-sized blocks. - Peter Luschny, Dec 06 2011

The n-th row of the triangle is formed by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probability of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms in the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k) as the (2k)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k). - Luca Onnis, Oct 29 2023

			Triangle begins:
  0
  1,  0
  0,  2,  0
  1,  0,  3,  0
  0,  4,  0,  4,  0
  1,  0, 10,  0,  5,  0
  0,  6,  0, 20,  0,  6,  0
  1,  0, 21,  0, 35,  0,  7,  0
  ...
  p[0](x) = 0;
  p[1](x) = 1
  p[2](x) = 2*x
  p[3](x) = 3*x^2 +  1
  p[4](x) = 4*x^3 +  4*x
  p[5](x) = 5*x^4 + 10*x^2 +  1
  p[6](x) = 6*x^5 + 20*x^3 +  6*x
  p[7](x) = 7*x^6 + 35*x^4 + 21*x^2 + 1
  p[8](x) = 8*x^7 + 56*x^5 + 56*x^3 + 8*x
.
Cf. the triangle of odd-numbered terms in rows of Pascal's triangle (A034867).
p[n] (k), n=0,1,...
k=0:  0, 1,  0,   1,    0,     1, ... A000035, (A059841)
k=1:  0, 1,  2,   4,    8,    16, ... A131577, (A000079)
k=2:  0, 1,  4,  13,   40,   121, ... A003462
k=3:  0, 1,  6,  28,  120,   496, ... A006516
k=4:  0, 1,  8,  49,  272,  1441, ... A005059
k=5:  0, 1, 10,  76,  520,  3376, ... A081199, (A016149)
k=6:  0, 1, 12, 109,  888,  6841, ... A081200, (A016161)
k=7:  0, 1, 14, 148, 1400, 12496, ... A081201, (A016170)
k=8:  0, 1, 16, 193, 2080, 21121, ... A081202, (A016178)
k=9:  0, 1, 18, 244, 2952, 33616, ... A081203, (A016186)
k=10: 0, 1, 20, 301, 4040, 51001, ... ......., (A016190)
.
p[n] (k), k=0,1,...
p[0]: 0,  0,   0,    0,    0,     0, ... A000004
p[1]: 1,  1,   1,    1,    1,     1, ... A000012
p[2]: 0,  2,   4,    6,    8,    10, ... A005843
p[3]: 1,  4,  13,   28,   49,    76, ... A056107
p[4]: 0,  8,  40,  120,  272,   520, ... A105374
p[5]: 1, 16, 121,  496, 1441,  3376, ...
p[6]: 0, 32, 364, 2016, 7448, 21280, ...

Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.
Luca Onnis, Animation of the Ehrenfest model.
Wikipedia, Ehrenfest model.

Cf. A119467.

# Polynomials: p_n(x)
p := proc(n,x) local k;
pow := (n,k) -> `if`(n=0 and k=0,1,n^k);
add((k mod 2)*binomial(n,k)*pow(x,n-k),k=0..n) end;
# Coefficients: a(n)
seq(print(seq(coeff(i!*coeff(series(exp(x*t)/csch(t), t,16),t,i),x,n), n=0..i)), i=0..8);

p[n_, x_] := Sum[Binomial[n, 2*k-1]*x^(n-2*k+1), {k, 0, n+2}]; row[n_] := CoefficientList[p[n, x], x] // Append[#, 0]&; Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
n = 15; "n-th row"
mat = Table[Table[0, {j, 1, n + 1}], {i, 1, n + 1}];
mat[[1, 2]] = 1;
mat[[n + 1, n]] = 1;
For[i = 2, i <= n, i++, mat[[i, i - 1]] = (i - 1)/n ];
For[i = 2, i <= n, i++, mat[[i, i + 1]] = (n - i + 1)/n];
mat // MatrixForm;
P2 = Dot[mat, mat];
R1 = Simplify[
  Eigenvectors[Transpose[P2]][[1]]/
   Total[Eigenvectors[Transpose[P2]][[1]]]]
R2 = Table[Dot[R1, Transpose[mat][[k]]], {k, 1, n + 1}]
even = R1*2^(n - 1) (* Luca Onnis, Oct 29 2023 *)

p_n(x) = Sum_{k=0..n} (k mod 2)*binomial(n,k)*x^(n-k).
E.g.f.: exp(x*t)/csch(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2+1)*(t^3/3!) + ...
The 'co'-polynomials with generating function exp(x*t)*sech(t) are the Swiss-Knife polynomials (A153641).

cp's OEIS Frontend

A121877 Numbers k such that (5^k - 3^k)/2 = A005059(k) is prime.

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A006516 a(n) = 2^(n-1)*(2^n - 1), n >= 0.

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A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

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A005058 a(n) = 5^n - 3^n.

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A081186 4th binomial transform of (1,0,1,0,1,...), A059841.

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A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

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A120612 For n>1, a(n) = 2a(n-1) + 15a(n-2); a(0)=1, a(1)=1.