Displaying 1-10 of 13 results found.
Period 2: Repeat [1,0]. a(n) = 1 - (n mod 2); Characteristic function of even numbers.
+10
245
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0
COMMENTS
When viewed as a triangular array, the row sum values are 0 1 1 1 2 3 3 3 4 5 5 5 6 ... ( A004525). This is the r=0 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found. Elementary Cellular Automata rule 77 produces this sequence. See Wolfram, Weisstein and Index links below. - Robert Price, Jan 30 2016
FORMULA
G.f.: 1/(1-x^2).
E.g.f.: cosh(x).
a(n) = (n+1) mod 2.
a(n) = 1/2 + (-1)^n/2. (End)
Additive with a(p^e) = 1 if p = 2, 0 otherwise.
E.g.f.: cosh(x) = 1 + x^2/(Q(0) - x^2); Q(k) = 8k + 2 + x^2/(1 + (2k + 1)*(2k + 2)/Q(k + 1)); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011 E.g.f.: cosh(x) = 1/2*Q(0); Q(k) = 1 + 1/(1 - x^2/(x^2 + (2k + 1)*(2k + 2)/Q(k + 1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011 E.g.f.: cosh(x) = E(0)/(1-x) where E(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013 For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 0;
1, 0, 1, 0;
1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 0;
1, 0, 1, 0, 1, 0, 1, 0;
1, 0, 1, 0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0;
...
MATHEMATICA
CoefficientList[Series[1/(1 - x^2), {x, 0, 104}], x] (* or *)
Table[QBinomial[n, 1, -1], {n, 1, 74}] (* John Keith, Jun 28 2021 *)
PROG
(PARI) a(n)=(n+1)%2; \\ or 1-n%2 as in NAME.
(Haskell)
a059841 n = (1 -) . (`mod` 2)
a059841_list = cycle [1, 0]
(Python)
CROSSREFS
One's complement of A000035 (essentially the same, but shifted once).
a(n) = 2^(n-1)*(1+2^n).
(Formerly M2849)
+10
74
1, 3, 10, 36, 136, 528, 2080, 8256, 32896, 131328, 524800, 2098176, 8390656, 33558528, 134225920, 536887296, 2147516416, 8590000128, 34359869440, 137439215616, 549756338176, 2199024304128, 8796095119360, 35184376283136, 140737496743936, 562949970198528
COMMENTS
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 Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_8. Example: a(1)=3 because in the cycle ABCDEFGH we have three walks of length 3 between A and B: ABAB, ABCB and AHAB. - Emeric Deutsch, Apr 01 2004 Smallest number containing in its binary representation two equal non-overlapping subwords of length n: A097295(a(n))=n and A097295(m)<n for m<a(n). - Reinhard Zumkeller, Aug 04 2004 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 either x equals y or x does not equal y. - Ross La Haye, Jan 02 2008 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). This is just a simpler statement of my previous comment for this sequence. - Ross La Haye, Jan 10 2008 a(n+1) written in base 2: 11, 1010, 100100, 10001000, 1000010000, ..., i.e., number 1, n times 0, number 1, n times 0 ( A163449(n)). - Jaroslav Krizek, Jul 27 2009 Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times sum_{k=0..n} binomial(n,k)*k^q, then A007582(x)= sum_{k=0..x-1} T(x,k)*2^k. - John M. Campbell, Nov 16 2011 a(n) gives the number of pairs (r, s) such that 0 <= r <= s <= (2^n)-1 that satisfy AND(r, s, XOR(r, s)) = 0. - Ramasamy Chandramouli, Aug 30 2012 Consider the quantum spin-1/2 chain with even number of sites L (physics, condensed matter theory). The spectrum of the Hamiltonian can be classified according to symmetries. If the only symmetry of the spin Hamiltonian is Parity, i.e., reflection with respect to the middle of the chain (see e.g. the transverse-field Ising model with open boundary conditions), then the dimension of the p=+1 parity sector is given by a(n) with n=L/2. - Marin Bukov, Mar 11 2016 a(n) is also the total number of words of length n, over an alphabet of four letters, of which one of them appears an even number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter odd case), and the Balakrishnan reference there. For the 1- to 11-letter cases, see the crossrefs. - Wolfdieter Lang, Jul 17 2017 a(n) is the number of nonisomorphic spanning trees of the cyclic snake formed with n+1 copies of the cycle on 4 vertices. A cyclic snake is a connected graph whose block-cutpoint is a path and all its n blocks are isomorphic to the cycle C_m. - Christian Barrientos, Sep 05 2024 Also, with offset 1, the cogrowth sequence of the dihedral group with 16 elements, D8 = <S,T | S^8, T^2, (ST)^2>. - Sean A. Irvine, Nov 06 2024
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: (1-3*x)/((1-2*x)*(1-4*x)). C(1+2^n, 2) where C(n, 2) is n-th triangular number A000217. E.g.f.: exp(3*x)*cosh(x). - Paul Barry, Apr 07 2003 a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*3^(n-2*k). - Paul Barry, May 08 2003 a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) + StirlingS2(n+1,4)). - Ross La Haye, Mar 01 2008 a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) + StirlingS2(n+1,4)). - Ross La Haye, Apr 02 2008 a(n) = Sum_{k=-floor(n/4)..floor(n/4)} binomial(2*n,n+4*k)/2. - Mircea Merca, Jan 28 2012 G.f.: Q(0)/2 where Q(k) = 1 + 2^k/(1 - 2*x/(2*x + 2^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013 a(n) = (1/3) * Sum_{k=2^n..2^(n+1)} k. - J. M. Bergot, Jan 26 2015 a(n+1) = 2*a(n) + 4^n. - Yuchun Ji, Mar 10 2017
MATHEMATICA
LinearRecurrence[{6, -8}, {1, 3}, 30] (* Harvey P. Dale, Apr 08 2013 *)
PROG
(PARI) a(n)=if(n<0, 0, 2^(n-1)*(1+2^n))
(PARI) a(n)=sum(k=-n\4, n\4, binomial(2*n+1, n+1+4*k))
CROSSREFS
The number of words of length n with m letters, one of them appearing an even number of times is for m = 1..11: A000035, A011782, A007051, A007582, A081186, A081187, A081188, A081189, A081190, A060531, A081192. - Wolfdieter Lang, Jul 17 2017
Numbers k such that (5^k - 3^k)/2 = A005059(k) is prime.
+10
53
13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, 128941, 147571, 182099, 866029
COMMENTS
All terms are primes. Their indices are listed in A123704. Corresponding primes are listed in A123705.
MATHEMATICA
Do[f=(5^n-3^n)/2; If[PrimeQ[f], Print[{n, f}]], {n, 1, 300}]
Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.
+10
31
3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789
COMMENTS
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
MATHEMATICA
Do[f=5^n+3^n; If[PrimeQ[f/2^3], Print[{n, f/2^3}]], {n, 1, 1231}]
Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)
+10
19
1, 1, 3, 5, 15, 33, 91, 221, 583, 1465, 3795, 9653, 24831, 63441, 162763, 416525, 1067575, 2733673, 7003971, 17938661, 45954543, 117709185, 301527355, 772364093, 1978473511
COMMENTS
Suppose that p=p(0)*x^n+p(1)*x^(n-1)+...+p(n-1)*x+p(n) is a polynomial of positive degree and that Q is a sequence of polynomials: q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k), for k=0,1,2,... The Q-downstep of p is the polynomial given by D(p)=p(0)*q(n-1,x)+p(1)*q(n-2,x)+...+p(n-1)*q(0,x)+p(n).
Since degree(D(p))<degree(p), the result of n applications of D is a constant, which we call the Q-residue of p. If p is a constant to begin with, we define D(p)=p.
Example: let p(x)=2*x^3+3*x^2+4*x+5 and q(k,x)=(x+1)^k.
D(p)=2(x+1)^2+3(x+1)+4(1)+5=2x^2+7x+14
D(D(p))=2(x+1)+7(1)+14=2x+23
D(D(D(p)))=2(1)+23=25;
the Q-residue of p is 25.
We may regard the sequence Q of polynomials as the triangular array formed by coefficients:
t(0,0)
t(1,0)....t(1,1)
t(2,0)....t(2,1)....t(2,2)
t(3,0)....t(3,1)....t(3,2)....t(3,3)
and regard p as the vector (p(0),p(1),...,p(n)). If P is a sequence of polynomials [or triangular array having (row n)=(p(0),p(1),...,p(n))], then the Q-residues of the polynomials form a numerical sequence.
Following are examples in which Q is the triangle given by t(i,j)=1 for 0<=i<=j:
Q.....P...................Q-residue of P
1.....1................... A000079, 2^n 1....(x+1)^n.............. A007051, (1+3^n)/2 1....(x+2)^n.............. A034478, (1+5^n)/2 1....(x+3)^n.............. A034494, (1+7^n)/2 More examples:
Q...........P.............Q-residue of P
(x+1)^n...(x+1)^n......... A000110, Bell numbers (k+1).....(k+1)........... A001906 (even-ind. Fib. nos.) (In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)
Changing the notation slightly leads to the Mathematica program below and the following formulation for the Q-downstep of p: first, write t(n,k) as q(n,k). Define r(k)=Sum{q(k-1,i)*r(k-1-i) : i=0,1,...,k-1} Then row n of D(p) is given by v(n)=Sum{p(n,k)*r(n-k) : k=0,1,...,n}.
FORMULA
Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015
EXAMPLE
First five rows of Q, coefficients of Fibonacci polynomials ( A049310): 1
1...0
1...0...1
1...0...2...0
1...0...3...0...1
To obtain a(4)=15, downstep four times:
D(x^4+3*x^2+1)=(x^3+x^2+x+1)+3(x+1)+1: (1,1,4,5) [coefficients]
DD(x^4+3*x^2+1)=D(1,1,4,5)=(1,2,11)
DDD(x^4+3*x^2+1)=D(1,2,11)=(1,14)
DDDD(x^4+3*x^2+1)=D(1,14)=15.
MATHEMATICA
q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n + 1, x];
p[n_, k_] := Coefficient[f[n, x], x, k]; (* A049310 *) v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 24}] (* A193649 *) TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]
CROSSREFS
Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).
2, 8, 34, 152, 706, 3368, 16354, 80312, 397186, 1972808, 9824674, 49005272, 244672066, 1222297448, 6108298594, 30531927032, 152630937346, 763068593288, 3815084686114, 19074648589592, 95370918425026, 476847618556328
FORMULA
G.f.: 1/(1-3*x) + 1/(1-5*x).
E.g.f.: exp(3*x) + exp(5*x). (End)
MATHEMATICA
Table[3^n + 5^n, {n, 0, 25}]
PROG
(SageMath) [3^n+5^n for n in range(41)] # G. C. Greubel, Jan 14 2024
CROSSREFS
Cf. A000051, A007689, A034472, A034474, A034491, A052539, A062394, A062395, A062396, A063376, A063481, A074600.. A074624, A081186.
A masked Pascal triangle.
+10
12
1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 5, 0, 10, 0, 1, 1, 0, 15, 0, 15, 0, 1, 0, 7, 0, 35, 0, 21, 0, 1, 1, 0, 28, 0, 70, 0, 28, 0, 1, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 1, 0, 45, 0, 210, 0, 210, 0, 45, 0, 1, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 1, 0, 66, 0, 495, 0, 924
COMMENTS
Row sums are A011782. Diagonal sums are F(n+1)*(1+(-1)^n)/2 (aerated version of A001519). Product by Pascal's triangle A007318 is A119468. Schur product of (1/(1-x),x/(1-x)) and (1/(1-x^2),x). Exponential Riordan array (cosh(x),x). Inverse is (sech(x),x) or A119879. - Paul Barry, May 26 2006 Rows give coefficients of polynomials p_n(x) = Sum_{k=0..n} (k+1 mod 2)*binomial(n,k)*x^(n-k) having e.g.f. exp(x*t)*cosh(t)= 1*(t^0/0!) + x*(t^1/1!) + (1+x^2)*(t^2/2!) + ... - Peter Luschny, Jul 14 2009 Inverse of the coefficient matrix of the Swiss-Knife polynomials in ascending order of x^i (reversed and aerated rows of A153641). - Peter Luschny, Jul 16 2012 Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*... is equal to A136630 but with the first row and column omitted. - Peter Bala, Jul 28 2014 The row polynomials SKv(n,x) = [(x+1)^n + (x-1)^n]/2 , with e.g.f. cosh(t)*exp(xt), are the umbral compositional inverses of the row polynomials of A119879 (basically the Swiss Knife polynomials SK(n,x) of A153641); i.e., umbrally SKv(n,SK(.,x)) = x^n = SK(n,SKv(.,x)). Therefore, this entry's matrix and A119879 are an inverse pair. Both sequences of polynomials are Appell sequences, i.e., d/dx P(n,x) = n * P(n-1,x) and (P(.,x)+y)^n = P(n,x+y). In particular, (SKv(.,0)+x)^n = SKv(n,x), reflecting that the first column has the e.g.f. cosh(t). The raising operator is R = x + tanh(d/dx); i.e., R SKv(n,x) = SKv(n+1,x). The coefficients of this operator are basically the signed and aerated zag numbers A000182, which can be expressed as normalized Bernoulli numbers. The triangle is formed by multiplying the n-th diagonal of the lower triangular Pascal matrix by the Taylor series coefficient a(n) of cosh(x). More relations for this type of triangle and its inverse are given by the formalism of A133314. - Tom Copeland, Sep 05 2015 The signed version of this matrix has the e.g.f. cos(t) e^{xt}, generating Appell polynomials that have only real, simple zeros and whose extrema are maxima above the x-axis and minima below and situated above and below the zeros of the next lower degree polynomial. The bivariate versions appear on p. 27 of Dimitrov and Rusev in conditions for entire functions that are cosine transforms of a class of functions to have only real zeros. - Tom Copeland, May 21 2020 The n-th row of the triangle is obtained 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 probabilities 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 of the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k-1) as the (2k-1)-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-1). - Luca Onnis, Oct 29 2023
REFERENCES
Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.
FORMULA
G.f.: (1-x*y)/(1-2*x*y-x^2+x^2*y^2);
T(n,k) = C(n,k)*(1+(-1)^(n-k))/2;
Column k has g.f. (1/(1-x^2))*(x/(1-x^2))^k*Sum_{j=0..k+1} binomial(k+1,j)*sin((j+1)*Pi/2)^2*x^j.
Column k has e.g.f. cosh(x)*x^k/k!. - Paul Barry, May 26 2006
EXAMPLE
Triangle begins
1,
0, 1,
1, 0, 1,
0, 3, 0, 1,
1, 0, 6, 0, 1,
0, 5, 0, 10, 0, 1,
1, 0, 15, 0, 15, 0, 1,
0, 7, 0, 35, 0, 21, 0, 1,
1, 0, 28, 0, 70, 0, 28, 0, 1,
0, 9, 0, 84, 0, 126, 0, 36, 0, 1,
1, 0, 45, 0, 210, 0, 210, 0, 45, 0, 1
p[0](x) = 1
p[1](x) = x
p[2](x) = 1 + x^2
p[3](x) = 3*x + x^3
p[4](x) = 1 + 6*x^2 + x^4
p[5](x) = 5*x + 10*x^3 + x^5
Connection with A136630: With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins /1 \/1 \/1 \ /1 \
|0 1 ||0 1 ||0 1 | |0 1 |
|1 0 1 ||0 0 1 ||0 0 1 |... = |1 0 1 |
|0 3 0 1 ||0 1 0 1 ||0 0 0 1 | |0 4 0 1 |
|1 0 6 0 1||0 0 3 0 1||0 0 1 0 1| |1 0 10 0 1|
|... ||... ||... | |... |
MAPLE
# Polynomials: p_n(x)
p := proc(n, x) local k, pow; pow := (n, k) -> `if`(n=0 and k=0, 1, n^k);
add((k+1 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)*cosh(t), t, 16), t, i), x, n), n=0..i)), i=0..8); # Peter Luschny, Jul 14 2009
MATHEMATICA
Table[Binomial[n, k] (1 + (-1)^(n - k))/2, {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 06 2015 *) 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}]
PROG
(Sage)
@CachedFunction
R = PolynomialRing(ZZ, 'x')
x = R.gen()
return R.one() if n==0 else R.sum(binomial(n, k)*x^(n-k) for k in range(0, n+1, 2))
(Haskell)
a119467 n k = a119467_tabl !! n !! k
a119467_row n = a119467_tabl !! n
a119467_tabl = map (map (flip div 2)) $
zipWith (zipWith (+)) a007318_tabl a130595_tabl
(Magma) /* As triangle */ [[Binomial(n, k)*(1 + (-1)^(n - k))/2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 26 2015
CROSSREFS
p[n](k), n=0,1,...
k= 0: 1, 0, 1, 0, 1, 0, ... A128174k= 1: 1, 1, 2, 4, 8, 16, ... A011782k= 2: 1, 2, 5, 14, 41, 122, ... A007051k= 3: 1, 3, 10, 36, 136, ... A007582k= 4: 1, 4, 17, 76, 353, ... A081186k= 5: 1, 5, 26, 140, 776, ... A081187k= 6: 1, 6, 37, 234, 1513, ... A081188k= 7: 1, 7, 50, 364, 2696, ... A081189k= 8: 1, 8, 65, 536, 4481, ... A081190k= 9: 1, 9, 82, 756, 7048, ... A060531k=10: 1, 10, 101, 1030, ... A081192p[n](k), k=0,1,...
p[3]: 0,4,14,36,76,140, .. A079908 (End)
For n>1, a(n) = 2*a(n-1) + 15*a(n-2); a(0)=1, a(1)=1.
+10
9
1, 1, 17, 49, 353, 1441, 8177, 37969, 198593, 966721, 4912337, 24325489, 122336033, 609554401, 3054149297, 15251614609, 76315468673, 381405156481, 1907542343057, 9536162033329, 47685459212513, 238413348924961, 1192108586037617, 5960417405949649
COMMENTS
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
FORMULA
Let M = the 2 X 2 matrix [1,4; 4,1], then a(n) = M^n * [1,0], left term.
a(n) = ( 5^n + (-1)^n * 3^n ) / 2.
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
EXAMPLE
a(4) = 353 = 2*49 + 15*17 = 2*a(3) + 15*a(2).
MATHEMATICA
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 *)
PROG
(PARI) concat(1, Vec((15*x+1)/(-15*x^2-2*x+1) + O(x^100))) \\ Colin Barker, Mar 12 2014
6th binomial transform of (1,0,1,0,1,.....), A059841.
+10
8
1, 6, 37, 234, 1513, 9966, 66637, 450834, 3077713, 21153366, 146120437, 1013077434, 7042713913, 49054856766, 342163294237, 2389039544034, 16692759230113, 116696726720166, 816114147588037, 5708984335850634
COMMENTS
a(n) is also the number of words of length n over an alphabet of seven letters, of which a chosen one appears an even number of times. See a comment in A007582, also for the crossrefs. for the 1- to 11- letter word cases. - Wolfdieter Lang, Jul 17 2017
FORMULA
a(n) = 12*a(n-1) -35*a(n-2), a(0)=1, a(1)=6.
G.f.: (1-6*x)/((1-5*x)*(1-7*x)).
E.g.f.: exp(6*x)*cosh(x).
a(n) = 5^n/2 + 7^n/2.
MATHEMATICA
CoefficientList[Series[(1 - 6 x) / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *) LinearRecurrence[{12, -35}, {1, 6}, 30] (* Harvey P. Dale, Mar 24 2016 *)
Primes of the form (3^n + 5^n)/2^3 = A074606(n)/8.
+10
2
19, 421, 10039, 95383574161, 2384331073699, 1925929944387235853055979210606894889560480247048440342330377620014353281101
COMMENTS
Corresponding numbers n such that (3^n + 5^n)/8 is prime are listed in A122853. All these numbers are primes. - Alexander Adamchuk, Sep 14 2006
MATHEMATICA
Do[f=5^n+3^n; If[PrimeQ[f/2^3], Print[{n, f/2^3}]], {n, 1, 1231}] (* Alexander Adamchuk, Sep 14 2006 *)
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