A081186 -id:A081186 - cp's OEIS frontend

A059841 Period 2: Repeat [1,0]. a(n) = 1 - (n mod 2); Characteristic function of even numbers.

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

Offset: 0

Alford Arnold, Feb 25 2001

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.
Successive binomial transforms of this sequence: A011782, A007051, A007582, A081186, A081187, A081188, A081189, A081190, A060531, A081192.
Characteristic function of even numbers: a(A005843(n))=1, a(A005408(n))=0. - Reinhard Zumkeller, Sep 29 2008
This sequence is the Euler transformation of A185012. - Jason Kimberley, Oct 14 2011
a(n) is the parity of n+1. - Omar E. Pol, Jan 17 2012
Read as partial sequences, we get to A000975. - Jon Perry, Nov 11 2014
Elementary Cellular Automata rule 77 produces this sequence.  See Wolfram, Weisstein and Index links below. - Robert Price, Jan 30 2016
Column k = 1 of A051159. - John Keith, Jun 28 2021
When read as a constant: decimal expansion of 10/99, binary expansion of 2/3. - Jason Bard, Aug 25 2025

			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;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Atsuto Seko, Atsushi Togo, and Isao Tanaka, Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential, arXiv:1901.02118 [physics.comp-ph], 2019.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,1).
Index entries for characteristic functions

One's complement of A000035 (essentially the same, but shifted once).
Cf. A033999 (first differences), A008619 (partial sums), A004525, A011782 (binomial transf.), A000975.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), this sequence (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7).

a059841 n = (1 -) . (`mod` 2)
a059841_list = cycle [1,0]
-- Reinhard Zumkeller, May 05 2012, Dec 30 2011

[0^(n mod 2): n in  [0..100]]; // Vincenzo Librandi, Nov 09 2014

seq(1-modp(n,2), n=0..150); # Muniru A Asiru, Apr 05 2018

CoefficientList[Series[1/(1 - x^2), {x, 0, 104}], x] (* or *)
Array[1/2 + (-1)^#/2 &, 105, 0] (* Michael De Vlieger, Feb 19 2019 *)
Table[QBinomial[n, 1, -1], {n, 1, 74}] (* John Keith, Jun 28 2021 *)
PadRight[{},120,{1,0}] (* Harvey P. Dale, Mar 06 2023 *)

PARI
```
a(n)=(n+1)%2; \\ or 1-n%2 as in NAME.
```

A059841(n)=!bittest(n,0) \\ M. F. Hasler, Jan 13 2012

def A059841(n): return 1 - (n & 1) # Chai Wah Wu, May 25 2022

a(n) = 1 - A000035(n). - M. F. Hasler, Jan 13 2012
From Paul Barry, Mar 11 2003: (Start)
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.
a(n) = Sum_{k=0..n} (-1)^k*A038137(n, k). - Philippe Deléham, Nov 30 2006
a(n) = Sum_{k=1..n} (-1)^(n-k) for n > 0. - William A. Tedeschi, Aug 05 2011
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
a(n) = A000035(n+1) = A008619(n) - A110654(n). - Wesley Ivan Hurt, Jul 20 2013

Better definition from M. F. Hasler, Jan 13 2012

Reinhard Zumkeller's Sep 29 2008 description added as a secondary name by Antti Karttunen, May 03 2022

A007582 a(n) = 2^(n-1)*(1+2^n).

Original entry on oeis.org

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

Offset: 0

Simon Plouffe

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)Reinhard Zumkeller, Aug 04 2004
a(n)^2 + (A006516(n))^2 = a(2n). E.g., a(3) = 36, A006516(3) = 28, a(6) = 2080. 36^2 + 28^2 = 2080. - Gary W. Adamson, Jun 17 2006
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
For n>0: A000120(a(n))=2, A023414(a(n))=2*(n-1), A087117(a(n))=n-1. - Reinhard Zumkeller, Jun 23 2009
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
a(n) for n >= 1 is a bisection of A001445(n+1). - Jaroslav Krizek, Aug 14 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
a(n) = A000217(2^n) = 2^(2n-1) + 2^(n-1) is the nearest triangular number above 2^(2n-1); cf. A006516, A233327. - Antti Karttunen, Feb 26 2014
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 = . - Sean A. Irvine, Nov 06 2024

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
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
S. Hong and J. H. Kwak, Regular fourfold coverings with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 168
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.
Mircea Merca, A Note on Cosine Power Sums, J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Index entries for linear recurrences with constant coefficients, signature (6,-8)

Cf. A000217, A049773, A049775.
Cf. A006516.
Cf. A134308.
Cf. A000225, A000392, A032263, A028243, A000079.
Cf. A102573.
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

[Binomial(2^n + 1, 2) : n in [0..30]]; // Wesley Ivan Hurt, Jul 03 2020

seq(binomial(-2^n, 2), n=0..23); # Zerinvary Lajos, Feb 22 2008

Table[ Binomial[2^n + 1, 2], {n, 0, 23}] (* Robert G. Wilson v, Jul 30 2004 *)
LinearRecurrence[{6,-8},{1,3},30] (* Harvey P. Dale, Apr 08 2013 *)

A007582(n):=2^(n-1)*(1+2^n)$ makelist(A007582(n),n,0,30); /* Martin Ettl, Nov 15 2012 */

PARI
```
a(n)=if(n<0,0,2^(n-1)*(1+2^n))
```

a(n)=sum(k=-n\4,n\4,binomial(2*n+1,n+1+4*k))

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.
Binomial transform of A007051. Inverse binomial transform of A081186. - Paul Barry, Apr 07 2003
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+1) = 4*a(n) - 2^n; see also A049775. a(n) = 2^(n-1)*A000051(n). - Philippe Deléham, Feb 20 2004
a(n) = 6*a(n-1) - 8*a(n-2). - Emeric Deutsch, Apr 01 2004
Row sums of triangle A134308. - Gary W. Adamson, Oct 19 2007
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) = A000079(n) + A006516(n). - Yosu Yurramendi, Aug 06 2008
a(n) = A028403(n+1) / 4. - Jaroslav Krizek, Jul 27 2009
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) = Sum_{k=1..2^n} k. - Joerg Arndt, Sep 01 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

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

Original entry on oeis.org

13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, 128941, 147571, 182099, 866029

Offset: 1

Alexander Adamchuk, Aug 31 2006, Oct 08 2006

All terms are primes. Their indices are listed in A123704.
Corresponding primes are listed in A123705.
If it exists, a(17) > 125000. - Robert Price, Aug 15 2011
If it exists, a(21) > 1000000. - Jon Grantham, Jul 29 2023

OEIS Wiki, Primes of the form (a^n+b^n)/(a+b) and (a^n-b^n)/(a-b).
Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A005058, A005059, A109347, A120612, A081186, A121824, A123704, A123705.

Do[f=(5^n-3^n)/2;If[PrimeQ[f],Print[{n,f}]],{n,1,300}]

forprime(p=2,1e4,if(ispseudoprime((5^p-3^p)>>1),print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = prime(A123704(n)).

More terms from Farideh Firoozbakht, Oct 11 2006
a(13)-a(16) from Robert Price, Aug 15 2011
a(17)-a(19) from Kellen Shenton, May 18 2022
a(20) from Jon Grantham, Jul 29 2023

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

A193649 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.)

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

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))

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

1....(2x+1)^n.............A007582

1....(3x+1)^n.............A081186

1....(2x+3)^n.............A081342

1....(3x+2)^n.............A081336

1.....A040310.............A193649

1....(x+1)^n+(x-1)^n)/2...A122983

1....(x+2)(x+1)^(n-1).....A057198

1....(1,2,3,4,...,n)......A002064

1....(1,1,2,3,4,...,n)....A048495

1....(n,n+1,...,2n).......A087323

1....(n+1,n+2,...,2n+1)...A099035

1....p(n,k)=(2^(n-k))*3^k.A085350

1....p(n,k)=(3^(n-k))*2^k.A090040

1....A008288 (Delannoy)...A193653

1....A054142..............A101265

1....cyclotomic...........A193650

1....(x+1)(x+2)...(x+n)...A193651

1....A114525..............A193662

More examples:

Q...........P.............Q-residue of P

(x+1)^n...(x+1)^n.........A000110, Bell numbers

(x+1)^n...(x+2)^n.........A126390

(x+2)^n...(x+1)^n.........A028361

(x+2)^n...(x+2)^n.........A126443

(x+1)^n.....1.............A005001

(x+2)^n.....1.............A193660

A094727.....1.............A193657

(k+1).....(k+1)...........A001906 (even-ind. Fib. nos.)

(k+1).....(x+1)^n.........A112091

(x+1)^n...(k+1)...........A029761

(k+1)......A049310........A193663

(In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)

A051162...(x+1)^n.........A193658

A094727...(x+1)^n.........A193659

A049310...(x+1)^n.........A193664

A075362....A075362........A193665

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}.

			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.

Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).

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}]]

Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015

A074606 a(n) = 3^n + 5^n.

Original entry on oeis.org

2, 8, 34, 152, 706, 3368, 16354, 80312, 397186, 1972808, 9824674, 49005272, 244672066, 1222297448, 6108298594, 30531927032, 152630937346, 763068593288, 3815084686114, 19074648589592, 95370918425026, 476847618556328

Offset: 0

Robert G. Wilson v, Aug 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (8,-15).

Cf. A000051, A007689, A034472, A034474, A034491, A052539, A062394, A062395, A062396, A063376, A063481, A074600..A074624, A081186.

[3^n + 5^n: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

Mathematica
```
Table[3^n + 5^n, {n, 0, 25}]
```

a(n)=3^n+5^n \\ Charles R Greathouse IV, Sep 28 2015

[3^n+5^n for n in range(41)] # G. C. Greubel, Jan 14 2024

From Mohammad K. Azarian, Jan 11 2009: (Start)
G.f.: 1/(1-3*x) + 1/(1-5*x).
E.g.f.: exp(3*x) + exp(5*x). (End)
a(n) = 8*a(n-1) - 15*a(n-2) with a(0)=2, a(1)=8. - Vincenzo Librandi, Jul 21 2010
a(n) = 2*A081186(n). - R. J. Mathar, Mar 10 2022

A119467 A masked Pascal triangle.

Original entry on oeis.org

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

Offset: 0

Paul Barry, May 21 2006

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

			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|
|...      ||...      ||...      |      |...       |
- _Peter Bala_, Jul 28 2014

Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Tom Copeland, Skipping over Dimensions, Juggling Zeros in the Matrix, 2020.
D. Dimitrov and P. Rusev, Zeros of entire Fourier transforms, East Journal on Approximations, Vol. 17, No. 1, p. 1-108, 2011.
Miguel Méndez and Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO], 2017, Section 4.3, Example 4.
Miguel A. Méndez and Rafael Sánchez Lamoneda, Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.
Luca Onnis, Animation of the Ehrenfest model.
Wikipedia, Ehrenfest model.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A131047, A153641, A162590.
From Peter Luschny, Jul 14 2009: (Start)
Cf. A034839, A162590.
p[n](k), n=0,1,...
k= 0: 1,  0,   1,    0,    1,   0, ... A128174
k= 1: 1,  1,   2,    4,    8,  16, ... A011782
k= 2: 1,  2,   5,   14,   41, 122, ... A007051
k= 3: 1,  3,  10,   36,  136,      ... A007582
k= 4: 1,  4,  17,   76,  353,      ... A081186
k= 5: 1,  5,  26,  140,  776,      ... A081187
k= 6: 1,  6,  37,  234, 1513,      ... A081188
k= 7: 1,  7,  50,  364, 2696,      ... A081189
k= 8: 1,  8,  65,  536, 4481,      ... A081190
k= 9: 1,  9,  82,  756, 7048,      ... A060531
k=10: 1, 10, 101, 1030,            ... A081192
p[n](k), k=0,1,...
p[0]: 1,1,1,1,1,1, ....... A000012
p[1]: 0,1,2,3,4,5, ....... A001477
p[2]: 1,2,5,10,17,26, .... A002522
p[3]: 0,4,14,36,76,140, .. A079908 (End)
Cf. A000182, A133314, A153641.

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
-- Reinhard Zumkeller, Mar 23 2014

/* As triangle */ [[Binomial(n, k)*(1 + (-1)^(n - k))/2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 26 2015

# 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

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}]
odd = R2*2^(n - 1) (* _Luca Onnis *)

@CachedFunction
def A119467_poly(n):
    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))
def A119467_row(n):
    return list(A119467_poly(n))
for n in (0..10) : print(A119467_row(n)) # Peter Luschny, Jul 16 2012

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
Let Pascal's triangle, A007318 = P; then this triangle = (1/2) * (P + 1/P). Also A131047 = (1/2) * (P - 1/P). - Gary W. Adamson, Jun 12 2007
Equals A007318 - A131047 since the zeros of the triangle are masks for the terms of A131047. Thus A119467 + A131047 = Pascal's triangle. - Gary W. Adamson, Jun 12 2007
T(n,k) = (A007318(n,k) + A130595(n,k))/2, 0<=k<=n. - Reinhard Zumkeller, Mar 23 2014

Edited by N. J. A. Sloane, Jul 14 2009

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

A081188 6th binomial transform of (1,0,1,0,1,.....), A059841.

Original entry on oeis.org

1, 6, 37, 234, 1513, 9966, 66637, 450834, 3077713, 21153366, 146120437, 1013077434, 7042713913, 49054856766, 342163294237, 2389039544034, 16692759230113, 116696726720166, 816114147588037, 5708984335850634

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A081187.

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

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. A007582, A081186, A081189.

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

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 *)

a(n)=5^n/2+7^n/2 \\ Charles R Greathouse IV, Oct 07 2015

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.

A121938 Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.

Original entry on oeis.org

19, 421, 10039, 95383574161, 2384331073699, 1925929944387235853055979210606894889560480247048440342330377620014353281101

Offset: 1

Zak Seidov, Sep 10 2006

Corresponding numbers k such that (3^k + 5^k)/8 is prime are listed in A122853. All these numbers are primes. - Alexander Adamchuk, Sep 14 2006

The next term is too large to include. - Alexander Adamchuk, Sep 14 2006

Cf. A074706, A122853, A081186, A079773, A121824, A121877, A005058, A005059, A121938, A109347.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}] (* Alexander Adamchuk, Sep 14 2006 *)

a(n) = (A122853(n)^3 + A122853(n)^5)/8. a(n) = A074606[A122853(n)]/8 = A081186[A122853(n)]/4. a(n) = A079773[A122853(n)]. - Alexander Adamchuk, Sep 14 2006

More terms from Alexander Adamchuk, Sep 14 2006

Showing 1-10 of 13 results. Next

cp's OEIS Frontend

A059841 Period 2: Repeat [1,0]. a(n) = 1 - (n mod 2); Characteristic function of even numbers.

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A007582 a(n) = 2^(n-1)*(1+2^n).

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

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

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A193649 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.)

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A074606 a(n) = 3^n + 5^n.

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A119467 A masked Pascal triangle.

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