A015518 -id:A015518 - cp's OEIS frontend

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A329008 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(3) as in A327321.

Original entry on oeis.org

1, 1, 7, 5, 61, 91, 547, 205, 4921, 7381, 44287, 33215, 398581, 597871, 3587227, 672605, 32285041, 48427561, 290565367, 217924025, 2615088301, 3922632451, 23535794707, 8825923015, 211822152361, 317733228541, 1906399371247, 1429799528435, 17157594341221

Offset: 1

Clark Kimberling, Nov 08 2019

nonn

a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)).

			See Example in A327321.

Cf. A015518, A327321, A329009, A329010.

c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@Variables /@ #1 &)[List @@ poly], 0], poly];
r = Sqrt[3]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]];  (* A327321 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329008 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329009 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329010 *)
(* Peter J. C. Moses, Nov 01 2019 *)
Numerator[CoefficientList[Normal[Series[1/((4 + x)*(4 - 3*x)), {x, 0, 30}]], x]] (* Vaclav Kotesovec, Mar 19 2022 *)

a(2*n - 1) = A015518(2*n - 1). - Vaclav Kotesovec, Mar 19 2022

A015588 Expansion of x/(1 - 10x - 3x^2).

Original entry on oeis.org

0, 1, 10, 103, 1060, 10909, 112270, 1155427, 11891080, 122377081, 1259444050, 12961571743, 133394049580, 1372825211029, 14128434259030, 145402818223387, 1496413485010960, 15400343304779761, 158492673502830490, 1631127764942644183, 16786755669934933300

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,3).

Cf. A000045, A015518.

[n le 2 select n-1 else 10*Self(n-1) + 3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2012

Join[{a=0,b=1},Table[c=10*b+3*a;a=b;b=c,{n,50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 26 2011 *)
LinearRecurrence[{10, 3}, {0, 1}, 30] (* Vincenzo Librandi, Nov 15 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-10*x-3*x^2))) \\ G. C. Greubel, Jan 06 2018

[lucas_number1(n,10,-3) for n in range(0, 20)] # Zerinvary Lajos, Apr 26 2009

a(n) = 10*a(n-1) + 3*a(n-2).

A092810 Binomial transform of a Jacobsthal trisection.

Original entry on oeis.org

1, 6, 54, 486, 4374, 39366, 354294, 3188646, 28697814, 258280326, 2324522934, 20920706406, 188286357654, 1694577218886, 15251194969974, 137260754729766, 1235346792567894, 11118121133111046, 100063090197999414, 900567811781994726, 8105110306037952534

Offset: 0

Paul Barry, Mar 10 2004

Binomial transform of A082311.

Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001045.

[1] cat [2*3^(2*n-1): n in [1..20]]; // Vincenzo Librandi, Jun 20 2015

Table[EulerPhi[9^n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

Vec((1-3*x)/(1-9*x) + O(x^30)) \\ Michel Marcus, Jun 18 2015

G.f.: (1-3*x)/(1-9*x).
E.g.f.: 2*exp(9*x)/3 + 1/3.
a(n) = 2*9^n/3 + 0^n/3.
a(n) = A054878(2n+1) - A054878(2n-1) + 0^n/3 = A015518(2n+1) - A015518(2n-1) + 0^n/3.
a(n) = 2*3^(2*n-1), for n>0. - Gionata Neri, Jun 18 2015

A097137 Convolution of 3^n and floor(n/2).

Original entry on oeis.org

0, 0, 1, 4, 14, 44, 135, 408, 1228, 3688, 11069, 33212, 99642, 298932, 896803, 2690416, 8071256, 24213776, 72641337, 217924020, 653772070, 1961316220, 5883948671, 17651846024, 52955538084, 158866614264, 476599842805

Offset: 0

Paul Barry, Jul 29 2004

nonn

a(n+1) gives partial sums of A033113 and second partial sums of A015518(n+1). Binomial transform of {0,0,1,1,4,4,16,16,...}.

Partial sums of floor(3^n/8) = round(3^n/8). - Mircea Merca, Dec 28 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Column k=3 of A368296.

Cf. A033113.

a:=[0,0,1,4];; for n in [5..30] do a[n]:=4*a[n-1]-2*a[n-2]-4*a[n-3] +3*a[n-4]; od; a; # G. C. Greubel, Jul 14 2019

[Round((3*3^n-4*n-4)/16): n in [0..30]]; // Vincenzo Librandi, Jun 25 2011

A097137 := proc(n) add( floor(3^i/8),i=0..n) ; end proc:

CoefficientList[Series[x^2/((1-x)^2(1-3x)(1+x)),{x,0,30}],x]  (* Harvey P. Dale, Mar 11 2011 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^2/((1-x)^2*(1-3*x)*(1+x)))) \\ G. C. Greubel, Jul 14 2019

(x^2/((1-x)^2*(1-3*x)*(1+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 14 2019

G.f.: x^2/((1-x)^2*(1-3*x)*(1+x)).
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) + 3*a(n-4).
a(n) = Sum_{k=0..n} floor((n-k)/2)*3^k = Sum_{k=0..n} floor(k/2)*3^(n-k).
From Mircea Merca, Dec 26 2010: (Start)
a(n) = round((3*3^n - 4*n - 4)/16) = floor((3*3^n - 4*n - 3)/16) = ceiling((3*3^n - 4*n - 5)/16) = round((3*3^n - 4*n - 3)/16).
a(n) = a(n-2) + (3^(n-1)-1)/2, n > 2. (End)
a(n) = (floor(3^(n+1)/8) - floor((n+1)/2))/2. - Seiichi Manyama, Dec 22 2023

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 5, 8, 5, 0, 1, 2, 6, 12, 16, 6, 0, 1, 2, 7, 16, 29, 32, 7, 0, 1, 2, 8, 20, 44, 70, 64, 8, 0, 1, 2, 9, 24, 61, 120, 169, 128, 9, 0, 1, 2, 10, 28, 80, 182, 328, 408, 256, 10, 0, 1, 2, 11, 32, 101, 256, 547, 896, 985, 512, 11

Offset: 0

Ralf Stephan, Oct 13 2004

			Square array, A(n, k), begins as:
  0, 1, 2,  3,  4,   5,    6,    7,     8, ... A001477;
  0, 1, 2,  4,  8,  16,   32,   64,   128, ... A000079;
  0, 1, 2,  5, 12,  29,   70,  169,   408, ... A000129;
  0, 1, 2,  6, 16,  44,  120,  328,   896, ... A002605;
  0, 1, 2,  7, 20,  61,  182,  547,  1640, ... A015518;
  0, 1, 2,  8, 24,  80,  256,  832,  2688, ... A063727;
  0, 1, 2,  9, 28, 101,  342, 1189,  4088, ... A002532;
  0, 1, 2, 10, 32, 124,  440, 1624,  5888, ... A083099;
  0, 1, 2, 11, 36, 149,  550, 2143,  8136, ... A015519;
  0, 1, 2, 12, 40, 176,  672, 2752, 10880, ... A003683;
  0, 1, 2, 13, 44, 205,  806, 3457, 14168, ... A002534;
  0, 1, 2, 14, 48, 236,  952, 4264, 18048, ... A083102;
  0, 1, 2, 15, 52, 269, 1110, 5179, 22568, ... A015520;
  0, 1, 2, 16, 56, 304, 1280, 6208, 27776, ... A091914;
Antidiagonal triangle, T(n, k), begins as:
  0;
  0,  1;
  0,  1,  2;
  0,  1,  2,  3;
  0,  1,  2,  4,  4;
  0,  1,  2,  5,  8,  5;
  0,  1,  2,  6, 12, 16,   6;
  0,  1,  2,  7, 16, 29,  32,   7;
  0,  1,  2,  8, 20, 44,  70,  64,   8;
  0,  1,  2,  9, 24, 61, 120, 169, 128,   9;
  0,  1,  2, 10, 28, 80, 182, 328, 408, 256,  10;

G. C. Greubel, Antidiagonals n = 0..50, flattened
Ralf Stephan, Prove or disprove. 100 Conjectures from the OEIS, #16, arXiv:math/0409509 [math.CO], 2004.

Rows m: A001477 (m=0), A000079 (m=1), A000129 (m=2), A002605 (m=3), A015518 (m=4), A063727 (m=5), A002532 (m=6), A083099 (m=7), A015519 (m=8), A003683 (m=9), A002534 (m=10), A083102 (m=11), A015520 (m=12), A091914 (m=13).
Columns q: A000004 (q=0), A000012 (q=1), A009056 (q=2), A008586 (q=3).
Main diagonal gives A357502.

A099173:= func< n,k | (&+[n^j*Binomial(k,2*j+1): j in [0..Floor(k/2)]]) >;
[A099173(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023

A[k_, n_]:= Which[k==0, n, n==0, 0, True, ((1+Sqrt[k])^n - (1-Sqrt[k])^n)/(2 Sqrt[k])]; Table[A[k-n, n]//Simplify, {k, 0, 12}, {n, 0, k}]//Flatten (* Jean-François Alcover, Jan 21 2019 *)

A(k,n)=sum(i=0,n\2,k^i*binomial(n,2*i+1))

def A099173(n,k): return sum( n^j*binomial(k, 2*j+1) for j in range((k//2)+1) )
flatten([[A099173(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023

A(n, k) = Sum_{i=0..floor(k/2)} n^i * C(k, 2*i+1) (array).
Recurrence: A(n, k) = 2*A(n, k-1) + (n-1)*A(n, k-2), with A(n, 0) = 0, A(n, 1) = 1.
T(n, k) = A(n-k, k) (antidiagonal triangle).
T(2*n, n) = A357502(n).
A(n, k) = ((1+sqrt(n))^k - (1-sqrt(n))^k)/(2*sqrt(n)). - Jean-François Alcover, Jan 21 2019

A109189 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps at level zero. (A Grand Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 4, 0, 1, 8, 4, 6, 0, 1, 16, 20, 6, 8, 0, 1, 46, 40, 36, 8, 10, 0, 1, 114, 128, 72, 56, 10, 12, 0, 1, 310, 324, 254, 112, 80, 12, 14, 0, 1, 822, 932, 654, 432, 160, 108, 14, 16, 0, 1, 2238, 2540, 1986, 1128, 670, 216, 140, 16, 18, 0, 1, 6094, 7164, 5546

Offset: 0

Emeric Deutsch, Jun 21 2005

Row sums yield the central trinomial coefficients (A002426). T(n,0)=A109190(n). sum(k*T(n,k),k=0..n)=A015518(n).

			T(4,1) = 4 because we have (h)uhd, (h)dhu, uhd(h) and dhu(h), where u=(1,1), d=(1,-1), h=(1,0) and the (1,0) steps at level 0 are shown between parentheses.
Triangle begins:
1;
0,1;
2,0,1;
2,4,0,1;
8,4,6,0,1;
16,20,6,8,0,1;

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A002426, A109190, A015518, A001006.

M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-t*z-2*z^2*M): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od;
# second Maple program:
b:= proc(x, y) option remember;
      `if`(abs(y)>x, 0, `if`(x=0, 1, expand(b(x-1, y)*
      `if`(y=0, t, 1) +b(x-1, y+1) +b(x-1, y-1))))
    end:
T:= n-> (p-> seq(coeff(p, t, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 05 2014

nn=10;m=(1-x-(1-2x-3x^2)^(1/2))/(2x^2);CoefficientList[Series[1/(1-y x-2x^2m),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Feb 05 2014 *)

G.f.= 1/(1-tz-2z^2*M), where M=1+zM+z^2*M^2=[1-z-sqrt(1-2z-3z^2)]/(2z^2) is the g.f. of the Motzkin numbers (A001006).

A111010 Primes of the form (3^k - (-1)^k)/4.

Original entry on oeis.org

2, 7, 61, 547, 398581, 23535794707, 82064241848634269407

Offset: 1

Cino Hilliard, Oct 02 2005

nonn

The next term is too large to include.
Is there an infinity of primes in this sequence?
All a(n), except a(1) = 2, are primes of the form (3^k + 1)/4. Corresponding numbers k such that (3^k + 1)/4 is prime are listed in A007658(n) = {3, 5, 7, 13, 23, 43, 281, 359, 487, 577, ...}. All such numbers k are primes. a(1) = 2 is the only prime of the form (3^k - 1)/4. - Alexander Adamchuk, Nov 19 2006

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

Amiram Eldar, Table of n, a(n) for n = 1..14 (terms 1..11 from Alexander Adamchuk, Nov 19 2006)

Cf. A007658, A015518.

Do[f=(3^n - (-1)^n)/4; If[PrimeQ[f],Print[{n,f}]],{n,1,577}] (* Alexander Adamchuk, Nov 19 2006 *)

primenum(n,k,typ) = /* k=mult,typ=1 num,2 denom. ouyput prime num or denom. */ { local(a,b,x,tmp,v); a=1;b=1; for(x=1,n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1,v=a,v=b); if(isprime(v),print1(v","); ) ); print(); print(a/b+.); }

Given a(0)=1, b(0)=1, then for i=1, 2, ..., a(i)/b(i) = (a(i-1) + 2*b(i-1)) /(a(i-1) + b(i-1)).

a(n) = A015518(A007658(n-1)) for n >= 2. - Amiram Eldar, Jul 04 2024

Edited by Alexander Adamchuk, Nov 19 2006

A114283 Sequence array for binomial transform of Jacobsthal numbers A001045(n+1).

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 18, 6, 2, 1, 54, 18, 6, 2, 1, 162, 54, 18, 6, 2, 1, 486, 162, 54, 18, 6, 2, 1, 1458, 486, 162, 54, 18, 6, 2, 1, 4374, 1458, 486, 162, 54, 18, 6, 2, 1, 13122, 4374, 1458, 486, 162, 54, 18, 6, 2, 1, 39366, 13122, 4374, 1458, 486, 162, 54, 18, 6, 2, 1

Offset: 0

Paul Barry, Nov 20 2005

Sequence array for A025192. Row sums are 3^n, A000244. Diagonal sums are A015518(n+1). Inverse is A114284.

			Triangle begins
1;
2,1;
6,2,1;
18,6,2,1;
54,18,6,2,1;
162,54,18,6,2,1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

a114283 n k = a114283_tabl !! n !! k
a114283_row n = a114283_tabl !! n
a114283_tabl = iterate
   (\row -> (sum $ zipWith (+) row $ reverse row) : row) [1]
-- Reinhard Zumkeller, Nov 27 2012

Riordan array ((1-x)/(1-3x), x).

A131049 (1/4) * (A007318^3 - A007318^(-1)).

Original entry on oeis.org

1, 2, 2, 7, 6, 3, 20, 28, 12, 4, 61, 100, 70, 20, 5, 182, 366, 300, 140, 30, 6, 547, 1274, 1281, 700, 245, 42, 7, 1640, 4376, 5096, 3416, 1400, 392, 56, 8, 4921, 14760, 19692, 15288, 7686, 2520, 588, 72, 9

Offset: 1

Gary W. Adamson, Jun 12 2007

Row sums = powers of 4: (1, 4, 16, 64, ...).

Left border = A015518: (1, 2, 7, 20, 61, 182, ...).

			First few rows of the triangle:
    1;
    2,    2;
    7,    6,    3;
   20,   28,   12,   4;
   61,  100,   70,  20,   5;
  182,  366,  300, 140,  30,  6;
  547, 1274, 1281, 700, 245, 42, 7;
  ...

Cf. A015518, A007318, A131047, A131048, A131050, A131051.

(1/4) * (P^3 - 1/P), where P = Pascal's triangle, A007318. Delete right border of zeros.

cp's OEIS Frontend

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

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A329008 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(3) as in A327321.

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Mathematica

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A015588 Expansion of x/(1 - 10*x - 3*x^2).

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Magma

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SageMath

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A092810 Binomial transform of a Jacobsthal trisection.

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Magma

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A097137 Convolution of 3^n and floor(n/2).

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A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2*x-(k-1)*x^2).

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A109189 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps at level zero. (A Grand Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).).

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A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A015588 Expansion of x/(1 - 10x - 3x^2).

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).