A005061 -id:A005061 - cp's OEIS frontend

A002250 a(n) = 4^n - 2*3^n.

-1, -2, -2, 10, 94, 538, 2638, 12010, 52414, 222778, 930478, 3840010, 15714334, 63920218, 258869518, 1045044010, 4208873854, 16921588858, 67944635758, 272553384010, 1092538058974, 4377125804698, 17529423925198, 70180457820010, 280910117637694, 1124205329623738, 4498515895713838

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-12).

Cf. A000244, A000302, A005061, A008776.

[4^n-2*3^n: n in [0..30]]; // Vincenzo Librandi, Jun 20 2013

Table[4^n - 2 3^n, {n, 0, 30}] (* or *) CoefficientList[Series[-(1 - 5 x) / ((1 - 3 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 20 2013 *)

a(n)=4^n-2*3^n \\ Charles R Greathouse IV, Jun 23 2020

From Bruno Berselli, Jan 25 2011: (Start)
G.f.: -(1-5*x)/((1-3*x)*(1-4*x)).
a(n) = 7*a(n-1) - 12*a(n-2) for n > 1. (End)
From Elmo R. Oliveira, Sep 15 2024: (Start)
E.g.f.: exp(3*x)*(exp(x) - 2).
a(n) = A000302(n) - A008776(n). (End)

A129736 Primes of the form 4^k - 3^k.

Original entry on oeis.org

7, 37, 14197, 17050729021, 332306984815842876487217260305275077

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..13
Bogley, William A.; Williams, Gerald Efficient finite groups arising in the study of relative asphericity. Math. Z. 284, No. 1-2, 507-535 (2016).
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Cf. A005061, A059801, A129733, A129737.

Filtered(List([1..100], n -> 4^n-3^n), IsPrime); # Muniru A Asiru, Feb 09 2018

[a: n in [0..300] | IsPrime(a) where a is 4^n-3^n]; // Vincenzo Librandi, Nov 23 2010

select(isprime, [seq(4^n - 3^n, n=0..100)]); # Muniru A Asiru, Feb 09 2018

fQ[n_] := If[PrimeQ[4^n - 3^n], 4^n - 3^n, Nothing]; Array[fQ, 300] (* Robert G. Wilson v, Feb 12 2018 *)

lista(nn) = for(k=1, nn, if(isprime(p=4^k-3^k), print1(p", "))) \\ Altug Alkan, Mar 03 2018

a(n) = A005061(A059801(n)). - Michel Marcus, Feb 12 2018

A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

Original entry on oeis.org

1, 0, 4, 2, 16, 14, 64, 74, 256, 350, 1024, 1562, 4096, 6734, 16384, 28394, 65536, 117950, 262144, 484922, 1048576, 1979054, 4194304, 8034314, 16777216, 32491550, 67108864, 131029082, 268435456, 527304974, 1073741824, 2118785834, 4294967296, 8503841150

Offset: 0

Paul Curtz, Nov 12 2009

Binomial transform of A077917, the signed variant of A127864.

Robert Israel, Table of n, a(n) for n = 0..3318
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Cf. A154383.

seq(2^n - (1 - (-1)^n)*3^((n-1)/2), n=0..100); # Robert Israel, Apr 11 2019

LinearRecurrence[{2, 3, -6}, {1, 0, 4}, 40] (* Harvey P. Dale, Nov 29 2011 *)

a(n) = A167936(n+1) - A167936(n).
a(2n) = A000302(n). a(2n+1) = 2*A005061(n).
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).
G.f.: (x-1)^2/((2*x-1)*(3*x^2-1)).
a(n+4) mod 9 = A153130(n+4) = A146501(n+2), n>=0.
E.g.f.: exp(2*x) - (2/sqrt(3))*sinh(sqrt(3)*x). - G. C. Greubel, Jun 27 2016

Edited and extended by R. J. Mathar, Feb 27 2010

Incorrect b-file corrected by Robert Israel, Apr 11 2019

A248337 a(n) = 6^n - 4^n.

Original entry on oeis.org

0, 2, 20, 152, 1040, 6752, 42560, 263552, 1614080, 9815552, 59417600, 358602752, 2160005120, 12993585152, 78095728640, 469111242752, 2816814940160, 16909479575552, 101491237191680, 609084862103552, 3655058928435200, 21932552593866752, 131604111656222720, 789659854309425152, 4738099863344906240, 28429162130022858752

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-24).

Cf. sequences of the form k^n - 4^n: -A000302 (k=0), -A024036 (k=1), -A020522 (k=2), -A005061 (k=3), A005060 (k=5), this sequence (k=6), A190542 (k=7), A059409 (k=8), A118004 (k=9), A248338 (k=10), A139742 (k=11), 8*A016159 (k=12).

Cf. A000079, A000302, A000400, A001047, A081199.

Magma
```
[6^n-4^n: n in [0..30]];
```

Table[6^n - 4^n, {n,0,30}]
CoefficientList[Series[(2 x)/((1-4 x)(1-6 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-24},{0,2},30] (* Harvey P. Dale, Aug 18 2024 *)

vector(20,n,6^(n-1)-4^(n-1)) \\ Derek Orr, Oct 05 2014

A248337=BinaryRecurrenceSequence(10,-24,0,2)
[A248337(n) for n in range(31)] # G. C. Greubel, Nov 11 2024

G.f.: 2*x/((1-4*x)*(1-6*x)).
a(n) = 10*a(n-1) - 24*a(n-2).
a(n) = 2^n*(3^n-2^n) =  A000079(n) * A001047(n) = A000400(n) - A000302(n).
a(n) = 2*A081199(n). - Bruno Berselli, Oct 05 2014
E.g.f.: 2*exp(5*x)*sinh(x). - G. C. Greubel, Nov 11 2024

More terms added by G. C. Greubel, Nov 11 2024

A269576 a(n) = Product_{i=1..n} (4^i - 3^i).

Original entry on oeis.org

1, 7, 259, 45325, 35398825, 119187843775, 1692109818073675, 99792176520894983125, 24195710911432718503470625, 23942309231057283642583777144375, 96180015123706384385790918441966041875

Offset: 1

Bob Selcoe, Mar 02 2016

nonn

In general, for sequences of the form a(n) = Product_{i=1..n} j^i-k^i, where j>k>=1 and n>=1: given probability p=(k/j)^n that an outcome will occur at the n-th stage of an infinite process, then r = 1 - a(n)/j^((n^2+n)/2) is the probability that the outcome has occurred at or before the n-th iteration. Here j=4 and k=3, so p=(3/4)^n and r = 1-a(n)/A053763(n+1). The limiting ratio of r is ~ 0.9844550.

Robert Israel, Table of n, a(n) for n = 1..57

Cf. A005061, A053763.
Cf. sequences of the form Product_{i=1..n}(j^i - 1): A005329 (j=2), A027871 (j=3), A027637 (j=4), A027872 (j=5), A027873 (j=6), A027875 (j=7),A027876 (j=8), A027877 (j=9), A027878 (j=10), A027879 (j=11), A027880 (j=12).
Cf. sequences of the form Product_{i=1..n}(j^i - k^1), k>1: A263394 (j=3, k=2), A269661 (j=5, k=4).

seq(mul(4^i-3^i,i=1..n),n=0..20); # Robert Israel, Jun 01 2023

Table[Product[4^i - 3^i, {i, n}], {n, 11}] (* Michael De Vlieger, Mar 07 2016 *)
FoldList[Times,Table[4^n-3^n,{n,20}]] (* Harvey P. Dale, Jul 30 2018 *)

a(n) = prod(k=1, n, 4^k-3^k); \\ Michel Marcus, Mar 05 2016

a(n) = Product_{i=1..n} A005061(i).
a(n) ~ c * 2^(n*(n+1)), where c = QPochhammer(3/4) = 0.015545038845451847... . - Vaclav Kotesovec, Oct 10 2016
a(n+3)/a(n+2) - 7 * a(n+2)/a(n+1) + 12 * a(n+1)/a(n) = 0. - Robert Israel, Jun 01 2023

A269731 Dimensions of the 3-polytridendriform operad TDendr_3.

Original entry on oeis.org

1, 7, 61, 595, 6217, 68047, 770149, 8939707, 105843409, 1273241431, 15517824973, 191202877411, 2377843390873, 29807864423071, 376255282112629, 4778240359795147, 61007205215610529, 782648075371992487, 10083436451634033757, 130413832663780730995, 1692599303723819234281, 22037570163808433691247, 287762084009227350367621

Offset: 1

N. J. A. Sloane, Mar 08 2016

Gheorghe Coserea, Table of n, a(n) for n = 1..512
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016; Adv. Appl. Math., 77, 3-85, 2016.

Cf. A001003, A005061, A001263, A126216, A269730, A269732.

I:=[1,7]; [n le 2 select I[n] else (7*(2*n-1)*Self(n-1)-(n-2)*Self(n-2))/(n+1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2016

Rest[CoefficientList[Series[(1 - 7*x - Sqrt[1 - 14*x + x^2])/(24*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 7]/(2*Sqrt[3]), {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 7, (n + 1) a[n] == 7 (2 n - 1) a[n-1] - (n - 2) a[n-2]}, a, {n, 25}] (* Vincenzo Librandi, Nov 29 2016 *)

A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
my(q=3); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016

my(q=3, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

a(n) = P_n(3), where P_n(x) is the polynomial associated with row n of triangle A126216 in order of decreasing powers of x.
Recurrence: (n+1)*a(n) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(24 + 14*sqrt(3)) * (7 + 4*sqrt(3))^n / (24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
A(x) = -serreverse(A005061(x))(-x). - Gheorghe Coserea, Sep 30 2017
From Peter Bala, Dec 25 2020: (Start)
a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 3.
a(n) = (1/(2*n+1)) * (1/(2*m*(m+1))) * ( Legendre_P(n+1,2*m+1) - Legendre_P(n-1,2*m+1) ) at m = 3. (End)

More terms from Gheorghe Coserea, Apr 23 2016

A327316 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = ((x+r)^n - (x+s)^n)/(r - s), where r = 3 and s = 2.

Original entry on oeis.org

1, 5, 2, 19, 15, 3, 65, 76, 30, 4, 211, 325, 190, 50, 5, 665, 1266, 975, 380, 75, 6, 2059, 4655, 4431, 2275, 665, 105, 7, 6305, 16472, 18620, 11816, 4550, 1064, 140, 8, 19171, 56745, 74124, 55860, 26586, 8190, 1596, 180, 9, 58025, 191710, 283725, 247080

Offset: 1

Clark Kimberling, Nov 01 2019

For every choice of integers r and s, the polynomials p(n,x) form a strong divisibility sequence. Thus, if r, s, and x are integers, then p(x,n) is a strong divisibility sequence. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)).

			First seven rows:
     1
     5      2
    19     15     3
    65     76    30     4
   211    325   190    50    5
   665   1266   975   380   75    6
  2059   4655  4431  2275  665  105   7

Cf. A001047 (x=0), A005061 (x=1), A005060 (x=2), A005062 (x=3), A081200 (x=1/2).

f[x_, n_] := ((x + r)^n - (x + s)^n)/(r - s);
r = 3; s = 2;
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327316 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327316 sequence *)

A343237 Triangle T obtained from the array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0, by reading antidiagonals upwards.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 15, 19, 7, 1, 1, 31, 65, 37, 9, 1, 1, 63, 211, 175, 61, 11, 1, 1, 127, 665, 781, 369, 91, 13, 1, 1, 255, 2059, 3367, 2101, 671, 127, 15, 1, 1, 511, 6305, 14197, 11529, 4651, 1105, 169, 17, 1

Offset: 0

Wolfdieter Lang, May 10 2021

This is the row reversed version of the triangle A047969(n, m). The corresponding array A047969 is a(n, k) = A(k, n) (transposed of array A).

A(n-1, k-1) = k^n - (k-1)^n gives the number of n-digit numbers with digits from K = {1, 2, 3, ..., k} such that any digit from K, say k, appears at least once. Motivated by a comment in A005061 by Enrique Navarrete, the instance k=4 for n >= 1 (the column 3 in array A), and the d = 3 (sub)-diagonal sequence of T for m >= 0.

			The array A begins:
n\k  0  1   2    3     4     5     6      7      8      9 ...
-------------------------------------------------------------
0:   1  1   1    1     1     1     1      1      1      1 ...
1:   1  3   5    7     9    11    13     15     17     19 ...
2:   1  7  19   37    61    91   127    169    217    271 ...
3:   1 15  65  175   369   671  1105   1695   2465   3439 ...
4:   1 31 211  781  2101  4651  9031  15961  26281  40951 ...
5:   1 63 665 3367 11529 31031 70993 144495 269297 468559 ...
...
The triangle T begins:
n\m   0    1     2     3     4     5    6    7   8  9 10 ...
-------------------------------------------------------------
0:    1
1:    1    1
2:    1    3     1
3:    1    7     5     1
4:    1   15    19     7     1
5:    1   31    65    37     9     1
6:    1   63   211   175    61    11    1
7:    1  127   665   781   369    91   13    1
8:    1  255  2059  3367  2101   671  127   15   1
9:    1  511  6305 14197 11529  4651 1105  169  17  1
10:   1 1023 19171 58975 61741 31031 9031 1695 217 19  1
...
Combinatorial interpretation (cf. A005061 by _Enrique Navarrete_)
The three digits numbers with digits from K ={1, 2, 3, 4} having at least one 4 are:
j=1 (one 4): 114, 141, 411; 224, 242, 422; 334, 343, 433; 124, 214, 142, 241, 412, 421; 134, 314, 143, 341, 413, 431; 234, 243, 423. That is,  3*3 + 3!*3 = 27 = binomial(3, 1)*(4-1)^(3-1) = 3*3^2;
j=2 (twice 4):  144, 414, 441;  244, 424, 442;  344, 434, 443; 3*3 = 9 = binomial(3, 2)*(4-1)^(3-2) = 3*3;
j=3 (thrice 4) 444; 1 = binomial(3, 3)*(4-1)^(3-3).
Together: 27 + 9 + 1 = 37 = A(2, 3) = T(5, 3).

Cf. A005061, A008292, A047969 (reversed), A045531 (central diagonal), A047970 (row sums of triangle).
Row sequences of array A (nexus numbers): A000012, A005408, A003215, A005917(k+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
Column sequences of array A: A000012, A000225(n+1), A001047(n+1), A005061(n+1), A005060(n+1), A005062(n+1), A016169(n+1), A016177(n+1), A016185(n+1), A016189(n+1), A016195(n+1), A016197(n+1).

egf := exp(exp(x)*y + x)*(exp(x)*y - y + 1): ser := series(egf, x, 12):
cx := n -> series(n!*coeff(ser, x, n), y, 12):
Arow := n -> seq(k!*coeff(cx(n), y, k), k=0..9):
for n from 0 to 5 do Arow(n) od; # Peter Luschny, May 10 2021

A[n_, k_] := (k + 1)^(n + 1) - k^(n + 1); Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 10 2021 *)

Array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0.
A(n-1, k-1) = Sum_{j=1} binomial(n, j)*(k-1)^(n-j) = Sum_{j=0} binomial(n, j)*(k-1)^(n-j) - (k-1)^n = (1+(k-1))^n - (k-1)^n = k^n - (k-1)^n (from the combinatorial comment on A(n-1, k-1) above).
O.g.f. row n of array A: RA(n, x) = P(n, x)/(1 - x)^n, with P(n, x) = Sum_{m=0..n} A008292(n+1, m+1)*x^m, (the Eulerian number triangle  A008292 has offset 1) for n >= 0. (See the Oct 26 2008 comment in A047969 by Peter Bala). RA(n, x) = polylog(-(n+1), x)*(1-x)/x. (For P(n, x) see the formula by Vladeta Jovovic, Sep 02 2002.)
E.g.f. of e.g.f.s of the rows of array A: EE(x, y) = exp(x)*(1 + y*(exp(x) - 1))*exp(y*exp(x)), that is A(n, k) = [y^k/k!][x^n/n!] EE(x, y).
Triangle T(n, m) = A(n-m, m) = (m+1)^(n-m+1) - m^(n-m+1), n >= 0, m = 0, 1, ..., n.
E.g.f.: -(exp(x)-1)/(x*exp(x)*y-x). - Vladimir Kruchinin, Nov 02 2022

A380024 a(n) = 4^n - 3^n - binomial(n,2)*3^(n-2).

Original entry on oeis.org

0, 1, 6, 28, 121, 511, 2152, 9094, 38563, 163729, 694282, 2934592, 12348541, 51697075, 215291356, 891989002, 3677964295, 15099277669, 61745907934, 251632677604, 1022414950465, 4143511249831, 16755357788176, 67628131638478, 272531374722091

Offset: 0

Enrique Navarrete, Feb 05 2025

a(n) is the number of words of length n defined on 4 letters where one of the letters is used at least once but not twice.

			For n=2, the 6 words on {0, 1, 2, 3} that use 0 at least once but not twice are 10, 01, 20, 02, 30, 03.

Index entries for linear recurrences with constant coefficients, signature (13,-63,135,-108).

Cf. A005061, A086443, A380651.

Array[4^#-3^#-Binomial[#,2]*3^(#-2)&,25,0] (* or *) LinearRecurrence[{13,-63,135,-108},{0,1,6,28},25] (* James C. McMahon, Feb 14 2025 *)

def A380024(n): return (1<<(n<<1))-((n*(n-1)>>1)+9)*3**(n-2) if n>1 else n # Chai Wah Wu, Feb 14 2025

E.g.f.: exp(3*x)*(exp(x)-(x^2)/2-1).

G.f.: x*(1 - 7*x + 13*x^2)/((1 - 3*x)^3*(1 - 4*x)). - Stefano Spezia, Mar 03 2025

A228213 Number of second differences of arrays of length n + 2 of numbers in 0..3.

Original entry on oeis.org

13, 103, 625, 3151, 14053, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807, 70274600998837

Offset: 1

R. H. Hardin, Aug 16 2013

nonn

			Some solutions for n=4:
..5....0...-5....3....4....1....3...-3...-6....2....2....1...-5....1....0....2
.-4....2....2...-3...-5....1....1....3....3...-1....3....1....3....4....3....1
..2....0...-1....5....5...-3...-5....1....1....4...-5....1....0...-6...-3...-1
..3...-2....2...-6...-2....0....3...-5....1...-5....4...-1...-3....6....3....0

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 3 of A228218.

Empirical: a(n) = 7*a(n-1) - 12*a(n-2) = A005061(n+2) for n>7.
Conjectures from Colin Barker, Sep 09 2018: (Start)
G.f.: x*(13 + 12*x + 60*x^2 + 12*x^3 - 504*x^4 - 1584*x^5 - 1728*x^6) / ((1 - 3*x)*(1 - 4*x)).
a(n) = 4^(2+n) - 3^(2+n) for n>5.
(End)

cp's OEIS Frontend

A002250 a(n) = 4^n - 2*3^n.

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A129736 Primes of the form 4^k - 3^k.

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A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

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A248337 a(n) = 6^n - 4^n.

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A269576 a(n) = Product_{i=1..n} (4^i - 3^i).

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A269731 Dimensions of the 3-polytridendriform operad TDendr_3.

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A327316 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = ((x+r)^n - (x+s)^n)/(r - s), where r = 3 and s = 2.

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