A005060 -id:A005060 - cp's OEIS frontend

A257285 a(n) = 45^n - 34^n.

1, 8, 52, 308, 1732, 9428, 50212, 263348, 1365892, 7026068, 35916772, 182729588, 926230852, 4681485908, 23608756132, 118849087028, 597466660612, 3000218204948, 15052630632292, 75469311591668, 378171191679172, 1894154493279188, 9483966605929252

Offset: 0

M. F. Hasler, May 03 2015

First differences of 5^n - 4^n = A005060.

a(n-1) is the number of numbers with n digits having the largest digit equal to 4. Note that this is independent of the base b>4. Equivalently, number of n-letter words over a 5-letter alphabet {a,b,c,d,e}, which must not start with the first letter of the alphabet, and in which the last letter of the alphabet must appear.

Index entries for linear recurrences with constant coefficients, signature (9,-20).

Cf. A005060. See also A000225, A027649, A255463, A257286 - A257289 and A088924.

[4*5^n-3*4^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Table[4 5^n - 3 4^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)

PARI
```
a(n)=4*5^n-3*4^n
```

From Vincenzo Librandi, May 04 2015: (Start)
G.f.: (1-x)/((1-4*x)*(1-5*x)).
a(n) = 9*a(n-1) - 20*a(n-2). - (End)
E.g.f.: exp(4*x)*(4*exp(x) - 3). - Stefano Spezia, Nov 15 2023

A280209 Numbers m such that 5^m - 4^m is not squarefree, but 5^d - 4^d is squarefree for every proper divisor d of m.

Original entry on oeis.org

2, 55, 171, 183, 203

Offset: 1

Juri-Stepan Gerasimov, Dec 28 2016

Where numbers m such that 5^m - 4^m is not squarefree: numbers of the form i*a(j) for i >= 1.
Numbers m such that (k+1)^m - k^m is not squarefree, but (k+1)^d - k^d is squarefree for every proper divisor d of m:
A237043 (k = 1): 6, 20, 21, 110, 136, 155, 253, 364, 602, 657, 812, 889, 979, 1081, ... a(15) >= 1207 - Max Alekseyev, Sep 28 2015;
A280203 (k = 2): 10, 11, 42, 52, 57, 203, 272, 497, ... a(9) > 497 - Charles R Greathouse IV, Dec 27 2016;
A280208 (k = 3): 4, 14, 55, 78, 111, 253, 342, 355, ... a(9) >= 431;
this sequence (k = 4): 2, 55, 171, 183, 203, ... a(6) >= 367;
A... (k = 5): 21, 22, 39, 136, 186, 203, 244, 333, ... a(9) >= 337;
A280307 (k = 6): 20, 26, 55, 68, 171, 258, 310, ... a(8) >= 323;
A... (k = 7): 3, 10, 55, 272, ... a(5) >= 289;
A... (k = 8): 20, 21, 22, 34, 93, 116, 138, 156, 166, 205, 253, ... a(12) >= 277;
A... (k = 9): 33, 38, 42, 78, 110, 155, ... a(7) >= 263;
A... (k = 10): 6, 14, 52, 68, 253, ... a(6) >= 263;
A... (k = 11): 20, 42, 46, 53, 114, 136, 156, 205, ... a(9) >= 251.
The smallest square of 5^m - 4^m as defined above are 9, 121, 361, 3721, 841. - Robert Price, Mar 07 2017
a(6) <= 465. 465, 955, 1027 are terms. - Chai Wah Wu, Jul 20 2020

			2 is in this sequence because 5^1 - 4^1 = 1 is squarefree where 1 is proper divisor of 2 and 5^2 - 4^2 = 9 = 3^2 is not squarefree.

Cf. A005060, A237043, A280203, A280208.

a(3)-a(5) from Jinyuan Wang, May 15 2020

A269732 Dimensions of the 4-polytridendriform operad TDendr_4.

Original entry on oeis.org

1, 9, 101, 1269, 17081, 240849, 3511741, 52515549, 801029681, 12414177369, 194922521301, 3094216933509, 49575333021801, 800645021406369, 13020241953611181, 213025792632813549, 3504075376813414241, 57914491106005287849, 961297812844696640581, 16017765308027639317269, 267831397282643166904601, 4492625888792276208945009, 75578709400747348254905501

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, A005060, A001263, A126216, A269730, A269731, A084769.

I:=[1,9]; [n le 2 select I[n] else (9*(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 - 9*x - Sqrt[1 - 18*x + x^2])/(40*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 9]/(2*Sqrt[5]), {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 9, (n+1) a[n] == 9 (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=4); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016

my(q=4, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

a(n) = P_n(4), 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) = 9*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(40 + 18*sqrt(5)) * (9 + 4*sqrt(5))^n / (40*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ phi^(6*n + 3) / (2^(5/2) * 5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017
A(x) = -serreverse(A005060(x))(-x). - Gheorghe Coserea, Sep 30 2017
O.g.f.: A(x) = (1 - sqrt(1 - 18*x + x^2) - 9*x)/(40*x). - Peter Bala, Jan 25 2018
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 = 4.
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 = 4. (End)

More terms from Gheorghe Coserea, Apr 23 2016

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

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

A269661 a(n) = Product_{i=1..n} (5^i - 4^i).

Original entry on oeis.org

1, 9, 549, 202581, 425622681, 4907003889249, 302963327126122509, 98490045052104040328301, 166544794872251942218390753281, 1451779137596368920662880897497387769, 64798450159010700654830227323217753649135349

Offset: 1

Bob Selcoe, Mar 02 2016

Cf. A005060, A109345.
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), A269576 (j=4, k=3).

[&*[ 5^k-4^k: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Mar 03 2016

Table[Product[5^i - 4^i, {i, n}], {n, 15}] (* Vincenzo Librandi, Mar 03 2016 *)
Table[5^(Binomial[n + 1, 2]) *QPochhammer[4/5, 4/5, n], {n, 1, 20}] (* G. C. Greubel, Mar 05 2016 *)
FoldList[Times,Table[5^n-4^n,{n,15}]] (* Harvey P. Dale, Aug 28 2018 *)

a(n) = prod(k=1, n, 5^k-4^k); \\ Michel Marcus, Mar 05 2016

a(n) = Product_{i=1..n} A005060(i).
a(n) = 5^(binomial(n+1,2))*(4/5;4/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Mar 05 2016
a(n) ~ c * 5^(n*(n+1)/2), where c = QPochhammer(4/5) = 0.00336800585242312126... . - Vaclav Kotesovec, Oct 10 2016

A061929 Triangle with n >= k >= 0 where a(n,k) = 2^k3^(n-k)(C(n+1,0)+C(n+1,1)+...C(n+1,k)).

Original entry on oeis.org

1, 3, 6, 9, 24, 28, 27, 90, 132, 120, 81, 324, 576, 624, 496, 243, 1134, 2376, 3024, 2736, 2016, 729, 3888, 9396, 13824, 14256, 11520, 8128, 2187, 13122, 35964, 60264, 70416, 63072, 47424, 32640, 6561, 43740, 134136, 252720, 331776, 330048, 268416

Offset: 0

Henry Bottomley, May 22 2001

			Rows start (1), (3,6), (9,24,68), (27,90,132,120) etc.

Row sums are 5^(n+1)-4^(n+1), i.e. A005060. Cf. A061930.

a(n, k) =A054143(n, k)*A036561(n, n-k)

A228214 Number of second differences of arrays of length n + 2 of numbers in 0..4.

Original entry on oeis.org

17, 177, 1429, 9705, 58141, 320481, 1688101, 8717049, 44633821, 227363409, 1153594261, 5835080169, 29443836301, 148292923329, 745759583941, 3745977788889, 18798608421181, 94267920012849, 472439111692021

Offset: 1

R. H. Hardin, Aug 16 2013

nonn

			Some solutions for n=4:
..8....5...-1...-2....4...-2....3....0...-2...-6....2....0....4...-5....0...-5
.-8....0....2....2...-5...-4...-4...-2....2....5....2...-2...-3....4....2....5
..5....0....0...-2...-1....7....3....2....4....0...-6....3...-1...-1...-6...-6
.-1...-2...-2....4....2...-3...-3....3...-7...-1....4...-3....6...-1....5....4

R. H. Hardin, Table of n, a(n) for n = 1..137

Column 4 of A228218.

Empirical: a(n) = 9*a(n-1) - 20*a(n-2) = A005060(n+2) for n>9.
Conjectures from Colin Barker, Sep 10 2018: (Start)
G.f.: x*(17 + 24*x + 176*x^2 + 384*x^3 - 624*x^4 - 8688*x^5 - 33408*x^6 - 66240*x^7 - 57600*x^8) / ((1 - 4*x)*(1 - 5*x)).
a(n) = 5^(2+n) - 4^(2+n) for  n>7.
(End)

A248343 a(n) = 10^n - 8^n.

Original entry on oeis.org

0, 2, 36, 488, 5904, 67232, 737856, 7902848, 83222784, 865782272, 8926258176, 91410065408, 931280523264, 9450244186112, 95601953488896, 964815627911168, 9718525023289344, 97748200186314752, 981985601490518016, 9855884811924144128

Offset: 0

Vincenzo Librandi, Oct 05 2014

Index entries for linear recurrences with constant coefficients, signature (18,-80).

Cf. A016185, A016186, A139746.

Cf. A000079, A001018, A005060, A011557.

Magma
```
[10^n-8^n: n in [0..30]];
```

Table[10^n - 8^n, {n, 0, 25}] (* or *) CoefficientList[Series[2 x/((1 - 8 x) (1 - 10 x)), {x, 0, 30}], x]
LinearRecurrence[{18,-80},{0,2},30] (* Harvey P. Dale, May 21 2018 *)

G.f.: 2*x/((1-8*x)*(1-10*x)).
a(n) = 18*a(n-1) - 80*a(n-2).
a(n) = 2^n*(5^n - 4^n) = A000079(n)*A005060(n) = A011557(n) - A001018(n).
a(n+1) = 2*A016186(n).
E.g.f.: 2*exp(9*x)*sinh(x). - Elmo R. Oliveira, Mar 31 2025

cp's OEIS Frontend

A257285 a(n) = 4*5^n - 3*4^n.

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A280209 Numbers m such that 5^m - 4^m is not squarefree, but 5^d - 4^d is squarefree for every proper divisor d of m.

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A269732 Dimensions of the 4-polytridendriform operad TDendr_4.

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

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

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Maple

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A269661 a(n) = Product_{i=1..n} (5^i - 4^i).

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A061929 Triangle with n >= k >= 0 where a(n,k) = 2^k*3^(n-k)*(C(n+1,0)+C(n+1,1)+...C(n+1,k)).

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A257285 a(n) = 45^n - 34^n.

A061929 Triangle with n >= k >= 0 where a(n,k) = 2^k3^(n-k)(C(n+1,0)+C(n+1,1)+...C(n+1,k)).