A047848 -id:A047848 - cp's OEIS frontend

A133494 Diagonal of the array of iterated differences of A047848.

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A047855 a(n) = A047848(7,n).

Original entry on oeis.org

1, 2, 12, 112, 1112, 11112, 111112, 1111112, 11111112, 111111112, 1111111112, 11111111112, 111111111112, 1111111111112, 11111111111112, 111111111111112, 1111111111111112, 11111111111111112, 111111111111111112, 1111111111111111112, 11111111111111111112, 111111111111111111112

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A001019(n-1) for n >= 1.
Range of A164898, apart from first term. - Reinhard Zumkeller, Aug 30 2009
a(n) is the number of integers less than or equal to 10^n, whose initial digit is 1. - Michel Marcus, Jul 04 2019
a(n) is 2^n represented in bijective base-2 numeration. - Alois P. Heinz, Aug 26 2019
This sequence proves both A028842 (numbers with prime product of digits) and A028843 (numbers with prime iterated product of digits) are infinite. Proof: Suppose either of those sequences is finite. Label as omega the supposed last term. Compute n = ceiling(log_10 omega) + 1. Then a(n) > omega. The product of digits of a(n) is 2, contradicting the assumption that omega is the final term of either A028842 or A028843. - Alonso del Arte, Apr 14 2020
For n >= 2, the concatenation of a(n) with 8*a(n) equals (3*R_n+3)^2, where R_n = A002275(n) is the repunit with n 1's; hence this sequence, except for {1,2}, is a subsequence of A115549. - Bernard Schott, Apr 30 2022

Ivan Panchenko, Table of n, a(n) for n = 0..200
Wikipedia, Bijective numeration.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A001019, A002275, A028842, A028843, A047848, A115549, A164898.

Cf. A002281, A062397.

[(10^n + 8)/9: n in [0..40]]; // G. C. Greubel, Jan 11 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=10*a[n-1]+1 od: seq(a[n]+1, n=0..18); # Zerinvary Lajos, Mar 20 2008

Join[{1}, Table[FromDigits[PadLeft[{2}, n, 1]], {n, 30}]] (* Harvey P. Dale, Apr 17 2013 *)
(10^Range[0, 29] + 8)/9 (* Alonso del Arte, Apr 12 2020 *)

a(n)=if(n==0,1,if(n==1,2,11*a(n-1)-10*a(n-2)))
for(i=0,10,print1(a(i),",")) \\ Lambert Klasen, Jan 28 2005

def A047855(n): return (pow(10,n) +8)//9
print([A047855(n) for n in range(41)]) # G. C. Greubel, Jan 11 2025

[gaussian_binomial(n,1,10)+1 for n in range(17)] # Zerinvary Lajos, May 29 2009

(List.fill(20)(10: BigInt)).scanLeft(1: BigInt)( * ).map(n => (n + 8)/9) // Alonso del Arte, Apr 12 2020

a(n) = (10^n + 8)/9. - Ralf Stephan, Feb 14 2004
a(0) = 1, a(1) = 2, a(n) = 11*a(n-1) - 10*a(n-2) for n > 1. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28 2005
G.f.: (1 - 9*x)/(1 - 11*x + 10*x^2). - Philippe Deléham, Oct 05 2009
a(n) = 10*a(n-1) - 8 (with a(0) = 1). - Vincenzo Librandi, Aug 06 2010
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(8 + exp(9*x))/9.
a(n) = (A062397(n) - A002281(n))/2. (End)

More terms from Harvey P. Dale, Apr 17 2013

A047851 a(n) = A047848(3,n).

Original entry on oeis.org

1, 2, 8, 44, 260, 1556, 9332, 55988, 335924, 2015540, 12093236, 72559412, 435356468, 2612138804, 15672832820, 94036996916, 564221981492, 3385331888948, 20311991333684, 121871948002100, 731231688012596, 4387390128075572, 26324340768453428, 157946044610720564, 947676267664323380

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A000351(n-1) for n >= 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000351, A047848.

[(6^n + 4)/5: n in [0..40]]; // G. C. Greubel, Jan 11 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=6*a[n-1]+1 od: seq(a[n]+1, n=0..20); # Zerinvary Lajos, Mar 20 2008

(6^Range[0,40] +4)/5 (* G. C. Greubel, Jan 11 2025 *)

def A047851(n): return (pow(6,n) + 4)//5
print([A047851(n) for n in range(41)]) # G. C. Greubel, Jan 11 2025

a(n) = (6^n + 4)/5. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 05 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 7*a(n-1) - 6*a(n-2) for n > 1.
G.f.: (1 - 5*x)/(1 - 7*x + 6*x^2). (End)
a(n) = 6*a(n-1) - 4 (with a(0)=1). - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(5*x) + 4)/5. - Elmo R. Oliveira, Aug 29 2024

a(21)-a(24) from Elmo R. Oliveira, Aug 29 2024

A047853 a(n) = A047848(5, n).

Original entry on oeis.org

1, 2, 10, 74, 586, 4682, 37450, 299594, 2396746, 19173962, 153391690, 1227133514, 9817068106, 78536544842, 628292358730, 5026338869834, 40210710958666, 321685687669322, 2573485501354570, 20587884010836554, 164703072086692426, 1317624576693539402, 10540996613548315210

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A000420(n-1) for n >= 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A000420, A047848, A226308.

[(8^n +6)/7: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=8*a[n-1]+1 od: seq(a[n]+1, n=0..18); # Zerinvary Lajos, Mar 20 2008

LinearRecurrence[{9, -8}, {1, 2}, 30] (* Harvey P. Dale, Dec 11 2016 *)
(8^Range[0,40] +6)/7 (* G. C. Greubel, Jan 12 2025 *)

def A047853(n): return (pow(8,n) +6)//7
print([A047853(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (8^n + 6)/7. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 05 2009: (Start)
a(0)=1, a(1)=2; a(n) = 9*a(n-1) - 8*a(n-2) for n>1.
G.f.: (1 - 7*x)/(1 - 9*x + 8*x^2). (End)
a(n) = 8*a(n-1) - 6 for n>0, a(0)=1. - Vincenzo Librandi, Aug 06 2010
a(n+1) = A226308(3*n). - Philippe Deléham, Feb 24 2014
E.g.f.: exp(x)*(6 + exp(7*x))/7. - Stefano Spezia, Oct 16 2023

More terms from Vladimir Joseph Stephan Orlovsky, Nov 07 2008

A047854 a(n) = A047848(6, n).

Original entry on oeis.org

1, 2, 11, 92, 821, 7382, 66431, 597872, 5380841, 48427562, 435848051, 3922632452, 35303692061, 317733228542, 2859599056871, 25736391511832, 231627523606481, 2084647712458322, 18761829412124891, 168856464709124012, 1519708182382116101, 13677373641439044902, 123096362772951404111

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A001018(n-1) for n >= 1.

Also, the cogrowth sequence of the 16-element group D4 X C2 = . - Sean A. Irvine, Nov 10 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A001018, A047848.

[(9^n +7)/8: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=9*a[n-1]+1 od: seq(a[n]+1, n=0..17); # Zerinvary Lajos, Mar 20 2008

a = {1}; ZZ = 1; Do[ZZ = ZZ + 3^(2x); AppendTo[a, ZZ], {x,0,40}]; a (* Zerinvary Lajos, Apr 03 2007 *)
(9^Range[0,40] +7)/8 (* G. C. Greubel, Jan 12 2025 *)

def A047854(n): return (pow(9,n) +7)//8
print([A047854(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (9^n + 7)/8. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 06 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 10*a(n-1) - 9*a(n-2) for n > 1.
G.f.: (1 - 8*x)/(1 - 10*x + 9*x^2). (End)
a(n) = 9*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(8*x) + 7)/8. - Elmo R. Oliveira, Aug 29 2024

a(18)-a(22) from Elmo R. Oliveira, Aug 29 2024

A047852 a(n) = A047848(4, n).

Original entry on oeis.org

1, 2, 9, 58, 401, 2802, 19609, 137258, 960801, 6725602, 47079209, 329554458, 2306881201, 16148168402, 113037178809, 791260251658, 5538821761601, 38771752331202, 271402266318409, 1899815864228858, 13298711049602001, 93090977347214002, 651636841430498009, 4561457890013486058

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A000400(n-1) for n >= 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000400, A047848.

[(7^n +5)/6: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=7*a[n-1]+1 od: seq(a[n]+1, n=0..19); # Zerinvary Lajos, Mar 20 2008

(7^Range[0,40] +5)/6 (* G. C. Greubel, Jan 12 2025 *)

def A047852(n): return (pow(7, n) + 5)//6
print([A047852(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (7^n + 5)/6. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 06 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 8*a(n-1) - 7*a(n-2) for n > 1.
G.f.: (1 - 6*x)/(1 - 8*x + 7*x^2). (End)
a(n) = 7*a(n-1) - 5, with a(0)=1. - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(6*x) + 5)/6. - Elmo R. Oliveira, Aug 29 2024

a(20)-a(23) from Elmo R. Oliveira, Aug 29 2024

A047856 a(n) = A047848(8, n).

Original entry on oeis.org

1, 2, 13, 134, 1465, 16106, 177157, 1948718, 21435889, 235794770, 2593742461, 28531167062, 313842837673, 3452271214394, 37974983358325, 417724816941566, 4594972986357217, 50544702849929378, 555991731349223149, 6115909044841454630, 67274999493256000921, 740024994425816010122

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A011557(n-1) for n >= 1.

G. C. Greubel, Table of n, a(n) for n = 0..955
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A011557, A047848.

[(11^n +9)/10: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=11*a[n-1]+1 od: seq(a[n]+1, n=0..16); # Zerinvary Lajos, Mar 20 2008

(11^Range[0,40] +9)/10 (* G. C. Greubel, Jan 12 2025 *)

def A047856(n): return (pow(11, n) + 9)//10
print([A047856(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (11^n + 9)/10. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 06 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 12*a(n-1) - 11*a(n-2) for n > 1.
G.f.: (1 - 10*x)/(1 - 12*x + 11*x^2). (End)
a(n) = 11*a(n-1) - 9, with a(0)=1. - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(10*x) + 9)/10. - Elmo R. Oliveira, Aug 30 2024

a(17)-a(21) from Elmo R. Oliveira, Aug 30 2024

A047857 a(n) = Sum_{k=0..n} A047848(k, n-k).

Original entry on oeis.org

1, 3, 8, 23, 73, 251, 920, 3573, 14695, 64047, 295792, 1445659, 7460349, 40539363, 231303192, 1381924345, 8623569739, 56078184471, 379232618512, 2662012084719, 19362915524849, 145719545817995, 1133022996552664, 9090156893772893, 75161929706243087, 639793220810832639

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..590

A047857:= func< n | n+1 + (&+[((k+1)^(n-k+2) -1)/k: k in [2..n+2]]) >;
[A047857(n): n in [0..30]]; // G. C. Greubel, Jan 17 2025

Table[Sum[((k+3)^(n-k) +k+1)/(k+2), {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, May 23 2021 *)

def A047857(n): return n+1 +sum((pow(k+1, n-k+2) -1)//k for k in range(2,n+3))
print([A047857(n) for n in range(31)]) # G. C. Greubel, Jan 17 2025

a(n) = n + 2 - H(n+2) + Sum_{k=2..n+2} (k+1)^(n-k+2)/k, where H(n) is the nth Harmonic number. - G. C. Greubel, Jan 17 2025

a(14) onward corrected by Sean A. Irvine, May 22 2021

More terms from G. C. Greubel, Jan 17 2025

A196791 a(n) = A047848(9, n).

Original entry on oeis.org

1, 2, 14, 158, 1886, 22622, 271454, 3257438, 39089246, 469070942, 5628851294, 67546215518, 810554586206, 9726655034462, 116719860413534, 1400638324962398, 16807659899548766, 201691918794585182, 2420303025535022174, 29043636306420266078, 348523635677043192926

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001021 (first differences).

Magma
```
[(12^n+10)/11: n in [0..20]];
```

LinearRecurrence[{13,-12},{1,2},30] (* Harvey P. Dale, Sep 07 2015 *)
(12^Range[0,40] +10)/11 (* G. C. Greubel, Jan 17 2025 *)

def A196791(n): return (pow(12, n) + 10)//11
print([A196791(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (12^n + 10)/11.
a(n) = 12*a(n-1) - 10, with a(0) = 1.
G.f.: (1-11*x)/((1-x)*(1-12*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(11*x) + 10)/11.
a(n) = 13*a(n-1) - 12*a(n-2) for n > 1. (End)

A196792 a(n) = A047848(10, n).

Original entry on oeis.org

1, 2, 15, 184, 2381, 30942, 402235, 5229044, 67977561, 883708282, 11488207655, 149346699504, 1941507093541, 25239592216022, 328114698808275, 4265491084507564, 55451384098598321, 720867993281778162, 9371283912663116095, 121826690864620509224, 1583746981240066619901

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (14,-13).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001022 (first differences).

Magma
```
[(13^n+11)/12: n in [0..20]];
```

(13^Range[0,40] +11)/12 (* G. C. Greubel, Jan 17 2025 *)

def A196792(n): return (pow(13, n) + 11)//12
print([A196792(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (13^n + 11)/12.
a(n) = 13*a(n-1) - 11, with a(0) = 1.
G.f.: (1-12*x)/((1-x)*(1-13*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(12*x) + 11)/12.
a(n) = 14*a(n-1) - 13*a(n-2) for n > 1. (End)

cp's OEIS Frontend

A133494 Diagonal of the array of iterated differences of A047848.

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A047855 a(n) = A047848(7,n).

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A047851 a(n) = A047848(3,n).

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A047853 a(n) = A047848(5, n).

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A047854 a(n) = A047848(6, n).

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A047852 a(n) = A047848(4, n).

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