A047855 -id:A047855 - cp's OEIS frontend

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

1, 2, 1, 5, 2, 1, 14, 6, 2, 1, 41, 22, 7, 2, 1, 122, 86, 32, 8, 2, 1, 365, 342, 157, 44, 9, 2, 1, 1094, 1366, 782, 260, 58, 10, 2, 1, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1, 29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1

Offset: 0

Clark Kimberling

			Array, A(n, k), begins as:
  1, 2,  5,  14,   41, ... = A007051.
  1, 2,  6,  22,   86, ... = A047849.
  1, 2,  7,  32,  157, ... = A047850.
  1, 2,  8,  44,  260, ... = A047851.
  1, 2,  9,  58,  401, ... = A047852.
  1, 2, 10,  74,  586, ... = A047853.
  1, 2, 11,  92,  821, ... = A047854.
  1, 2, 12, 112, 1112, ... = A047855.
  1, 2, 13, 134, 1465, ... = A047856.
  1, 2, 14, 158, 1886, ... = A196791.
  1, 2, 15, 184, 2381, ... = A196792.
Downward antidiagonals, T(n, k), begins as:
      1;
      2,     1;
      5,     2,     1;
     14,     6,     2,     1;
     41,    22,     7,     2,     1;
    122,    86,    32,     8,     2,    1;
    365,   342,   157,    44,     9,    2,   1;
   1094,  1366,   782,   260,    58,   10,   2,   1;
   3281,  5462,  3907,  1556,   401,   74,  11,   2,  1;
   9842, 21846, 19532,  9332,  2802,  586,  92,  12,  2, 1;
  29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A047857 (row sums), A196793 (main diagonal).

A:= func< n,k | ((n+3)^k +n+1)/(n+2) >; // array A047848
[A(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2025

A[n_, k_]:= ((n+3)^k +n+1)/(n+2);
A047848[n_, k_]:= A[k,n-k];
Table[A047848[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 11 2025 *)

def A(n,k): return (pow(n+3,k) +n+1)//(n+2) # array A047848
print(flatten([[A(k,n-k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 11 2025

A(n, k) = ((n+3)^k + n + 1)/(n+2). - Ralf Stephan, Feb 14 2004
From G. C. Greubel, Jan 11 2025: (Start)
T(n, k) = ((k+3)^(n-k) + k + 1)/(k+2) (antidiagonal triangle).
T(n, n) = A196793(n).
Sum_{k=0..n} T(n, k) = A047857(n). (End)

A061076 a(n) is the sum of the products of the digits of all the numbers from 1 to n.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 90, 92, 96, 102, 110, 120, 132, 146, 162, 180, 180, 183, 189, 198, 210, 225, 243, 264, 288, 315, 315, 319, 327, 339, 355, 375, 399, 427, 459, 495, 495, 500, 510, 525, 545, 570, 600, 635

Offset: 1

Amarnath Murthy, Apr 14 2001

What is the asymptotic behavior of this sequence? a(n) = a(n+1) for almost all n. A weak upper bound: a(n) << n^1.91. - Charles R Greathouse IV, Jan 13 2012
A check was done for k in {i^j | 1 <= i <= 10 AND 1 <= j <= 100}. For all these values, a(k) < k^1.733. Another check for k in {i^j | 101 <= i <= 110 AND 101 <= j <= 200} gave a(k) < k^1.65324. For k in {i | 10^6 <= i <= 10^7}, a(k) < k^1.6534. So I ask: is it true that a(n) < n^1.733 and a(n) -> n^(1.65323 + o(1)), or about n^(log(45)/log(10) + o(1))? - David A. Corneth, May 17 2016
For n = 10^(k-1), the closed-form formula from Mihai Teodor (see Formula section) gives a(n) = (45^k - 45)/44, so lim_{n->oo} log(a(n))/log_10(n) = log(45) = 3.80666248977.... - Jon E. Schoenfield, Apr 10 2022
For k >= 1, a(10^k-1) = a(10^k) = ... = a(10*R_k) where R = A002275; so there is a run of 10*R_{k-1} + 2 = A047855(k) consecutive terms equal to (45/44)*(45^k-1) when n runs from 10^k-1 up to 10*R_k, this is because those numbers have one or more 0's. Example: first runs with 2, 12, 112, 1112, ... consecutive terms equal to 45, 2070, 93195, 4193820, ... start at 9, 99, 999, 9999, ... and end at 10, 110, 1110, 11110, ... - Bernard Schott, Oct 18 2022

			a(9) = a(10) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1*0 = 1+2+3+4+5+6+7+8+9 = 45.

Amarnath Murthy, Smarandache friendly numbers and a few more sequences, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.

Daria Micovic, Table of n, a(n) for n = 1..10000
Amarnath Murthy, Smarandache friendly numbers and a few more sequences, vixra, 2014.
Luca Onnis, On the general Smarandache's sigma product of digits, arXiv:2203.07227 [math.GM], 2022.

Cf. A002275, A007954, A037123, A047855.

A007954:= n -> convert(convert(n,base,10),`*`):
ListTools:-PartialSums(map(A007954,[$1..100])); # Robert Israel, May 17 2016

Accumulate[Times@@IntegerDigits[#]& /@ Range[100]]

pd(n) = my(d = digits(n)); prod(i=1, #d, d[i]);
a(n) = sum(k=1, n, pd(k)); \\ Michel Marcus, Feb 01 2015

a(n) = {n=digits(n); p=1; d=#n; for(i=1,#n,if(n[i]==0,d=i-1;break));
(45/44) * (45^(#n-1)-1) + sum(i=1,d,p*=n[i]; p * (n[i]-1) * (45/44) * (45^(#n -i) - 45^(#n-i-1)) / 2)+p*(d==#n)} \\ David A. Corneth, May 17 2016

from math import prod
def A061076(n): return sum(prod(int(d) for d in str(i)) for i in range(1,n+1)) # Chai Wah Wu, Mar 21 2022

def A061076(n):
    p = 0
    i = 0
    while i < n + 1:
        p += prod(int(digit) for digit in str(i))
        i += 1
    return p # Daria Micovic, Apr 13 2016

a(n) = Sum_{k = 1..n} (product of the digits of k).
a(10^k-1) = (45/44)*(45^k-1). - Giovanni Resta, Oct 18 2012
From Robert Israel, May 17 2016: (Start)
Partial sums of A007954.
G.f.: (1-x)^(-1) * Sum_{n>=0} Product_{j=0..n} Sum_{k=1..9} k * x^(k*10^j).
G.f. satisfies A(x) = (x + 2*x^2 + ... + 9*x^9)*(1+(1-x^10)*A(x^10))/(1-x).
(End)
Let b(1), b(2), ..., b(k) be the digits of the base-10 expansion of n: n = b(1)*10^(k-1) + b(2)*10^(k-2) + ... + b(k). Then a(n) = b(1)*b(2)*...*b(k) + (45^k-45)/44 + (1/2)*Sum_{i=1..k} b(1)*b(2)*...*b(i)*(b(i)-1)*45^(k-i). - Mihai Teodor, Apr 09 2022

Corrected and extended by Matthew Conroy, Apr 16 2001

A093134 A Jacobsthal trisection.

Original entry on oeis.org

1, 0, 8, 56, 456, 3640, 29128, 233016, 1864136, 14913080, 119304648, 954437176, 7635497416, 61083979320, 488671834568, 3909374676536, 31274997412296, 250199979298360, 2001599834386888, 16012798675095096, 128102389400760776, 1024819115206086200, 8198552921648689608

Offset: 0

Paul Barry, Mar 23 2004

Counts closed walks at a vertex of the complete graph on 9 nodes K_9.

Second binomial transform is A047855.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,8).

Other sequences with a(n+1) = 8^n - a(n) are A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A015565. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Cf. A047855.

[(8^n/9+8*(-1)^n/9): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

k=0;lst={1, k};Do[k=8^n-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(8^n + 8*(-1)^n)/9, {n,0,30}] (* or *) LinearRecurrence[{7,8}, {1,0}, 30] (* G. C. Greubel, Jan 06 2018 *)

for(n=0,30, print1((8^n + 8*(-1)^n)/9, ", ")) \\ G. C. Greubel, Jan 06 2018

G.f.: (1-7*x)/(1 - 7*x - 8*x^2).
a(n) = (8^n + 8*(-1)^n)/9.
a(n) = 8*A001045(3*n-3)/3.
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(-x)*(exp(9*x) + 8)/9.
a(n) = 7*a(n-1) + 8*a(n-2) for n > 1. (End)

A337718 Numbers that can be written as (m + product of digits of m) for some m.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 50, 54, 55, 56, 58, 60, 62, 65, 66, 67, 68, 70, 74, 75, 78, 80, 81, 85, 86, 88, 89, 90, 92, 94, 95, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109

Offset: 1

Bernard Schott, Sep 16 2020

Every integer that contains a digit 0 is a term (A011540).
When R_m with m >= 1 is in A002275, then R_m + 1 is a term (A047855 \ {1}).
Near similar:
-> Not-Colombian (A176995) are numbers that can be written as (m + sum of digits of m) for some m.
-> Bogotá numbers (A336826) are numbers that can be written as (m * product of digits of m) for some m.

			10 = 5 + 5 = 10 + (1*0) and 22 = 16 + (1*6) are terms.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Subsequences: A011540, A047855 \ {1}.
Range of A230099.
Cf. A176995 (not Colombian), A336826 (Bogotá numbers).

m = 100; Select[Union[Table[n + Times @@ IntegerDigits[n], {n, 0, m}]], # <= m &] (* Amiram Eldar, Sep 16 2020 *)

isok(m) = {if (m==0, return (1)); for (k=1, m,  if (k+vecprod(digits(k)) == m, return (1)););} \\ Michel Marcus, Sep 17 2020

from math import prod
def b(n): return n + prod(map(int, str(n)))
def aupto(n): return sorted(set(b(m) for m in range(n+1) if b(m) <= n))
print(aupto(109)) # Michael S. Branicky, Jan 09 2023

A098406 a(n) = (10^n + 17)/9.

Original entry on oeis.org

2, 3, 13, 113, 1113, 11113, 111113, 1111113, 11111113, 111111113, 1111111113, 11111111113, 111111111113, 1111111111113, 11111111111113, 111111111111113, 1111111111111113, 11111111111111113, 111111111111111113, 1111111111111111113, 11111111111111111113, 111111111111111111113

Offset: 0

Klaus Brockhaus, Sep 07 2004

A097683 gives numbers k such that a(k) is prime.

			a(5) = (100000 + 17)/9 = 11113.

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

Cf. A097683, A047855, A002275.

Cf. A002282, A062397.

FromDigits/@Table[PadLeft[{3},n,1],{n,20}] (* Harvey P. Dale, Jun 18 2011 *)

PARI
```
for(n=1,18,print1(((10^n)+17)/9,","))
```

a(1) = 3; a(n) = a(n-1) + 10^(n-1).
a(1) = 3; a(n) = 10*a(n-1) - 17.
a(n) = A047855(n)+1 = A002275(n)+2.
G.f.: (2-19*x)/((10*x-1)*(x-1)). - R. J. Mathar, Jan 27 2017
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(x)*(exp(9*x) + 17)/9.
a(n) = A062397(n) - A002282(n).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1. (End)

a(0) from Ivan Panchenko, Nov 02 2013

A378865 a(n) is the smallest positive integer k such that n*k uses none of the digits of n, or 0 if no such k exists.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 2, 4, 4, 3, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 3, 2, 2, 0, 2, 4, 2, 2, 2, 2, 4, 2, 12, 0, 4, 2, 2, 2, 2, 2, 2, 2, 2, 0, 4, 5, 3, 2, 2, 2, 2, 3, 2, 0, 4, 2, 2, 3, 4, 2, 2, 2, 2, 0, 3, 2, 2, 3

Offset: 1

Gonzalo Martínez, Dec 09 2024

It is observed that a(10*m) = 0 and a(100*m + 5) = 0 for all positive integers m.

If m is a repdigit number (A010785) that does not have the digit 9, then a(m) = 2 and if m = 99...9, with t 9's, then a(m) = 11...2, i.e., (t - 1) 1's followed by 2, since 99...9 * 11...2 equals (t - 1) 1's followed by (t - 1) 8's, where k = 11...2 is the smallest number with this property. In other words, a(A002283(m)) = A047855(m), for all positive integers m.

			a(12) = 3, since 12*1 = 12, 12*2 = 24 have digits in common with 12, while 12*3 = 36 does not.

Dominic McCarty, Table of n, a(n) for n = 1..1000 (synthesized from _Michel Marcus_'s data at A074157)

Cf. A074157, A010785, A002283, A047855.

a(n) = A074157(n)/n.

A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 3, 23, 223, 2223, 22223, 222223, 2222223, 22222223, 222222223, 2222222223, 22222222223, 222222222223, 2222222222223, 22222222222223, 222222222222223, 2222222222222223, 22222222222222223, 222222222222222223, 2222222222222222223, 22222222222222222223, 222222222222222222223

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 2*A001045(3*n)/3 + (-1)^n.
Partial sums of A093136.
A convex combination of 10^n and 1.
In general the second binomial transform of k*Jacobsthal(3*n)/3 + (-1)^n is 1, 1+k, 1+11*k, 1+111*k, ... This is the case for k=2.
Essentially the same as A091628 (cf. 2nd formula). - Georg Fischer, Oct 06 2018
a(n) is 3^n represented in bijective base-3 numeration. - Alois P. Heinz, Aug 26 2019

Wikipedia, Bijective numeration.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A001045, A002279, A047855, A062397, A091628, A093136.

a(n) = (2*10^n + 7)/9.
a(n) = 10*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 02 2010
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(7 + 2*exp(9*x))/9.
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = (A062397(n) - A002279(n))/2. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A115549 Numbers k such that the concatenation of k with 8*k gives a square.

Original entry on oeis.org

3, 12, 28, 63, 112, 278, 1112, 2778, 11112, 27778, 111112, 277778, 1111112, 2777778, 4938272, 7716050, 11111112, 12802888, 13151250, 13504288, 13862002, 14224392, 14591458, 14963200, 15339618, 15720712, 16106482, 16496928, 16892050, 17291848, 17696322, 18105472

Offset: 1

Giovanni Resta, Jan 25 2006

If k = 10*R_m + 2, with m >= 1, then the concatenation of k with 8*k equals (30*R_m + 6)^2, so A047855 \ {1,2} is a subsequence. - Bernard Schott, Apr 09 2022
Numbers k such that A009470(k) is a square. - Michel Marcus, Apr 09 2022
The numbers 28, 278, 2778, ..., 2*10^k + 7*(10^k - 1)/9 + 1, ..., k >= 1, are terms, because the concatenation forms the squares 28224 = 168^2, 2782224 = 1668^2, 277822224 = 16668^2, ..., (10^m + 2*(10^m - 1)/3 + 2)^2, m >= 2, ... - Marius A. Burtea, Apr 10 2022

			3_24 = 18^2.
11112_88896 = 33336^2.

Marius A. Burtea, Table of n, a(n) for n = 1..176

Cf. A002275, A047855, A009470, A102567, A106497, A115527-A115556.

[n:n in [1..20000000]|IsSquare(Seqint(Intseq(8*n) cat Intseq(n)))]; // Marius A. Burtea, Apr 10 2022

isok(k) = issquare(eval(Str(k, 8*k))); \\ Michel Marcus, Apr 09 2022

More terms from Marius A. Burtea, Apr 13 2022

A164898 First differences of numbers having only odd digits in their decimal representation.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 112, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 112, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12, 2, 2, 2, 2, 12

Offset: 1

Reinhard Zumkeller, Aug 30 2009

a(n) = A014261(n+1) - A014261(n);
all terms are of the form (10^k + 8)/9, see A047855;
a(n) > 2 iff A014261(n) mod 10 = 9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014261.

R[1]:= [1,3,5,7,9]:
for d from 2 to 3 do
  R[d]:= map(t -> seq(10*t+i,i=[1,3,5,7,9]),R[d-1])
od:
B:= map(op,[seq(R[d],d=1..3)]):
B[2..-1]-B[1..-2]; # Robert Israel, Jan 24 2022

Differences[Select[Range[1000],Count[IntegerDigits[#],?EvenQ]==0&]] (* _Harvey P. Dale, Jan 03 2019 *)

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)

cp's OEIS Frontend

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

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A061076 a(n) is the sum of the products of the digits of all the numbers from 1 to n.

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A093134 A Jacobsthal trisection.

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A337718 Numbers that can be written as (m + product of digits of m) for some m.

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Mathematica

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A098406 a(n) = (10^n + 17)/9.

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A378865 a(n) is the smallest positive integer k such that n*k uses none of the digits of n, or 0 if no such k exists.

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A093135 Expansion of g.f. (1-8*x)/((1-x)*(1-10*x)).

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A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).