A047855 -id:A047855 - cp's OEIS frontend

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

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)

A196793 a(n) = A047848(n, n).

Original entry on oeis.org

1, 2, 7, 44, 401, 4682, 66431, 1111112, 21435889, 469070942, 11488207655, 311505013052, 9267595563617, 300239975158034, 10523614159962559, 396861212733968144, 16024522975978953761, 689852631578947368422, 31544039619835776489479

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300

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

Magma
```
[((n+3)^n+n+1)/(n+2): n in [0..20]]
```

A196793:=n->((n+3)^n+n+1)/(n+2); seq(A196793(k), k=0..50); # Wesley Ivan Hurt, Nov 08 2013

Table[((n+3)^n+n+1)/(n+2), {n,0,50}] (* Wesley Ivan Hurt, Nov 08 2013 *)

def A196793(n): return (pow(n+3, n) +n+1)//(n+2)
print([A196793(n) for n in range(51)]) # G. C. Greubel, Jan 17 2025

a(n) = ((n+3)^n + n + 1)/(n+2).

A305753 A base-3/2 sorted Fibonacci sequence that starts with a(0) = 0 and a(1) = 1. The terms are interpreted as numbers written in base 3/2. To get a(n+2), add a(n) and a(n+1), write the result in base 3/2 and sort the "digits" into increasing order, omitting all zeros.

Original entry on oeis.org

0, 1, 1, 2, 2, 12, 12, 112, 112, 1112, 1112, 11112, 11112, 111112, 111112, 1111112, 1111112, 11111112, 11111112, 111111112, 111111112, 1111111112, 1111111112, 11111111112, 11111111112, 111111111112, 111111111112, 1111111111112, 1111111111112, 11111111111112, 11111111111112

Offset: 0

Tanya Khovanova and PRIMES STEP Senior group, Jun 09 2018

In base 10, the corresponding sequence is A069638 and is periodic.

			Write decimal numbers as x_10, base-3/2 numbers as x_b (see A024629).
We have a(1) = 1, a(2) = 2 (in both bases).
Adding, we get 1+2 = 3_10 = 20_b, and sorting the digits gives a(3) = 2_b = 2_10.
Adding 2 and 2 we get 4_10 = 21_b, and sorting the digits gives a(4) = 12_b = (7/2)_10.
Adding 2 and 7/2 we get (11/2)_10 = 201_b, and sorting the digits gives a(5) = 12_b = (7/2)_10.
Adding (7/2)_10 and (7/2)_10 we get 7_10 = 211_b, and sorting the digits gives a(6) = 112_b = (23/4)_10.
Adding (7/2)_10 and (23/4)_10 we get (37/4)_10 = 2011_b, and sorting the digits gives a(7) = 112_b = (23/4)_10.
And so on.

Colin Barker, Table of n, a(n) for n = 0..1000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A000045, A024629, A069638, A237575, A305880.

This is A047855 with terms repeated. - N. J. A. Sloane, Jun 19 2018

concat(0, Vec(x*(1 - 3*x)*(1 + 3*x) / ((1 - x)*(1 - 10*x^2)) + O(x^40))) \\ Colin Barker, Jun 19 2018

From Colin Barker, Jun 14 2018: (Start)
Generating function: x*(1 - 3*x)*(1 + 3*x) / ((1 - x)*(1 - 10*x^2)).
a(n) = (10^(n/2) + 80) / 90 for n>0.
a(n) = (10^((n-1)/2) + 8) / 9 for n>0.
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>4.
(End)

Edited by N. J. A. Sloane, Jun 22 2018

A309908 a(n) is 2^n represented in bijective base-9 numeration.

Original entry on oeis.org

1, 2, 4, 8, 17, 35, 71, 152, 314, 628, 1357, 2725, 5551, 12212, 24424, 48848, 98797, 218715, 438531, 878162, 1867334, 3845668, 7792447, 16694895, 34499911, 69121922, 149243944, 299487988, 619987187, 1342185385, 2684381781, 5478773672, 11968657454, 24148425918

Offset: 0

Alois P. Heinz, Aug 21 2019

Differs from A001357 first at n = 16: a(16) = 98797 < 108807 = A001357(16).

			a(10) =  1357_bij9 =       9*(9*(9*1+3)+5)+7 =  1024 = 2^10.
a(16) = 98797_bij9 = 9*(9*(9*(9*9+8)+7)+9)+7 = 65536 = 2^16.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995, 27-29.
Eric Weisstein's World of Mathematics, Zerofree
Wikipedia, Bijective numeration

Cf. A000079, A001357, A047855, A052382, A093135, A214676, A325203.

b:= proc(n) local d, l, m; m:= n; l:= "";
      while m>0 do d:= irem(m, 9, 'm');
        if d=0 then d:=9; m:= m-1 fi; l:= d, l
      od; parse(cat(l))
    end:
a:= n-> b(2^n):
seq(a(n), n=0..33);

a(n) = A052382(2^n) = A052382(A000079(n)).

A073557 Number of Fibonacci numbers F(k), k <= 10^n, whose initial digit is 1.

Original entry on oeis.org

3, 30, 301, 3011, 30103, 301031, 3010300, 30103001, 301029995, 3010299957, 30102999568

Offset: 1

Shyam Sunder Gupta, Aug 15 2002

			a(2) = 30 because there are 30 Fibonacci numbers up to 10^2 whose initial digit is 1.

Cf. A000045, A047855 (numbers of integers <= 10^n, whose initial digit is 1).

default(realprecision, 10^4); m=log((1+sqrt(5))/2);
lista(nn) = {my(d=log(10)/m, r=log(sqrt(5))/m, s=log(5-sqrt(5))/m, t=0, u=1); for(n=1, nn, u=10*u; while(sJinyuan Wang, Feb 21 2020

Limit_{n->infinity} a(n)/10^n = log(2), where the base is 10. - Robert Gerbicz, Sep 05 2002

More terms from Robert Gerbicz, Sep 05 2002
a(9)-a(10) from Jinyuan Wang, Feb 21 2020
a(11) from Sean A. Irvine, Dec 04 2024

A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 4, 2, 2, 0, 8, 4, 2, 2, 0, 16, 8, 4, 2, 2, 0, 32, 16, 8, 4, 2, 2, 0, 64, 32, 16, 8, 4, 2, 2, 0, 128, 64, 32, 16, 8, 4, 2, 2, 0, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 2048, 1024

Offset: 0

Philippe Deléham, Oct 05 2009

			Triangle begins :
1,
0,2,
0,2,2,
0,4,2,2,
0,8,4,2,2,
0,16,8,4,2,2,
0,32,16,8,4,2,2,
0,64,32,16,8,4,2,2,
0,128,64,32,16,8,4,2,2,
0,256,128,64,32,16,8,4,2,2,
0,512,256,128,64,32,16,8,4,2,2,

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A084247(n), A000007(n), A000079(n), A001787(n+1), A166060(n), A165665(n), A083585(n) for x= -1, 0, 1, 2, 3, 4, 5 respectively. Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A040000(n), A000079(n), A095121(n), A047851(n), A047853(n), A047855(n) for x = 0, 1, 2, 3, 4, 5 respectively.

G.f.: (1-2*x+x*y)/((-1+2*x)*(x*y-1)). - R. J. Mathar, Aug 11 2015

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Philippe Deléham, Oct 07 2009

			Triangle begins :
1 ;
0,2 ;
0,1,2 ;
0,1,1,2 ;
0,1,1,1,2 ;
0,1,1,1,1,2 ;
0,1,1,1,1,1,2 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A166122(n), A166114(n), A084222(n), A084247(n), A000034(n), A040000(n), A000027(n+1), A000079(n), A007051(n), A047849(n), A047850(n), A047851(n), A047852(n), A047853(n), A047854(n), A047855(n), A047856(n) for x= -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^k= A000007(n), A000027(n+1), A033484(n), A134931(n), A083597(n) for x= 0,1,2,3,4 respectively.
T(n,k)= A166065(n,k)/2^(n-k).
G.f.: (1-x+x*y)/(1-x-x*y+x^2*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

A176088 Table T(n,k) = ceiling(10^n/(10^k-1)), n >= 0, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 12, 1, 1, 112, 2, 1, 1, 1112, 11, 1, 1, 1, 11112, 102, 2, 1, 1, 1, 111112, 1011, 11, 1, 1, 1, 1, 1111112, 10102, 101, 2, 1, 1, 1, 1, 11111112, 101011, 1002, 11, 1, 1, 1, 1, 1, 111111112, 1010102, 10011, 101, 2, 1, 1, 1, 1, 1, 1111111112, 10101011, 100101

Offset: 0

Rick L. Shepherd, Apr 10 2010

For n+1 >= k, minimal number of k-digit base 10 numbers totaling an (n+1)-digit sum.
Column 1 of the table, T(n,1) = 1, 2, 12, 112, 1112, ..., is A047855.
T(n,k) = 1 for k >= n+1.
T(i,i) = 2 for i > 0. Generally, for all m >= 1 and i >= 0, T(2m-1+i,m+i) = 10^(m-1) + 1.

			The table begins:
.........1,........1,.......1,......1,.....1,....1,...1,..1,.1,1,1,1,...
.........2,........1,.......1,......1,.....1,....1,...1,..1,.1,1,1,1,...
........12,........2,.......1,......1,.....1,....1,...1,..1,.1,1,1,1,...
.......112,.......11,.......2,......1,.....1,....1,...1,..1,.1,1,1,1,...
......1112,......102,......11,......2,.....1,....1,...1,..1,.1,1,1,1,...
.....11112,.....1011,.....101,.....11,.....2,....1,...1,..1,.1,1,1,1,...
....111112,....10102,....1002,....101,....11,....2,...1,..1,.1,1,1,1,...
...1111112,...101011,...10011,...1001,...101,...11,...2,..1,.1,1,1,1,...
..11111112,..1010102,..100101,..10002,..1001,..101,..11,..2,.1,1,1,1,...
.111111112,.10101011,.1001002,.100011,.10001,.1001,.101,.11,.2,1,1,1,...
1111111112,101010102,10010011,1000101,100002,10001,1001,101,11,2,1,1,...
...

Cf. A047855.

T(n,k) = if(n>=0 && k>=1, ceil(10^n/(10^k-1)))

A308945 Number of totient numbers, phi(k), k <= 10^n, whose initial digit is 1.

Original entry on oeis.org

2, 20, 213, 2152, 21594, 216009, 2159776, 21595522, 215951111, 2159507603, 21595061256, 215950604593

Offset: 1

Frank M Jackson, Jul 02 2019

The probability that a totient number starts with an initial 1 does not obey Benford's law however it does appear to tend to a constant value. In a sample of 10^9 totient numbers the distribution of initial digits 1 - 9 is approx. 21.595%, 20.774%, 16.457%, 12.682%, 7.904%, 6.633%, 5.505%, 4.634%, 3.816%.

			a(1)=2 as the first 10 totient numbers are {1, 1, 2, 2, 4, 2, 6, 4, 6, 4} and the occurrence of numbers with an initial 1 is 2.

Wikipedia, Benford's law

Cf. A000010, A047855, A073517, A073557.

lst1={}; Do[lst=Table[0, {n, 1, 9}]; Do[++lst[[First@IntegerDigits@EulerPhi[n]]], {n, 1, 10^m}]; AppendTo[lst1, lst[[1]]], {m, 1, 7}]; lst1

a(n) = {k=0; for(j=1, 10^n, if(digits(eulerphi(j))[1]==1, k++)); k} \\ Jinyuan Wang, Jul 04 2019

a(10)-a(12) from Giovanni Resta, Jul 04 2019

cp's OEIS Frontend

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

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A196793 a(n) = A047848(n, n).

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A305753 A base-3/2 sorted Fibonacci sequence that starts with a(0) = 0 and a(1) = 1. The terms are interpreted as numbers written in base 3/2. To get a(n+2), add a(n) and a(n+1), write the result in base 3/2 and sort the "digits" into increasing order, omitting all zeros.

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Extensions

A309908 a(n) is 2^n represented in bijective base-9 numeration.

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Examples

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Maple

Formula

A073557 Number of Fibonacci numbers F(k), k <= 10^n, whose initial digit is 1.

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Examples

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PARI

Formula

Extensions

A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A176088 Table T(n,k) = ceiling(10^n/(10^k-1)), n >= 0, k >= 1, read by antidiagonals.

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A308945 Number of totient numbers, phi(k), k <= 10^n, whose initial digit is 1.

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