A257850 -id:A257850 - cp's OEIS frontend

A368062 Numbers k such that k = A257850(k) + A257297(k).

0, 36, 655, 1258, 6208, 12508, 45715, 65455, 75385, 125008, 235297, 1250008, 2857144, 3214288, 4210528, 6545455, 6792453, 12500008, 34615386, 47058824, 87671233, 125000008, 654545455, 1250000008, 9529411765, 12500000008, 39130434783, 45714285715, 65454545455, 75384615385

Offset: 1

Nicolas Bělohoubek, Dec 10 2023

			     0 = 0*0   +   0*0;
    36 = 3*6   +   3*6;
   655 = 6*55  +  65*5;
  6208 = 6*208 + 620*8;
  ...

Kevin Ryde, Table of n, a(n) for n = 1..700
Nicolas Bělohoubek, C# program
Nicolas Bělohoubek, Subsequences of form A-(B)-C
Kevin Ryde, PARI/GP Code

Cf. A257850, A257297.

Select[Range[0,10^6], Part[digits=IntegerDigits[#],1]FromDigits[Drop[digits,1]] + FromDigits[Drop[digits,-1]]Part[digits,Length[digits]] == # &] (* Stefano Spezia, Dec 10 2023 *)

PARI
```
\\ See links.
```

def ok(n):
    if n < 10: return n == 0
    s = str(n)
    return n == int(s[0])*int(s[1:]) + (n%10)*(n//10)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Dec 10 2023

# faster for generating initial segment of sequence
from itertools import count, islice
def agen(): # generator of terms
    yield 0
    for digits in count(2):
        for first in range(1, 10):
            base = first*10**(digits-1)
            for rest in range(10**(digits-1)):
                n = base + rest
                if first*rest + (n%10)*(n//10) == n:
                    yield n
            print("...", digits, first, time()-time0, alst)
print(list(islice(agen(), 18))) # Michael S. Branicky, Dec 10 2023

a(24)-a(30) from Michael S. Branicky, Dec 10 2023

A142150 The nonnegative integers interleaved with 0's.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 0, 12, 0, 13, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18, 0, 19, 0, 20, 0, 21, 0, 22, 0, 23, 0, 24, 0, 25, 0, 26, 0, 27, 0, 28, 0, 29, 0, 30, 0, 31, 0, 32, 0, 33, 0, 34, 0, 35, 0, 36, 0, 37, 0, 38, 0, 39, 0, 40, 0, 41, 0, 42, 0, 43, 0

Offset: 0

Reinhard Zumkeller, Jul 15 2008

Number of vertical pairs in a wheel with n equal sections. - Wesley Ivan Hurt, Jan 22 2012
Number of even terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013
Also the result of writing n-1 in base 2 and multiplying the last digit with the number with its last digit removed. See A115273 and A257844-A257850 for generalization to other bases. - M. F. Hasler, May 10 2015
Also follows the rule: a(n+1) is the number of terms that are identical with a(n) for a(0..n-1). - Marc Morgenegg, Jul 08 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Reinhard Zumkeller, Logical Convolutions
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000004, A000007, A000027, A000217, A000326, A001057, A001477, A003817, A008805, A027656, A086099, A142149, A142151, A162610, A191967, A195034, A209297.

a142150 = uncurry (*) . (`divMod` 2) . (+ 1)
a142150_list = scanl (+) 0 a001057_list
-- Reinhard Zumkeller, Apr 02 2012

[n*(1+(-1)^n)/4 : n in [0..100]]; // Wesley Ivan Hurt, Aug 21 2014

&cat[[n, 0]: n in [0..50]]; // Vincenzo Librandi, Oct 31 2016

A142150:=n->n*(1+(-1)^n)/4: seq(A142150(n), n=0..100); # Wesley Ivan Hurt, Aug 21 2014

Table[Mod[Floor[n^2/2], n], {n, 200}] (* Enrique Pérez Herrero, Jul 29 2009 *)
Riffle[Range[0, 50], 0] (* Paolo Xausa, Feb 08 2024 *)

a(n)=!bittest(n,0)*n>>1 \\ M. F. Hasler, May 10 2015

def A142150(n): return (n+1>>1)*(n&1^1) # Chai Wah Wu, Jan 19 2023

a(n) = XOR{k AND (n-k): 0<=k<=n}.
a(n) = (n/2)*0^(n mod 2); a(2*n)=n and a(2*n+1)=0.
a(n) = floor(n^2/2) mod n. - Enrique Pérez Herrero, Jul 29 2009
a(n) = A027656(n-2). - Reinhard Zumkeller, Nov 05 2009
a(n) = Sum_{k=0..n} (k mod 2)*((n-k) mod 2). - Reinhard Zumkeller, Nov 05 2009
a(n+1) = A000217(n) mod A000027(n+1) = A000217(n) mod A001477(n+1). - Edgar Almeida Ribeiro (edgar.a.ribeiro(AT)gmail.com), May 19 2010
From Bruno Berselli, Oct 19 2010: (Start)
a(n) = n*(1+(-1)^n)/4.
G.f.: x^2/(1-x^2)^2.
a(n) = 2*a(n-2)-a(n-4) for n > 3.
Sum_{i=0..n} a(i) = (2*n*(n+1)+(2*n+1)*(-1)^n-1)/16 (see A008805). (End)
a(n) = -a(-n) = A195034(n-1)-A195034(-n-1). - Bruno Berselli, Oct 12 2011
a(n) = A000326(n) - A191967(n). - Reinhard Zumkeller, Jul 07 2012
a(n) = Sum_{i=1..n} floor((2*i-n)/2). - Wesley Ivan Hurt, Aug 21 2014
a(n-1) = floor(n/2)*(n mod 2), where (n mod 2) is the parity of n, or remainder of division by 2. - M. F. Hasler, May 10 2015
a(n) = A158416(n) - 1. - Filip Zaludek, Oct 30 2016
E.g.f.: x*sinh(x)/2. - Ilya Gutkovskiy, Oct 30 2016
a(n) = A000007(a(n-1)) + a(n-2) for n > 1. - Nicolas Bělohoubek, Oct 06 2024

A115273 a(n) = floor(n/3)*(n mod 3).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 0, 2, 4, 0, 3, 6, 0, 4, 8, 0, 5, 10, 0, 6, 12, 0, 7, 14, 0, 8, 16, 0, 9, 18, 0, 10, 20, 0, 11, 22, 0, 12, 24, 0, 13, 26, 0, 14, 28, 0, 15, 30, 0, 16, 32, 0, 17, 34, 0, 18, 36, 0, 19, 38, 0, 20, 40, 0, 21, 42, 0, 22, 44, 0, 23, 46, 0, 24, 48, 0, 25, 50, 0, 26, 52, 0, 27, 54, 0, 28, 56

Offset: 0

Zak Seidov, Jan 18 2006

Three arithmetic progressions interlaced: a(1)=1,2,0 and d=a(n+1)-a(n)=1,2,0. Cf. A115274(n) = n+a(n), n=1,2,3,...

Cf. A115274.

Cf. A142150 (the base 2 analog), A257844, ..., A257850.

[Floor(n/3)*(n mod 3): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Table[Floor[n/3]*Mod[n, 3], {n, 0, 86}] (* Extended to offset 0 by M. F. Hasler, May 11 2015 *)

a(n, b=3)=(n=divrem(n, b))[1]*n[2] \\ M. F. Hasler, May 10 2015

from math import prod
def A115273(n): return prod(divmod(n,3)) # Chai Wah Wu, Jan 19 2023

a(3*k+1) = k, a(3*k+2) = 2*k, a(3*k+3) = 0, k >= 1.

G.f.: x^4*(2*x+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, May 11 2015

a(0)-a(3) and cross-references added by M. F. Hasler, May 11 2015

A257844 a(n) = floor(n/4) * (n mod 4).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 0, 2, 4, 6, 0, 3, 6, 9, 0, 4, 8, 12, 0, 5, 10, 15, 0, 6, 12, 18, 0, 7, 14, 21, 0, 8, 16, 24, 0, 9, 18, 27, 0, 10, 20, 30, 0, 11, 22, 33, 0, 12, 24, 36, 0, 13, 26, 39, 0, 14, 28, 42, 0, 15, 30, 45, 0, 16, 32, 48, 0, 17, 34, 51, 0, 18, 36

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 4, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A142150 (the base-2 analog), A115273, A257845 - A257850.

[Floor(n/4)*(n mod 4) : n in [0..100]]; // Wesley Ivan Hurt, Jun 22 2015

I:=[0,0,0,0,0,1,2,3]; [n le 8 select I[n] else 2*Self(n-4)-Self(n-8): n in [1..100]]; // Vincenzo Librandi, Jun 23 2015

A257844:=n->floor(n/4)*(n mod 4): seq(A257844(n), n=0..100); # Wesley Ivan Hurt, Jun 22 2015

Table[Floor[n/4] Mod[n, 4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 22 2015 *)

PARI
```
a(n,b=4)=(n=divrem(n,b))[1]*n[2]
```

concat([0,0,0,0,0], Vec(x^5*(3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

def A257844(n): return (n>>2)*(n&3) # Chai Wah Wu, Jan 27 2023

a(n) = 2*a(n-4) - a(n-8), n > 8. - Colin Barker, May 11 2015
G.f.: x^5*(3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, May 11 2015
a(n) = (3 - 2*(-1)^((2*n - 1 + (-1)^n)/4) - (-1)^n)*(2*n - 3 + 2*(-1)^((2*n - 1 + (-1)^n)/4) + (-1)^n)/16. - Wesley Ivan Hurt, Jun 22 2015

A088117 Let the decimal expansion of n be abcd...; then a(n) = (abcd... + bacd... + cabd... + dabc... + ...) + (abcd... + bcad... + cdab... + ...) + ... . That is, a(n) = sum over all the digit strings of the product (number obtained by deleting a digit string) (deleted digit string).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 11

Offset: 0

Amarnath Murthy, Sep 25 2003

Each substring is used in only product.

a(n) = 0 for 0 <= n <= 10.

			a(1234) = (1*234 + 2*134 + 3*124 + 4*123) + (12*34 + 23*14) = 2096.
a(12345) = (1*2345 + 2*1345 + 3*1245 + 4*1235 + 5*1234) + (12*345 + 15*234 + 23*145 + 34*125 + 45*123) = 40650.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A007954, A088116.

Different from A035930, A171765, A257850.

a:= n-> (s-> add(add(parse(s[i..j])*parse(cat(s[1..i-1],
    s[j+1..length(s)])), i=1..j), j=1..length(s)-1))(""||n):
seq(a(n), n=0..120);  # Alois P. Heinz, May 22 2021

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007

a(0)=0 inserted and examples corrected by Alois P. Heinz, May 22 2021

A257297 a(n) = (initial digit of n) * (n with initial digit removed).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 1, 2, 3

Offset: 0

M. F. Hasler, May 10 2015

The initial 100 terms match those of A035930 (maximal product of any two numbers whose concatenation is n) and also those of A171765 (product of digits of n, or 0 for n<10), and except for initial terms, also A007954 (product of decimal digits of n) and A115300 (greatest digit of n * least digit of n).
Iterations of this map always end in 0, since a(n) < n. Sequence A257299 lists numbers for which these iterations reach 0 in exactly 9 steps, with the additional constraint of having each time a different initial digit.
If "initial" is replaced by "last" in the definition (A257850), then we get the same values up to a(100), but (10, 20, 30, ...) for n = 101, 102, 103, ..., again different from each of the 4 other sequences mentioned in the first comment. - M. F. Hasler, Sep 01 2021

			For n<10, a(n) = n*0 = 0, since removing the initial and only digit leaves nothing, i.e., zero (by convention).
a(10) = 1*0 = 0, a(12) = 1*2 = 2, ..., a(20) = 2*0 = 0, a(21) = 2*1 = 2, a(22) = 2*2 = 4, ...
a(99) = 9*9 = 81, a(100) = 1*00 = 0, a(101) = 1*01 = 1, ..., a(123) = 1*23, ...

Cf. A257850, A007954, A035930, A080464, A115300, A169669, A111707, A088117, A187844.

a:= n-> `if`(n<10, 0, (s-> parse(s[1])*parse(s[2..-1]))(""||n)):
seq(a(n), n=0..120);  # Alois P. Heinz, Feb 12 2024

Table[Times@@FromDigits/@TakeDrop[IntegerDigits@n,1],{n,0,103}] (* Giorgos Kalogeropoulos, Sep 03 2021 *)

apply( {A257297(n)=vecprod(divrem(n,10^logint(n+!n,10)))}, [0..111]) \\ Edited by M. F. Hasler, Sep 01 2021

def a(n): s = str(n); return 0 if len(s) < 2 else int(s[0])*int(s[1:])
print([a(n) for n in range(104)]) # Michael S. Branicky, Sep 01 2021

For 1 <= m <= 9 and n < 10^k, a(m*10^k + n) = m*n.

a(101..103) corrected by M. F. Hasler, Sep 01 2021

A330633 The concatenation of the products of every pair of consecutive digits of n (with a(n) = 0 for 0 <= n <= 9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0

Offset: 0

Scott R. Shannon, Dec 21 2019

If the decimal expansion of n is d_1 d_2 ... d_k then a(n) is the number formed by concatenating the decimal numbers d_1*d_2, d_2*d_3, ..., d_{k-1}*d_k.

Due to the fact that for two digit numbers the sequence is simply the multiplication of those two numbers, this sequence matches numerous others for the first 100 terms. See the sequences in the cross references. The terms begin to differ beyond n = 100.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A007954, A088117, A040115, A053392, A035930, A257850, A080464, A171765, A169669, A257297.

read("transforms") :
A330633 := proc(n)
    local dgs,L,i ;
    if n <=9 then
        0;
    else
        dgs := ListTools[Reverse](convert(n,base,10)) ;
        L := [] ;
        for i from 2 to nops(dgs) do
            L := [op(L), op(i-1,dgs)*op(i,dgs)] ;
        end do:
        digcatL(L) ;
    end if;
end proc: # R. J. Mathar, Jan 11 2020

Array[If[Or[# == 0, IntegerLength@ # == 1], 0, FromDigits[Join @@ IntegerDigits[Times @@ # & /@ Partition[IntegerDigits@ #, 2, 1]]]] &, 81, 0] (* Michael De Vlieger, Dec 23 2019 *)

a(n) = my(d=digits(n), s="0"); for (k=1, #d-1, s=concat(s, d[k]*d[k+1])); eval(s); \\ Michel Marcus, Apr 28 2020

a(10) = 0 as 1 * 0 = 0.
a(29) = 18 as 2 * 9 = 18.
a(100) = 0 as 1 * 0 = 0 and 0 = 0 = 0, and '00' is reduced to 0.
a(110) = 10 as 1 * 1 = 1 and 1 * 0 = 0. This is the first term that differs from A007954 and A171765, the multiplication of all digits of n.

A257845 a(n) = floor(n/5) * (n mod 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 0, 3, 6, 9, 12, 0, 4, 8, 12, 16, 0, 5, 10, 15, 20, 0, 6, 12, 18, 24, 0, 7, 14, 21, 28, 0, 8, 16, 24, 32, 0, 9, 18, 27, 36, 0, 10, 20, 30, 40, 0, 11, 22, 33, 44, 0, 12, 24, 36, 48, 0, 13, 26, 39, 52, 0, 14, 28, 42, 56

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 5, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{0,0,0,0,0,0,1,2,3,4},80] (* Harvey P. Dale, Aug 15 2021 *)

PARI
```
a(n,b=5)=(n=divrem(n,b))[1]*n[2]
```

concat([0,0,0,0,0,0], Vec(x^6*(4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

a(n) = 2*a(n-5)-a(n-10). - Colin Barker, May 11 2015

G.f.: x^6*(4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, May 11 2015

A257849 a(n) = floor(n/9) * (n mod 9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 2, 4, 6, 8, 10, 12, 14, 16, 0, 3, 6, 9, 12, 15, 18, 21, 24, 0, 4, 8, 12, 16, 20, 24, 28, 32, 0, 5, 10, 15, 20, 25, 30, 35, 40, 0, 6, 12, 18, 24, 30, 36, 42, 48, 0, 7, 14, 21, 28, 35, 42, 49, 56, 0, 8, 16, 24, 32, 40, 48, 56, 64, 0

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 9, multiply the last digit by the number with its last digit removed.

See A142150(n-1) for the base 2 analog, and A115273, A257844 - A257850 for the base 3 - base 10 variants.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

[Floor(n/9)*(n mod 9): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Table[Floor[n/9] Mod[n, 9], {n, 100}] (* Vincenzo Librandi, May 11 2015 *)

PARI
```
A257849(n)=n\9*(n%9)
```

concat([0,0,0,0,0,0,0,0,0,0], Vec(x^10*(8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2*(x^6+x^3+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

from math import prod
def A257849(n): return prod(divmod(n,9)) # Chai Wah Wu, Jan 19 2023

[floor(n/9)*(n % 9)  for n in (0..80)]; # Bruno Berselli, May 11 2015

a(n) = 2*a(n-9)-a(n-18). - Colin Barker, May 11 2015

G.f.: x^10*(8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2*(x^6+x^3+1)^2). - Colin Barker, May 11 2015

A257846 a(n) = floor(n/6) * (n mod 6).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 0, 2, 4, 6, 8, 10, 0, 3, 6, 9, 12, 15, 0, 4, 8, 12, 16, 20, 0, 5, 10, 15, 20, 25, 0, 6, 12, 18, 24, 30, 0, 7, 14, 21, 28, 35, 0, 8, 16, 24, 32, 40, 0, 9, 18, 27, 36, 45, 0, 10, 20, 30, 40, 50, 0, 11, 22, 33, 44, 55, 0, 12, 24

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 6, multiply the last digit by the number with its last digit removed.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257850.

Table[Floor[n/6]*Mod[n, 6], {n, 120}] (* Michael De Vlieger, May 11 2015 *)

PARI
```
a(n,b=6)=(n=divrem(n,b))[1]*n[2]
```

concat([0, 0, 0, 0, 0, 0, 0], Vec(x^7*(5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, May 11 2015

a(n) = 2*a(n-6)-a(n-12). - Colin Barker, May 11 2015

G.f.: x^7*(5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2). - Colin Barker, May 11 2015

cp's OEIS Frontend

A368062 Numbers k such that k = A257850(k) + A257297(k).

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A142150 The nonnegative integers interleaved with 0's.

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A115273 a(n) = floor(n/3)*(n mod 3).

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A257844 a(n) = floor(n/4) * (n mod 4).

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A088117 Let the decimal expansion of n be abcd...; then a(n) = (a*bcd... + b*acd... + c*abd... + d*abc... + ...) + (ab*cd... + bc*ad... + cd*ab... + ...) + ... . That is, a(n) = sum over all the digit strings of the product (number obtained by deleting a digit string) * (deleted digit string).

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A257297 a(n) = (initial digit of n) * (n with initial digit removed).

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Programs

Maple

Mathematica

A088117 Let the decimal expansion of n be abcd...; then a(n) = (abcd... + bacd... + cabd... + dabc... + ...) + (abcd... + bcad... + cdab... + ...) + ... . That is, a(n) = sum over all the digit strings of the product (number obtained by deleting a digit string) (deleted digit string).