A081594 -id:A081594 - cp's OEIS frontend

A028897 If n = Sum c_i 10^i then a(n) = Sum c_i 2^i.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 4

Offset: 0

N. J. A. Sloane

For n<100, this is the same result as "If n = Sum c_i 10^i then a(n) = Sum c_i (i+1)". - Henry Bottomley, Apr 20 2001
n_2 in the notation of A122618.
Left inverse of A007088 (binary numbers), cf. formula from Karttunen. - M. F. Hasler, Jun 13 2023

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Robin C. Yu, Decibinary Numbers, on Hackerrank.com.
Index entries for sequences related to decimal expansion of n

Cf. A007088, A122618.
Differs from A081594 and A244158 for the first time at n = 100, which here is a(100) = 4.
Cf. A249873, A072170.
See A322000 for integers ordered according to the value of a(n).

a028897 0 = 0
a028897 n = 2 * a028897 n' + d where (n', d) = divMod n 10
-- Reinhard Zumkeller, Nov 06 2014

a[n_ /; n < 10] := n; a[n_] := a[n] = If[Mod[n, 10] != 0, a[n-1] + 1, 2*a[n/10]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 02 2016 *)

a(n)=if(n<1,0,if(n%10,a(n-1)+1,2*a(n/10)))

A028897(n)=fromdigits(digits(n),2) \\ M. F. Hasler, Feb 14 2019
(MIT/GNU Scheme) (define (A028897 n) (let loop ((z 0) (i 0) (n n)) (if (zero? n) z (loop (+ z (* (modulo n 10) (expt 2 i))) (1+ i) (floor->exact (/ n 10)))))) ;; Antti Karttunen, Jun 22 2014

a(n) = 2*a(floor(n/10)) + (n mod 10). - Henry Bottomley, Apr 20 2001
a(0) = 0, a(n) = 2*a(n/10) if n == 0 (mod 10), a(n) = a(n-1)+1 otherwise. - Benoit Cloitre, Dec 21 2002
For all n, a(A007088(n)) = n. - Antti Karttunen, Jun 22 2014

More terms from Erich Friedman.

Terms up to n = 100 added by Antti Karttunen, Jun 22 2014

A081582 Pascal-(1,7,1) array.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 97, 25, 1, 1, 33, 241, 241, 33, 1, 1, 41, 449, 1161, 449, 41, 1, 1, 49, 721, 3297, 3297, 721, 49, 1, 1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1, 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1, 1, 73, 1921, 22121, 108353, 192969, 108353, 22121, 1921, 73, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A017077, A081593, A081594. Coefficients of the row polynomials in the Newton basis are given by A013614.

			Rows begin
  1,  1,   1,    1,     1, ... A000012;
  1,  9,  17,   25,    33, ... A017077;
  1, 17,  97,  241,   449, ... A081593;
  1, 25, 241, 1161,  3297, ...
  1, 33, 449, 3297, 14721, ...
Triangle begins:
  1;
  1,  1;
  1,  9,    1;
  1, 17,   17,    1;
  1, 25,   97,   25,     1;
  1, 33,  241,  241,    33,    1;
  1, 41,  449, 1161,   449,   41,    1;
  1, 49,  721, 3297,  3297,  721,   49,  1;
  1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1;

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A143683 (m = 8).

Cf. A015519, A017077, A081593.

A081582:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081582(n,k,7): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[ Hypergeometric2F1[-k, k-n, 1, 8], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 8).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

T(n,k) = Sum_{j = 0..n-k} binomial(n-k,j)*binomial(k,j)*8^j.
Riordan array (1/(1 - x), x*(1 + 7*x)/(1 - x)).
Square array T(n, k) defined by T(n, 0) = T(0, k)=1, T(n, k) = T(n, k-1) + 7*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1 + 7*x)^k/(1 - x)^(k+1).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 8). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(8*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 16*x + 64*x^2/2) = 1 + 17*x + 97*x^2/2! + 241*x^3/3! + 449*x^4/4! + 721*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n, k) = A015519(n+1). - G. C. Greubel, May 26 2021

A081502 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 3x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 21, 22, 23, 24, 25, 26, 27, 28, 29

Offset: 0

N. J. A. Sloane, Apr 22 2003

Eswaran observes that n is divisible by 7 iff repeated application of a ends at the number 7.

a(n) is divisible by 7 iff n is divisible by 7: e.g., a(7) = a(14) = a(21) = 7, a(28) = a(35) = a(42) = 14 etc. - Zak Seidov, Mar 19 2014

R. Eswaran, Test of divisibility of the number 7, Abstracts Amer. Math. Soc., 23 (No. 2, 2002), #974-00-5, p. 275.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Different from A028898 for n>=100 (e.g. a(111) = 34, A029989(111) = 13).

Cf. A081503, A081594, A081595, A081596, A081597, A081598, A081599, A081600.

A081502 := proc(n)
    local x,y ;
    y := modp(n,10) ;
    x := iquo(n,10) ;
    3*x+y ;
end proc:
seq(A081502(n),n=0..120) ; # R. J. Mathar, Oct 03 2014

Table[n - 7 * Floor[n / 10], {n, 0, 100}] (* Joshua Oliver, Dec 04 2019 *)

a(n) = 3*(n\10) + (n % 10); \\ Michel Marcus, Mar 19 2014

a(n) = [3,1]*divrem(n,10); \\ Kevin Ryde, Dec 04 2019

G.f.: -x*(6*x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1) / (x^11-x^10-x+1). - Colin Barker, Mar 19 2014

a(n) = n-7*floor(n/10). - Wesley Ivan Hurt, May 12 2016

A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 5

Offset: 0

Antti Karttunen, Jun 22 2014

This sequence converts any number from various "Catalan Base number systems" (when represented as decimal numbers) back to the integer the numeral represents: e.g. we have a(A014418(n)) = n and a(A244159(n)) = n (except for the latter this is eventually broken by the shortcomings of the decimal representation used, while for the former it works for all n, because no digits larger than 3 will ever appear in the terms of A014418).
A197433 is similar, but replaces 2^k with A000108(k+1) in binary expansion of n.
For 1- and 2-digit numbers the same as A156230. - R. J. Mathar, Jun 27 2014

Index entries for sequences related to decimal expansion of n

Cf. A000108, A014418, A244159, A197433, A244155, A244156, A244157.

Differs from A028897 and A081594 for the first time at n=100, which here is a(100) = 5.

A244158 := proc(n)
    local dgs,k ;
    dgs := convert(n,base,10) ;
    add( op(k,dgs)*A000108(k),k=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jan 31 2015

A081593 Third row of Pascal-(1,7,1) array A081582.

Original entry on oeis.org

1, 17, 97, 241, 449, 721, 1057, 1457, 1921, 2449, 3041, 3697, 4417, 5201, 6049, 6961, 7937, 8977, 10081, 11249, 12481, 13777, 15137, 16561, 18049, 19601, 21217, 22897, 24641, 26449, 28321, 30257, 32257, 34321, 36449, 38641, 40897, 43217, 45601, 48049, 50561, 53137

Offset: 0

Paul Barry, Mar 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017077, A081582, A081594.

[1-16*n+32*n^2: n in [0..40]]; // Vincenzo Librandi, Aug 09 2013

CoefficientList[Series[(1 + 7 x)^2 / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{3,-3,1},{1,17,97},50] (* Harvey P. Dale, Aug 30 2025 *)

a(n)=32*n^2-16*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 1 - 16*n + 32*n^2.
G.f.: (1+7*x)^2/(1-x)^3.
From Elmo R. Oliveira, Jun 09 2025: (Start)
E.g.f.: exp(x)*(1 + 16*x + 32*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

A156230 Sum of the products of the digits of n and the positions of the digits m in n starting from the last digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 3

Offset: 1

Cino Hilliard, Feb 06 2009

Starts to differ from A081594 when n>=100. [From R. J. Mathar, Feb 19 2009]

			a(19) = 9*1 + 1*2 = 11.

A156230 := proc(n)
    local dgs;
    dgs := convert(n,base,10) ;
    add(i*op(i,dgs),i=1..nops(dgs)) ;
end proc:
seq(A156230(n),n=1..100) ; # R. J. Mathar, Dec 02 2018

pdn[n_]:=Module[{idn=IntegerDigits[n]},Total[idn Range[Length[idn],1,-1]]]; Array[pdn,80] (* Harvey P. Dale, Aug 22 2012 *)

gr(n) = v=Vec((rev(n)));sum(x=1,length(v),x*eval(v[x]))
gr1(n) = for(j=1,n,print1(gr(j)","))
rev(str) = /* Get the reverse of the input string
*/ {
local(tmp,s,j);
tmp = Vec(Str(str));
s="";
forstep(j=length(tmp),1,-1,
s=concat(s,tmp[j]));
return(s)
}

Let n = d(1)d(2)...d(m) where d(1),d(2),...,d(m) are the digits of n. Then a(n) = m*d1+(m-1)*d2+...+d(m).

Changed the description, formula and Pari code Cino Hilliard, Feb 08 2009

More terms to separate from A322001. - R. J. Mathar, Dec 02 2018

A322001 Digits of n interpreted in factorial base: a(Sum d_k10^k) = Sum d_kk!

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6

Offset: 0

M. F. Hasler, Nov 27 2018

More terms than usual are given to distinguish the sequence from A081594, A028897 and A244158, which agree up to a(99). The last two correspond to k! replaced by 2^k resp. Catalan(k).

This is a left inverse to A007623 (factorial base representation of n): A322001(A007623(n)) = n for all n >= 0. One could imagine variants which have a(n) = 0 or a(n) = -1 if n is not a term of A007623. Restricted to the range of A007623, it is also a right inverse to A007623, at least up to the 10 digit terms, beyond which A007623 becomes non-injective.

Cf. A007623 (right inverse), A081594, A028897, A244158.

a[n_] := Module[{d=Reverse@IntegerDigits[n]}, Sum[d[[i]]*i!, {i,1,Length[d]}]]; Array[a, 100, 0] (* Amiram Eldar, Nov 28 2018 *)

A322001(n)=sum(i=1,#n=Vecrev(digits(n)),n[i]*i!) \\ M. F. Hasler, Nov 27 2018

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A028897 If n = Sum c_i 10^i then a(n) = Sum c_i 2^i.

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A081582 Pascal-(1,7,1) array.

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A081502 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 3x+y.

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A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

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A081593 Third row of Pascal-(1,7,1) array A081582.

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A156230 Sum of the products of the digits of n and the positions of the digits m in n starting from the last digit.

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A322001 Digits of n interpreted in factorial base: a(Sum d_k*10^k) = Sum d_k*k!

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A322001 Digits of n interpreted in factorial base: a(Sum d_k10^k) = Sum d_kk!