A093134 -id:A093134 - cp's OEIS frontend

A140690 A positive integer n is included if n written in binary can be subdivided into a number of runs all of equal-length, the first run from the left consisting of all 1's, the next run consisting of all 0's, the next run consisting of all 1's, the next run consisting of all 0's, etc.

1, 2, 3, 5, 7, 10, 12, 15, 21, 31, 42, 51, 56, 63, 85, 127, 170, 204, 240, 255, 341, 455, 511, 682, 819, 992, 1023, 1365, 2047, 2730, 3276, 3640, 3855, 4032, 4095, 5461, 8191, 10922, 13107, 16256, 16383, 21845, 29127, 31775, 32767, 43690, 52428, 61680

Offset: 1

Leroy Quet, Jul 11 2008

Also: numbers of the form (2^s-1)*[4^{s*(k+1)}-1]/(4^s-1) or 2^s(2^s-1)*[4^{s*(k+1)}-1]/(4^s-1), s>=1, k>=0. Subsequences are, with the possible exception of terms at n=0, A002450(n), A043291(n), A015565(2n), A093134(2n+1), A000225(n), A020522(n). [R. J. Mathar, Aug 04 2008]
From Emeric Deutsch, Jan 25 2018: (Start)
Also the indices of the compositions having equal parts.
We define the index of a composition to be the positive integer whose binary form has run-lengths (i.e. runs of 1's, runs of 0's, etc., from left to right) equal to the parts of the composition. Example: the composition [1,1,3,1] has index 46 since the binary form of 46 is 101110. The integer 992 is in the sequence since its binary form is 1111100000 and the composition [5,5] has equal parts. The integer 100 is not in the sequence since its binary form is 1100100 and the composition [2,2,1,2] does not have equal parts.
The command c(n) from the Maple program yields the composition having index n. (End)

			819 in binary is 1100110011. The runs of 0's and 1's are (11)(00)(11)(00)(11). Each run (alternating 1's and 0's) is the same length. So 819 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A101211, A298644.

import Data.Set (singleton, deleteFindMin, insert)
a140690 n = a140690_list !! (n-1)
a140690_list = f $ singleton (1, 1, 2) where
   f s | k == 1 = m : f (insert (2*b-1, 1, 2*b) $ insert (b*m, k+1, b) s')
       | even k    = m : f (insert (b*m+b-1, k+1, b) s')
       | otherwise = m : f (insert (b*m, k+1, b) s')
       where ((m, k, b), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 21 2014

Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 62000 do if nops(convert(c(n), set)) = 1 then A := `union`(A, {n}) else  end if end do: A; # most of the Maple program is due to W. Edwin Clark. - Emeric Deutsch, Jan 25 2018

Select[Range[62000],Length[Union[Length/@Split[IntegerDigits[#,2]]]]==1&] (* Harvey P. Dale, Mar 22 2012 *)

Terms beyond 42 from R. J. Mathar, Aug 04 2008

A015585 a(n) = 9a(n-1) + 10a(n-2).

Original entry on oeis.org

0, 1, 9, 91, 909, 9091, 90909, 909091, 9090909, 90909091, 909090909, 9090909091, 90909090909, 909090909091, 9090909090909, 90909090909091, 909090909090909, 9090909090909091, 90909090909090909, 909090909090909091, 9090909090909090909, 90909090909090909091

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_11. Example: a(2)=9 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJK are: ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB and AKB. - Emeric Deutsch, Apr 01 2004
Beginning with n=1 and a(1)=1, these are the positive integers whose balanced base-10 representations (A097150) are the first n digits of 1,-1,1,-1,.... Also, a(n) = (-1)^(n-1)*A014992(n) = |A014992(n)| for n >= 1. - Rick L. Shepherd, Jul 30 2004
General form: k=10^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (9,10).

Cf. A014992 (q-integers for q=-10), A097150.
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Cf. A094028, A098610, A098611.

[Round(10^n/11): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

k=0;lst={k};Do[k=10^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{9,10},{0,1},30] (* Harvey P. Dale, Aug 08 2014 *)

a(n)=10^n\/11 \\ Charles R Greathouse IV, Jun 24 2011

[lucas_number1(n,9,-10) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

[abs(gaussian_binomial(n,1,-10)) for n in range(0,19)] # Zerinvary Lajos, May 28 2009

a(n) = 9*a(n-1) + 10*a(n-2).
From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 10^(n-1) - a(n-1).
G.f.: x/(1 - 9x - 10x^2). (End)
From Henry Bottomley, Sep 17 2004: (Start)
a(n) = round(10^n/11).
a(n) = (10^n - (-1)^n)/11.
a(n) = A098611(n)/11 = 9*A094028(n+1)/A098610(n). (End)
E.g.f.: exp(-x)*(exp(11*x) - 1)/11. - Elmo R. Oliveira, Aug 17 2024

Extended by T. D. Noe, May 23 2011

A015577 a(n+1) = 8a(n) + 9a(n-1), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 8, 73, 656, 5905, 53144, 478297, 4304672, 38742049, 348678440, 3138105961, 28242953648, 254186582833, 2287679245496, 20589113209465, 185302018885184, 1667718169966657, 15009463529699912, 135085171767299209, 1215766545905692880, 10941898913151235921

Offset: 0

Olivier Gérard

Binomial transform is A011557, with a leading zero. - Paul Barry, Jul 09 2003
Number of walks of length n between any two distinct nodes of the complete graph K_10. Example: a(2) = 8 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJ are: ACB, ADB, AEB, AFB, AGB, AHB, AIB and AJB. - Emeric Deutsch, Apr 01 2004
The ratio a(n+1)/a(n) converges to 9 as n approaches infinity. - Felix P. Muga II, Mar 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (8,9).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[Round(9^n/10): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

seq(round(9^n/10),n=0..25); # Mircea Merca, Dec 28 2010

k=0;lst={k};Do[k=9^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(9^n - (-1)^n)/10, {n,0,30}] (* or *) LinearRecurrence[{8,9}, {0,1}, 30] (* G. C. Greubel, Jan 06 2018 *)

a[0]:0$
a[n]:=9^(n-1)-a[n-1]$
A015577(n):=a[n]$
makelist(A015577(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A015577_vec(N=20)=Vec(O(x^N)+1/(1-8*x-9*x^2), -N-1) \\ M. F. Hasler, Jun 14 2008, edited Oct 25 2019

for(n=0,30, print1((9^n - (-1)^n)/10, ", ")) \\ G. C. Greubel, Jan 06 2018

apply( {A015577(n)=9^n\/10}, [0..25]) \\ M. F. Hasler, Oct 25 2019

[lucas_number1(n,8,-9) for n in range(0, 19)] # Zerinvary Lajos, Apr 25 2009

From Paul Barry, Jul 09 2003: (Start)
G.f.: x/((1+x)*(1-9*x)).
E.g.f. exp(4*x)*sinh(5*x)/5.
a(n) = (9^n - (-1)^n)/10. (End)
a(n) = 9^(n-1)-a(n-1). - Emeric Deutsch, Apr 01 2004
a(n) = round(9^n/10). - Mircea Merca, Dec 28 2010

Extended by T. D. Noe, May 23 2011

A015592 a(n) = 10a(n-1) + 11a(n-2).

Original entry on oeis.org

0, 1, 10, 111, 1220, 13421, 147630, 1623931, 17863240, 196495641, 2161452050, 23775972551, 261535698060, 2876892678661, 31645819465270, 348104014117971, 3829144155297680, 42120585708274481, 463326442791019290, 5096590870701212191, 56062499577713334100

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_12. Example: a(2)=10 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJKL are ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB, AKB and ALB. - Emeric Deutsch, Apr 01 2004

General form: k=11^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,11).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[-(1/12)*(-1)^n+(1/12)*11^n: n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

k=0;lst={k};Do[k=11^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)

[lucas_number1(n,10,-11) for n in range(0, 18)] # Zerinvary Lajos, Apr 26 2009

a(n) = 11^(n-1) - a(n-1). G.f.: x/(1 - 10x - 11x^2). - Emeric Deutsch, Apr 01 2004
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(5*x)*sinh(6*x)/6.
a(n) = (11^n - (-1)^n)/12. (End)

A015609 a(n) = 11a(n-1) + 12a(n-2).

Original entry on oeis.org

0, 1, 11, 133, 1595, 19141, 229691, 2756293, 33075515, 396906181, 4762874171, 57154490053, 685853880635, 8230246567621, 98762958811451, 1185155505737413, 14221866068848955, 170662392826187461

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_13. Example: a(2)=11 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJKLM are ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB, AKB, ALB and AMB. - Emeric Deutsch, Apr 01 2004

General form: k=12^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (11,12).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585, A015592. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(1/13)*(12^n-(-1)^n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

CoefficientList[Series[x/(1-11*x-12*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{11,12}, {0,1}, 30] (* G. C. Greubel, Dec 30 2017 *)

x='x+O('x^30); concat([0], Vec(x/(1-11*x-12*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,11,-12) for n in range(0, 18)] # Zerinvary Lajos, Apr 27 2009

[abs(gaussian_binomial(n,1,-12)) for n in range(0,18)] # Zerinvary Lajos, May 28 2009

From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 12^(n-1) - a(n-1).
G.f.: x/(1 - 11*x - 12*x^2). (End)
E.g.f.: exp(-x)*(exp(13*x) - 1)/13. - Stefano Spezia, Mar 11 2020

A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 7, 42, 301, 2100, 14707, 102942, 720601, 5044200, 35309407, 247165842, 1730160901, 12111126300, 84777884107, 593445188742, 4154116321201, 29078814248400, 203551699738807, 1424861898171642, 9974033287201501, 69818233010410500, 488727631072873507, 3421093417510114542

Offset: 0

Ilya Gutkovskiy, Apr 13 2016

In general, the ordinary generating function for the recurrence b(n) = k^n - b(n-1), where n>0 and b(0)=0, is k*x/((1 + x)*(1 - k*x)). This recurrence gives the closed form b(n) = k*(k^n - (-1)^n)/(k + 1).

			a(2) = 7^2 - a(2-1) = 49 - 7 = 42.
a(4) = 7^4 - a(4-1) = 2401 - 301 = 2100.

Index entries for linear recurrences with constant coefficients, signature (6,7).

Cf. A000420, A015552.

Cf. similar sequences with the recurrence b(n) = k^n - b(n-1): A125122 (k=1), A078008 (k=2), A054878 (k=3), A109499 (k=4), A109500 (k=5), A109501 (k=6), this sequence (k=7), A093134 (k=8), A001099 (k=n).

LinearRecurrence[{6, 7}, {0, 7}, 30]
Table[7 (7^n - (-1)^n)/8, {n, 0, 30}]

vector(50, n, n--; 7*(7^n-(-1)^n)/8) \\ Altug Alkan, Apr 13 2016

for n in range(0,10**2):print((int)((7*(7**n-(-1)**n))/8))
# Soumil Mandal, Apr 14 2016

O.g.f.: 7*x/(1 - 6*x - 7*x^2).
E.g.f.: (7/8)*(exp(7*x) - exp(-x)).
a(n) = 6*a(n-1) + 7*a(n-2).
a(n) = 7*(7^n - (-1)^n)/8.
a(n) = 7*A015552(n).
Sum_{n>0} 1/(a(n) + a(n-1)) = 1/6 = A020793.
Limit_{n->oo} a(n-1)/a(n) = 1/7 = A020806.

cp's OEIS Frontend

A140690 A positive integer n is included if n written in binary can be subdivided into a number of runs all of equal-length, the first run from the left consisting of all 1's, the next run consisting of all 0's, the next run consisting of all 1's, the next run consisting of all 0's, etc.

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A015585 a(n) = 9*a(n-1) + 10*a(n-2).

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A015577 a(n+1) = 8*a(n) + 9*a(n-1), a(0) = 0, a(1) = 1.

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A015592 a(n) = 10*a(n-1) + 11*a(n-2).

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Magma

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A015609 a(n) = 11*a(n-1) + 12*a(n-2).

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Magma

Mathematica

PARI

Sage

Sage

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A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

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A015585 a(n) = 9a(n-1) + 10a(n-2).

A015577 a(n+1) = 8a(n) + 9a(n-1), a(0) = 0, a(1) = 1.

A015592 a(n) = 10a(n-1) + 11a(n-2).

A015609 a(n) = 11a(n-1) + 12a(n-2).