A125630 -id:A125630 - cp's OEIS frontend

A126646 a(n) = 2^(n+1) - 1.

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647

Offset: 0

Aleksandar M. Janjic and Milan Janjic, Feb 08 2007, Feb 13 2007

a(n) is the number of integers k less than 10^n such that the decimal representation of k lacks the digits 1,2,3,4,5,6 and 7 and at least one of the digits 8,9.
Partial sums of the powers of 2 (A000079).
a(n) is the number of elements (all m-dimensional faces) in an n-dimensional simplex (0 <= m <= n). - Sergey Pavlov, Aug 15 2015
A261461(a(n)) != A261922(a(n)). - Reinhard Zumkeller, Sep 17 2015
a(n) is the total number of matches in a knockout tournament with 2^n players. - Paul Duckett, Dec 12 2022

			a(8) = 2^9 - 1 = 511.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Wikipedia, Simplex Elements (see last column of table).
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Essentially the same as A000225.
Cf. A125630, A125945, A125947, A125948, A125940, A125909, A125908, A125880, A125897, A125904, A125858.
Cf. A000079, A168604.
Cf. A261461, A261922.

a126646 = (subtract 1) . (2 ^) . (+ 1)
a126646_list = iterate ((+ 1) . (* 2)) 1
-- Reinhard Zumkeller, Sep 17 2015

[2^(n+1)-1: n in [0.. 35]]; // Vincenzo Librandi, Aug 20 2015

A126646:=n->2*2^n-1; seq(A126646(n), n=0..50); # Wesley Ivan Hurt, Dec 02 2013

Table[2^(n+1) - 1, {n, 0, 50}] (* Wesley Ivan Hurt, Dec 02 2013 *)
LinearRecurrence[{3,-2},{1,3},40] (* Harvey P. Dale, Mar 23 2018 *)

first(m)=vector(m,i,i--;2^(i+1)-1) /* Anders Hellström, Aug 19 2015 */

def A126646(n): return (1<Chai Wah Wu, Mar 18 2024

[2^(n+1) -1 for n in (0..50)] # G. C. Greubel, Mar 31 2021

a(n-1)^2 + a(n) = a(2n) + 1, a square. - Vincenzo Librandi and Ralf Stephan, Nov 23 2010
G.f.: 1/ ( (1-2*x)*(1-x) ). - R. J. Mathar, Dec 02 2013
a(n) = 3*a(n-1) - 2*a(n-2), n > 1. - Wesley Ivan Hurt, Aug 21 2015
E.g.f.: 2*exp(2*x) - exp(x). - G. C. Greubel, Mar 31 2021

A125858 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of digits 1,2,3,4,5,6,7,8,9.

Original entry on oeis.org

10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 999637120, 9980041600, 99381289600, 985729744000, 9726841354240, 95404977568000, 929690189228800, 8999055703648000, 86532737999167360, 826798452380099200, 7852626768025993600

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 03 2007

Note that the first eight terms of the sequence are powers of 10.

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

Cf. A125630.

f:=n->9*9^n-36*8^n+84*7^n-126*6^n+126*5^n-84*4^n+36*3^n-9*2^n+1;

f[n_] := (9*9^n - 36*8^n + 84*7^n - 126*6^n + 126*5^n - 84*4^n + 36*3^n - 9*2^n + 1); Array[f, 18] (* Robert G. Wilson v, May 31 2009 *)
(* or *) f[n_] := Sum[ -(-1)^k*Binomial[9, k] (10 - k)^n, {k, 9}]; Array[f, 18] (* Robert G. Wilson v, May 31 2009 *)

vector(100, n, 9*9^n-36*8^n+84*7^n-126*6^n+126*5^n-84*4^n+36*3^n-9*2^n+1) \\ Colin Barker, Feb 23 2015

a(n) = 9*9^n-36*8^n+84*7^n-126*6^n+126*5^n-84*4^n+36*3^n-9*2^n+1.

Incorrect g.f. removed by Georg Fischer, May 15 2019

A125880 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of the digits 1,2 and at least one of the digits 3,4,5,6,7,8,9.

Original entry on oeis.org

10, 98, 946, 9026, 85330, 800738, 7463746, 69054146, 633214450, 5746812578, 51574432546, 457575310466, 4014480866770, 34850492651618, 299615359353346, 2553264555537986, 21587837904314290, 181256003521421858, 1512536914179602146, 12553861394003656706

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 03 2007, Feb 13 2007

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Cf. A125630.

f:=n->14*8^n-49*7^n+91*6^n-105*5^n+77*4^n-35*3^n+9*2^n-1;

Table[14*8^n-49*7^n+91*6^n-105*5^n+77*4^n-35*3^n+9*2^n-1,{n,20}] (* James C. McMahon, Dec 22 2024 *)

vector(100, n, 14*8^n-49*7^n+91*6^n-105*5^n+77*4^n-35*3^n+9*2^n-1) \\ Colin Barker, Feb 23 2015

a(n) = 14*8^n-49*7^n+91*6^n-105*5^n+77*4^n-35*3^n+9*2^n-1.

G.f.: -2*x*(20160*x^7 -54792*x^6 +53420*x^5 -28436*x^4 +8441*x^3 -1439*x^2 +131*x -5) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)). - Colin Barker, Feb 23 2015

A125904 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of the digits 1,2,3,4 and at least one of the digits 5,6,7,8,9.

Original entry on oeis.org

10, 100, 1000, 9976, 98920, 971440, 9420400, 90005176, 846101080, 7822770880, 71163913600, 637585923976, 5633160763240, 49148451899920, 424055315011600, 3623013325068376, 30689130457473400, 258015134469182560, 2155143129458730400

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 03 2007

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Cf. A125630.

f:=n->20*8^n-70*7^n+120*6^n-125*5^n+84*4^n-36*3^n+9*2^n-1;

vector(100, n, 20*8^n-70*7^n+120*6^n-125*5^n+84*4^n-36*3^n+9*2^n-1) \\ Colin Barker, Feb 23 2015

a(n) = 20*8^n-70*7^n+120*6^n-125*5^n+84*4^n-36*3^n+9*2^n-1.

G.f.: -2*x*(20160*x^7 -54792*x^6 +53362*x^5 -28337*x^4 +8392*x^3 -1430*x^2 +130*x -5) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)). - Colin Barker, Feb 23 2015

A125908 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digits 1 and 2 and at least one of digits 3,4,5,6,7,8,9.

Original entry on oeis.org

8, 64, 512, 4096, 32768, 262144, 2092112, 16595776, 130437728, 1013866624, 7788438512, 59145432256, 444357721088, 3306242197504, 24389881261712, 178578361769536, 1299058046034848, 9397253451942784, 67653687455953712, 485065987257543616

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 04 2007

			a(8) = 16595776.

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A125630.

f:=n->7*7^n-21*6^n+35*5^n-35*4^n+21*3^n-7*2^n+1;

Vec(-8*x*(630*x^6-1476*x^5+1457*x^4-664*x^3+162*x^2-20*x+1)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 7*7^n-21*6^n+35*5^n-35*4^n+21*3^n-7*2^n+1.

G.f.: -8*x*(630*x^6 -1476*x^5 +1457*x^4 -664*x^3 +162*x^2 -20*x +1)/((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)). - Colin Barker, Feb 22 2015

A125909 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digit 1, at least one of the digits 2,3 and at least one of digits 4,5,6,7,8,9.

Original entry on oeis.org

9, 79, 681, 5791, 48729, 405919, 3340521, 27094111, 216288249, 1699187359, 13147825161, 100334472031, 756309350169, 5639967562399, 41669245538601, 305413957523551, 2223312590034489, 16091187568891039, 115885120813664841, 831075819022712671

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 04 2007

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A125630.

f:=n->12*7^n-36*6^n+55*5^n-50*4^n+27*3^n-8*2^n+1;

Vec(-x*(5040*x^6 -11988*x^5 +11944*x^4 -5479*x^3 +1367*x^2 -173*x+9) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 12*7^n-36*6^n+55*5^n-50*4^n+27*3^n-8*2^n+1.

G.f.: -x*(5040*x^6 -11988*x^5 +11944*x^4 -5479*x^3 +1367*x^2 -173*x+9) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)). - Colin Barker, Feb 22 2015

A125945 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of digits 1,2, at least one of digits 3,4 and at least one of digits 5,6,7,8,9.

Original entry on oeis.org

10, 96, 892, 8076, 71500, 619236, 5241652, 43355436, 350740540, 2780210676, 21641505412, 165814383996, 1253254845580, 9362782430916, 69259450905172, 508062802935756, 3700662365318620, 26794126632247956, 193018363703408932, 1384505203965202716

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 04 2007

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A125630.

f:=n->20*7^n-60*6^n+85*5^n-70*4^n+34*3^n-9*2^n+1;

a(n)=20*7^n-60*6^n+85*5^n-70*4^n+34*3^n-9*2^n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 20*7^n-60*6^n+85*5^n-70*4^n+34*3^n-9*2^n+1.

G.f.: -2*x*(2520*x^6 -6054*x^5 +6063*x^4 -2794*x^3 +712*x^2 -92*x +5) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)). - Colin Barker, Feb 23 2015

A125947 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digit 1, at least one of digits 2,3,4,5 and at least one of digits 6,7,8,9.

Original entry on oeis.org

9, 81, 729, 6513, 57369, 495921, 4194969, 34689393, 280607769, 2224214961, 17313344409, 132651929073, 1002605145369, 7490229758001, 55407572177049, 406450276733553, 2960529995462169, 21435301615525041, 154414691892116889, 1107604165960750833

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 04 2007

			a(8) = 34689393.

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A125630.

f:=n->16*7^n-48*6^n+68*5^n-56*4^n+28*3^n-8*2^n+1;

Table[ 16*7^n-48*6^n+68*5^n-56*4^n+28*3^n-8*2^n+1, {n, 20}] (* James C. McMahon, Dec 23 2024 *)

Vec(-3*x*(1680*x^6 -3988*x^5 +3968*x^4 -1819*x^3 +453*x^2-57*x+3) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 16*7^n-48*6^n+68*5^n-56*4^n+28*3^n-8*2^n+1.

G.f.: -3*x*(1680*x^6 -3988*x^5 +3968*x^4 -1819*x^3 +453*x^2-57*x+3) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)). - Colin Barker, Feb 22 2015

A125948 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of digits 1,2,3, at least one of digits 4,5,6 and at least one of digits 7,8,9.

Original entry on oeis.org

10, 100, 982, 9388, 86950, 778780, 6748822, 56713708, 463661830, 3700940860, 28941318262, 222422713228, 1684544507110, 12602134130140, 93308618633302, 684912371803948, 4990975002030790, 36147245625290620, 260449744313893942

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 04 2007

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A125630.

f:=n->27*7^n-81*6^n+108*5^n-81*4^n+36*3^n-9*2^n+1;

Table[27*7^n-81*6^n+108*5^n-81*4^n+36*3^n-9*2^n+1,{n,19}] (* James C. McMahon, Dec 23 2024 *)

vector(100, n, 27*7^n-81*6^n+108*5^n-81*4^n+36*3^n-9*2^n+1) \\ Colin Barker, Feb 23 2015

a(n) = 27*7^n-81*6^n+108*5^n-81*4^n+36*3^n-9*2^n+1.

G.f.: -2*x*(2520*x^6 -6012*x^5 +5990*x^4 -2754*x^3 +701*x^2 -90*x +5) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)). - Colin Barker, Feb 23 2015

A125897 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of the digits 1,2,3 and at least one of the digits 4,5,6,7,8,9.

Original entry on oeis.org

10, 100, 994, 9796, 95650, 924820, 8845714, 83575396, 778945090, 7156197940, 64800104434, 578648865796, 5100362368930, 44424892053460, 382839350691154, 3268062540952996, 27665402010407170, 232490197792427380, 1941315874498269874

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 03 2007

Colin Barker, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Cf. A125630.

f:=n->18*8^n-63*7^n+111*6^n-120*5^n+83*4^n-36*3^n+9*2^n-1;

Table[18*8^n-63*7^n+111*6^n-120*5^n+83*4^n-36*3^n+9*2^n-1, {n, 19}]  (* James C. McMahon, Dec 22 2024 *)

vector(100, n, 18*8^n-63*7^n+111*6^n-120*5^n+83*4^n-36*3^n+9*2^n-1) \\ Colin Barker, Feb 23 2015

a(n) = 18*8^n-63*7^n+111*6^n-120*5^n+83*4^n-36*3^n+9*2^n-1.

G.f.: -2*x*(20160*x^7 -54792*x^6 +53344*x^5 -28304*x^4 +8374*x^3 -1427*x^2 +130*x -5) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)). - Colin Barker, Feb 23 2015

cp's OEIS Frontend

A126646 a(n) = 2^(n+1) - 1.

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A125858 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of digits 1,2,3,4,5,6,7,8,9.

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A125880 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of the digits 1,2 and at least one of the digits 3,4,5,6,7,8,9.

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A125904 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of the digits 1,2,3,4 and at least one of the digits 5,6,7,8,9.

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A125908 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digits 1 and 2 and at least one of digits 3,4,5,6,7,8,9.

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A125909 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digit 1, at least one of the digits 2,3 and at least one of digits 4,5,6,7,8,9.

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A125945 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks at least one of digits 1,2, at least one of digits 3,4 and at least one of digits 5,6,7,8,9.

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