A125908 -id:A125908 - 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

A126644 a(n) = 33^n - 32^n + 1.

Original entry on oeis.org

4, 16, 58, 196, 634, 1996, 6178, 18916, 57514, 174076, 525298, 1582036, 4758394, 14299756, 42948418, 128943556, 387027274, 1161475036, 3485211538, 10457207476, 31374768154, 94130595916, 282404370658, 847238277796

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 2007

Previous name was: 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 at least one of digits 7,8,9.

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x, 1) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 2) x equals y. Then a(n) = |R|. [Ross La Haye, Mar 19 2009]

			a(8) = 18916.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A001047, A125630, A125948, A125947, A125946, A125945, A125940, A125909, A125908, A125880, A125897, A125904, A125858.

Maple
```
f:=n->3*3^n-3*2^n+1;
```

LinearRecurrence[{6,-11,6},{4,16,58},30] (* Harvey P. Dale, Sep 14 2018 *)

a(n) = 3*3^n - 3*2^n + 1; \\ Michel Marcus, Nov 30 2015

a(n) = 3*3^n - 3*2^n + 1.
a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3). G.f.: -2*x*(3*x^2-4*x+2) / ((x-1)*(2*x-1)*(3*x-1)). [Colin Barker, Dec 10 2012]
a(n) = 3*A001047(n) + 1. - Hugo Pfoertner, Nov 22 2022

New name from Hugo Pfoertner, Nov 22 2022

A126645 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 and 5, at least one of digits 6,7 and at least one of digits 8,9.

Original entry on oeis.org

5, 21, 77, 261, 845, 2661, 8237, 25221, 76685, 232101, 700397, 2109381, 6344525, 19066341, 57264557, 171924741, 516036365, 1548633381, 4646948717, 13942943301, 41833024205, 125507461221, 376539160877, 1129651037061, 3389020220045, 10167194877861

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 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 (6,-11,6).

Cf. A125630, A125948, A125947, A125946, A125945, A125940, A125909, A125908, A125880, A125897, A125904, A125858.

Maple
```
f:=n->4*3^n-4*2^n+1;
```

Vec(-x*(6*x^2-9*x+5)/((x-1)*(2*x-1)*(3*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 4*3^n-4*2^n+1.
a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3). - Colin Barker, Feb 22 2015
G.f.: -x*(6*x^2-9*x+5) / ((x-1)*(2*x-1)*(3*x-1)). - Colin Barker, Feb 22 2015

A126627 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks digits 1,2,3 and at least one of digits 4,5,6,7,8,9.

Original entry on oeis.org

7, 49, 343, 2401, 16807, 116929, 803383, 5432161, 36120007, 236404609, 1525601623, 9726181921, 61371928807, 383929313089, 2384606035063, 14723095123681, 90457525939207, 553507860826369, 3375536272503703, 20528377102849441, 124556950506727207

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 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 (21,-175,735,-1624,1764,-720).

Cf. A125630, A125948, A125947, A125946, A125945, A125910, A125909, A125908, A125880, A125897, A125904, A125858.

f:=n->6*6^n-15*5^n+20*4^n-15*3^n+6*2^n-1;

LinearRecurrence[{21,-175,735,-1624,1764,-720},{7,49,343,2401,16807,116929},30] (* Harvey P. Dale, Aug 02 2017 *)

vector(100, n, 6*6^n-15*5^n+20*4^n-15*3^n+6*2^n-1) \\ Colin Barker, Feb 23 2015

a(n) = 6*6^n-15*5^n+20*4^n-15*3^n+6*2^n-1.

G.f.: -x*(720*x^5 -1764*x^4 +1372*x^3 -539*x^2 +98*x -7) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)). - Colin Barker, Feb 23 2015

A126628 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks digits 1 and 2, at least one of digits 3,4 and at least one of digits 5,6,7,8,9.

Original entry on oeis.org

8, 62, 470, 3506, 25718, 184682, 1294910, 8867186, 59423078, 390804602, 2529567950, 16157024066, 102070798838, 639011269322, 3970835898590, 24524390352146, 150705922308998, 922285972770842, 5624983337550830, 34210314230099426, 207580309651649558

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 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 (21,-175,735,-1624,1764,-720).

Cf. A125630, A125948, A125947, A125946, A125945, A125910, A125909, A125908, A125880, A125897, A125904, A125858.

f:=n->10*6^n-25*5^n+30*4^n-20*3^n+7*2^n-1;

CoefficientList[Series[-2*(360*x^5 - 882*x^4 + 697*x^3 - 284*x^2 + 53*x - 4)/((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 22 2022 *)

vector(100, n, 10*6^n-25*5^n+30*4^n-20*3^n+7*2^n-1) \\ Colin Barker, Feb 23 2015

a(n) = 10*6^n-25*5^n+30*4^n-20*3^n+7*2^n-1.
G.f.: -2*x*(360*x^5 -882*x^4 +697*x^3 -284*x^2 +53*x -4) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)). - Colin Barker, Feb 23 2015
a(n) = 21*a(n-1)-175*a(n-2)+735*a(n-3)-1624*a(n-4)+1764*a(n-5)-720*a(n-6). - Wesley Ivan Hurt, Jun 22 2022

A126631 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, at least one of digits 4,5 and at least one of digits 6,7,8,9.

Original entry on oeis.org

9, 77, 633, 5021, 38409, 283277, 2019033, 13963901, 94144809, 621444077, 4031587833, 25787305181, 163054382409, 1021372934477, 6349128459033, 39222102764861, 241061530639209, 1475385002210477, 8998880800344633, 54732125638998941

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 2007

			a(8) = 13963901.

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 (21,-175,735,-1624,1764,-720).

Cf. A125630, A125948, A125947, A125946, A125945, A125910, A125909, A125908, A125880, A125897, A125904, A125858.

f:=n->16*6^n-40*5^n+44*4^n-26*3^n+8*2^n-1;

LinearRecurrence[{21,-175,735,-1624,1764,-720},{9,77,633,5021,38409,283277},30] (* Harvey P. Dale, Oct 14 2016 *)

Vec(-x*(720*x^5-1764*x^4+1412*x^3-591*x^2+112*x-9)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 16*6^n-40*5^n+44*4^n-26*3^n+8*2^n-1.

G.f.: -x*(720*x^5-1764*x^4+1412*x^3-591*x^2+112*x-9) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Feb 22 2015

A126632 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, at least one of digits 4,5,6 and at least one of digits 7,8,9.

Original entry on oeis.org

9, 79, 669, 5431, 42189, 314119, 2251629, 15625591, 105563469, 697683559, 4529641389, 28986744151, 183339095949, 1148652643399, 7141191155949, 44118519949111, 271168742599629, 1659705919705639, 10123331198091309, 61571999920648471

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 2007

			a(8) = 15625591.

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 (21,-175,735,-1624,1764,-720).

Cf. A125630, A125948, A125947, A125946, A125945, A125910, A125909, A125908, A125880, A125897, A125904, A125858.

f:=n->18*6^n-45*5^n+48*4^n-27*3^n+8*2^n-1;

LinearRecurrence[{21,-175,735,-1624,1764,-720},{9, 79, 669, 5431, 42189, 314119},20] (* James C. McMahon, Dec 26 2024 *)

Vec(-x*(720*x^5-1764*x^4+1408*x^3-585*x^2+110*x-9) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 18*6^n-45*5^n+48*4^n-27*3^n+8*2^n-1.

G.f.: -x*(720*x^5-1764*x^4+1408*x^3-585*x^2+110*x-9) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Feb 22 2015

A126633 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, at least one of digits 5,6 and at least one of digits 7,8,9.

Original entry on oeis.org

10, 94, 832, 6946, 54880, 412714, 2975752, 20722306, 140285200, 928323034, 6031661272, 38617025266, 244322679520, 1531014308554, 9519483716392, 58816232361826, 361524350929840, 2212804949145274, 13497228660885112

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 2007

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A125630, A125948, A125947, A125946, A125945, A125910, A125909, A125908, A125880, A125897, A125904, A125858.

A126633:=n->24*6^n-60*5^n+62*4^n-33*3^n+9*2^n-1; seq(A126633(n), n=1..20);

Table[24*6^n - 60*5^n + 62*4^n - 33*3^n + 9*2^n - 1, {n, 20}] (* Wesley Ivan Hurt, May 03 2014 *)
LinearRecurrence[{21,-175,735,-1624,1764,-720},{10,94,832,6946,54880,412714},30] (* Harvey P. Dale, May 05 2018 *)

a(n) = 24*6^n-60*5^n+62*4^n-33*3^n+9*2^n-1.

G.f.: -2*x*(360*x^5-882*x^4+713*x^3-304*x^2+58*x-5) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, May 04 2014

A126634 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digits 1,2,3,4 and at least one of digits 5,6,7,8,9.

Original entry on oeis.org

6, 36, 216, 1296, 7656, 44136, 248016, 1362096, 7338456, 38927736, 203958816, 1058224896, 5448329256, 27880971336, 141993797616, 720419919696, 3644189320056, 18390164454936, 92630272564416, 465876904526496, 2340309918950856, 11745320884258536

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 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 (15,-85,225,-274,120).

Cf. A125630, A125948, A125947, A125946, A125945, A125910, A125909, A125908, A125880, A125897, A125904, A125858.

Maple
```
f:=n->5*5^n-10*4^n+10*3^n-5*2^n+1;
```

LinearRecurrence[{15,-85,225,-274,120},{6,36,216,1296,7656},30] (* Harvey P. Dale, Apr 01 2018 *)

Vec(-6*x*(20*x^4-39*x^3+31*x^2-9*x+1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 5*5^n-10*4^n+10*3^n-5*2^n+1.

G.f.: -6*x*(20*x^4-39*x^3+31*x^2-9*x+1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Feb 22 2015

A126635 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digits 1,2,3, at least one of digits 4,5 and at least one of digits 6,7,8,9.

Original entry on oeis.org

7, 47, 307, 1943, 11827, 69287, 392707, 2166743, 11703187, 62168327, 325983907, 1692105143, 8714154547, 44600020967, 227161443907, 1152585909143, 5830444893907, 29423488811207, 148206112628707, 745396075770743, 3744474953809267, 18792450661083047

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 08 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 (15,-85,225,-274,120).

Cf. A125630, A125948, A125947, A125946, A125945, A125910, A125909, A125908, A125880, A125897, A125904, A125858.

Maple
```
f:=n->8*5^n-16*4^n+14*3^n-6*2^n+1;
```

LinearRecurrence[{15,-85,225,-274,120},{7,47,307,1943,11827},30] (* Harvey P. Dale, Dec 31 2021 *)

Vec(-x*(120*x^4-242*x^3+197*x^2-58*x+7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 8*5^n-16*4^n+14*3^n-6*2^n+1.

G.f.: -x*(120*x^4-242*x^3+197*x^2-58*x+7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Feb 22 2015

cp's OEIS Frontend

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

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A126644 a(n) = 3*3^n - 3*2^n + 1.

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A126645 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 and 5, at least one of digits 6,7 and at least one of digits 8,9.

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A126627 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks digits 1,2,3 and at least one of digits 4,5,6,7,8,9.

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A126628 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks digits 1 and 2, at least one of digits 3,4 and at least one of digits 5,6,7,8,9.

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A126631 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, at least one of digits 4,5 and at least one of digits 6,7,8,9.

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A126632 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, at least one of digits 4,5,6 and at least one of digits 7,8,9.

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A126644 a(n) = 33^n - 32^n + 1.