A125630 -id:A125630 - cp's OEIS frontend

A125946 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 3,4,5 and at least one of digits 6,7,8,9.

10, 98, 940, 8798, 80140, 709238, 6096100, 50950718, 415060060, 3305238278, 25807024660, 198131841038, 1499550640780, 11213044626518, 82997777543620, 609099122145758, 4437879770746300, 32138240678881958, 231547934781860980, 1661033550903240878

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->24*7^n-72*6^n+98*5^n-76*4^n+35*3^n-9*2^n+1;

Table[24*7^n-72*6^n+98*5^n-76*4^n+35*3^n-9*2^n+1,{n, 20}]  (* James C. McMahon, Dec 22 2024 *)

vector(100, n, 24*7^n-72*6^n+98*5^n-76*4^n+35*3^n-9*2^n+1) \\ Colin Barker, Feb 23 2015

a(n) = 24*7^n-72*6^n+98*5^n-76*4^n+35*3^n-9*2^n+1.

G.f.: -2*x*(2520*x^6 -6042*x^5 +6043*x^4 -2783*x^3 +708*x^2 -91*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

A125910 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 and at least one of digits 5,6,7,8,9.

Original entry on oeis.org

9, 81, 723, 6381, 55539, 475461, 3993243, 32857101, 264890019, 2094889941, 16282118763, 124625344221, 941303216499, 7029057066021, 51980086628283, 381227207181741, 2776407821318979, 20100192515299701, 144786930345697803, 1038495372200033661

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 04 2007

			a(8) = 32857101.

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->15*7^n-45*6^n+65*5^n-55*4^n+28*3^n-8*2^n+1;

Vec(-3*x*(1680*x^6 -3976*x^5 +3946*x^4 -1807*x^3 +451*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) = 15*7^n-45*6^n+65*5^n-55*4^n+28*3^n-8*2^n+1.

G.f.: -3*x*(1680*x^6 -3976*x^5 +3946*x^4 -1807*x^3 +451*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

A077220 a(1) = 1; thereafter a(n) is smallest number not occurring earlier such that a(n-1)+a(n) is a triangular number.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 3, 7, 8, 13, 15, 21, 24, 31, 5, 10, 11, 17, 19, 26, 29, 16, 20, 25, 30, 36, 42, 49, 56, 22, 14, 41, 37, 18, 27, 28, 38, 40, 51, 54, 66, 39, 52, 53, 67, 69, 84, 87, 33, 45, 46, 32, 23, 43, 35, 70, 50, 55, 65, 71, 34, 44, 47, 58, 62, 74, 79, 57, 48, 72, 64, 89

Offset: 1

Amarnath Murthy, Nov 03 2002

Conjectured to be a permutation of the natural numbers (cf. A099130). The first few fixed points are: 1, 2, 19, 92, 220, 467, 556, 616, 690, 842.

			n=5: ss={1,2,4,6}; triangular numbers > 6 are 10,15,21; but 10-6=4 is in ss, hence a(5)=15-6=9;
n=6, ss={1,2,4,6,9}; triangular numbers > 9 are 10,15,21; but 10-9=4 and 15-9=6 are in ss, hence a(6)=21-9=12 etc.

Cf. A000217, A125595, A125630, A099130.

tr=Table[n(n+1)/2, {n, 100}]; s={1}; a=1; Do[Do[tk=tr[[k]]; If[tk > a, b=tk-a; If[FreeQ[s, b], AppendTo[s, b]; a=b; Break[]]], {k, 100}], {99}]; s (* Zak Seidov, Jul 12 2010 *)

v=[1]; n=1; while(n<100, if(ispolygonal(n+v[#v],3)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v \\ Derek Orr, Jun 08 2015

More terms from Reinhard Zumkeller, Sep 28 2004
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007
Further edited by N. J. A. Sloane, Jul 11 2010, Jul 19 2010

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

cp's OEIS Frontend

A125946 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 3,4,5 and at least one of digits 6,7,8,9.

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A125910 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 and at least one of digits 5,6,7,8,9.

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A077220 a(1) = 1; thereafter a(n) is smallest number not occurring earlier such that a(n-1)+a(n) is a triangular number.

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