A125910 -id:A125910 - cp's OEIS frontend

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.

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

A126629 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 3,4,5 and at least one of digits 6,7,8,9.

Original entry on oeis.org

8, 64, 506, 3916, 29498, 215524, 1527506, 10528876, 70841738, 467044084, 3027621506, 19356463036, 122355512378, 766290978244, 4762898595506, 29420807536396, 180813134269418, 1106606890266004, 6749433735297506, 41050188511748956, 249087606867080858

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->12*6^n-30*5^n+34*4^n-21*3^n+7*2^n-1;

LinearRecurrence[{21,-175,735,-1624,1764,-720},{8,64,506,3916,29498,215524},30] (* Harvey P. Dale, Sep 26 2019 *)

vector(100, n, 12*6^n-30*5^n+34*4^n-21*3^n+7*2^n-1) \\ Colin Barker, Feb 23 2015

a(n) = 12*6^n-30*5^n+34*4^n-21*3^n+7*2^n-1.

G.f.: -2*x*(360*x^5 -882*x^4 +695*x^3 -281*x^2 +52*x -4) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)). - Colin Barker, Feb 23 2015

A126718 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, at least one of digits 6,7 and at least one of digits 8,9.

Original entry on oeis.org

7, 43, 235, 1171, 5467, 24403, 105595, 447091, 1864027, 7686163, 31440955, 127865011, 517788187, 2090186323, 8417944315, 33843570931, 135890057947, 545108340883, 2185079263675, 8754257900851, 35058860433307, 140360940805843, 561820285607035

Offset: 1

Aleksandar M. Janjic and Milan Janjic, Feb 13 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A125910, A125945, A125946, A125947, A125948, A125880, A125630, A125987, A125904, A125858, A125909, A125908, A126646, A126645, A126644, A126643, A126642, A126641, A126640, A126639, A126635, A126634, A126633, A126632, A126631, A126628, A126627.

[8*4^n-12*3^n+6*2^n-1: n in [1..30]]; // Vincenzo Librandi, May 31 2011

Maple
```
a:=n->8*4^n-12*3^n+6*2^n-1;
```

LinearRecurrence[{10,-35,50,-24},{7, 43, 235, 1171},23] (* James C. McMahon, Dec 27 2024 *)

Vec(-x*(24*x^3-50*x^2+27*x-7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

a(n) = 8*4^n - 12*3^n + 6*2^n - 1.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4). - Colin Barker, Feb 22 2015
G.f.: -x*(24*x^3 - 50*x^2 + 27*x - 7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Feb 22 2015

A258800 The number of zeroless decimal numbers whose digital sum is n.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960, 4190597000, 8372936304, 16729373488, 33425781248

Offset: 0

Robert G. Wilson v, Jun 10 2015

If you were to include decimal numbers that contain any number of zeros, then a(n) would be infinity. If on the other hand, you limit the number of zeros to some number, then a(n) is finite.

			a(0) = 0 since there exists no decimal number lacking a zero whose digital sum is zero.
a(1) = 1 since there exists only one zeroless decimal number whose digital sum is one and that number is 1.
a(2) = 2 since there exist only two zeroless decimal numbers whose digital sum is two and they are 2 & 11.
a(3) = 4 since there exist only four zeroless decimal numbers whose digital sum is three and they are 3, 21, 12 & 111.
a(4) = 8 since there exist only eight zeroless decimal numbers whose digital sum is four and they are 4, 31, 13, 22, 211, 121, 112 & 1111.

Cf. A104144.
Cf. A125630, A125858, A125858, A125880, A125897, A125904, A125908, A125909, A125910, A125945, A125946, A125947, A125948, A126627, A126628, A126629, A126631, A126632, A126633, A126634, A126635, A126639, A126640, A126641, A126642, A126643, A126644, A126645, A126646, A126718.
Cf. A211072.

CoefficientList[ Series[-1 + 1/(1 - x (1 + x + x^2) (1 + x^3 + x^6)), {x, 0, 36}], x]

a(n) = A104144(n+8) for n>0.

G.f.: -(x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)/(-1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9) = -1 + 1/(1-x(1 + x + x^2)(1 + x^3 + x^6)).

cp's OEIS Frontend

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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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.

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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.

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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.

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A126629 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 3,4,5 and at least one of digits 6,7,8,9.