A328348 -id:A328348 - cp's OEIS frontend

A328350 Let S be any integer in the range 6 <= S <= 24. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and contains at most three distinct nonzero digits d1, d2, d3 such that d1+d2+d3 = S.

0, 1, 34, 1033, 31030, 931021, 27930994, 837930913, 25137930670, 754137929941, 22624137927754, 678724137921193, 20361724137901510, 610851724137842461, 18325551724137665314, 549766551724137133873, 16492996551724135539550, 494789896551724130756581, 14843696896551724116407674, 445310906896551724073360953

Offset: 0

Pierre-Alain Sallard, Oct 13 2019

This sequence is the building block for the calculation of the sums of positive integers whose decimal expansion contains only three distinct, nonzero digits: see attached pdf document.

			For n=2, the sum of all positive integers whose decimal expansions consist of at most n=2 of the digits 5, 6 or 7, i.e., the sum 5+6+7+55+56+57+65+66+67+75+76+77, is equal to a(2)*(5+6+7) = 612.
The formula is valid for any other choice of three distinct digits. Another example: again with n=2, but let's say with the digits 1, 2 and 3, the sum 1+2+3+11+12+13+21+22+23+31+32+33 is equal to a(2)*(1+2+3) = 204.

Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (34,-123,90).

Cf. A328348, A328351, A328352, A328353, A328354, A328355, A328356.

Array[(20*30^# - 29*3^# + 9)/522 &, 20, 0] (* or *)
LinearRecurrence[{34, -123, 90}, {0, 1, 34}, 20] (* Paolo Xausa, Apr 29 2024 *)

[(20*30**n-29*3**n+9)//522 for n in range(20)]

a(n) = (20*30^n - 29*3^n + 9)/522.
From Stefano Spezia, Oct 17 2019: (Start)
O.g.f.: x/(1 - 34*x + 123*x^2 - 90*x^3).
E.g.f.: (1/522)*(9*exp(x) - 29*exp(3*x) + 20*exp(30*x)).
a(n) = 34*a(n-1) - 123*a(n-2) + 90*a(n-3) for n > 2. (End)
a(n) = 31*a(n-1) - 30*a(n-2) + 3^n for n > 1. - Pierre-Alain Sallard, Dec 15 2019

Needs editing. - N. J. A. Sloane, Dec 12 2019

A328351 Let S be any integer in the range 10 <= S <= 30. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and contains at most four distinct nonzero digits d1, d2, d3 and d4 such that d1+d2+d3+d4=S.

Original entry on oeis.org

0, 1, 45, 1821, 72925, 2917341, 116695005, 4667805661, 186712248285, 7468490018781, 298739601100765, 11949584045428701, 477983361822740445, 19119334472931987421, 764773378917368975325, 30590935156695116926941, 1223637406267806108733405, 48945496250712250075959261

Offset: 0

Pierre-Alain Sallard, Oct 13 2019

This sequence is the building block for the calculation of the sums of positive integers whose decimal notation only uses four distinct, nonzero digits: see the attached pdf document.

			For n=2, the sum of all positive integers whose decimal notation is only made of the 4,5,6 or 7 digit with at most n=2 such digits, i.e., the sum 4+5+6+7+44+45+46+47+54+55+56+57+64+65+66+67+74+75+76+77, is equal to (4+5+6+7)*a(2)=990.
The formula is valid for any other quadruple, as soon as the four digits are different from each other. Another example: always with n=2 but let's say with the 1,2,3 and 4 digits, the sum 1+2+3+4+11+12+13+14+21+22+23+24+31+32+33+34+41+42+43+44 is equal to a(2)*(1+2+3+4) = 450.

Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (45,-204,160).

Cf. A328348, A328350, A328352, A328353, A328354, A328355, A328356.

CoefficientList[Series[x/(1 - 45 x + 204 x^2 - 160 x^3), {x, 0, 17}], x] (* Michael De Vlieger, Dec 23 2019 *)

[(30*40**n-39*4**n+9)//1053 for n in range(20)]

a(n) = (30*40^n-39*4^n+9)/1053.
From Stefano Spezia, Oct 15 2019: (Start)
G.f.: x/(1 - 45*x + 204*x^2 - 160*x^3).
E.g.f.: (1/351)*exp(x)*(3 - 13*exp(3*x) + 10*exp(39*x)).
a(n) = 45*a(n-1) - 204*a(n-2) + 160*a(n-3) for n > 2.
(End)
a(n) = 41*a(n-1) - 40*a(n-2) + 4^(n-1) for n > 1. - Pierre-Alain Sallard, Dec 22 2019

A328352 Similar to A328350, but for 5 digits rather then 3.

Original entry on oeis.org

0, 1, 56, 2831, 141706, 7086081, 354307956, 17715417331, 885770964206, 44288548698581, 2214427437370456, 110721371880729831, 5536068594097526706, 276803429705181511081, 13840171485260601432956, 692008574263037701042331, 34600428713151923199089206, 1730021435657596350689323581

Offset: 0

Pierre-Alain Sallard, Nov 26 2019

This sequence is the building block for the calculation of the sums of positive integers whose decimal expansion only uses five distinct, nonzero digits: see the attached pdf document.

			For n=2, the sum of all positive integers whose decimal notation is only made of the 3,4,5,6 and 7 digit with at most n=2 such digits, i.e. the sum 3+4+5+6+7+33+34+35+36+37+43+44+45+46+47+53+54+55+56+57+63+64+65+66+67+73+74+75+76+77 is equal to a(2)*(3+4+5+6+7) = 56*25 = 1400.
The formula is valid for any other 5-tuple of digits, as soon as the 5 digits are different from each other. Always with n=2 but let's say with the 5,6,7,8 and 9 digits, the sum 5+6+7+8+9+55+56+57+58+59+65+66+67+68+69+75+76+77+78+79+85+86+87+88+89+95+96+97+98+99 is equal to a(2)*(5+6+7+8+9) = 56*35 = 1960.

Pierre-Alain Sallard, Table of n, a(n) for n = 0..50
Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (56, -305, 250).

Cf. A328348, A328350, A328351, A328353, A328354, A328355, A328356.

[(40*50**n-49*5**n+9)//1764 for n in range(12)]

a(n) = (40*50^n - 49*5^n + 9) / 1764.
a(n) = 51*a(n-1) - 50*a(n-2) + 5^(n-1) for n > 1.
G.f.: x / (1 - 56*x + 305*x^2 - 250*x^3).
a(n) = 56*a(n-1) - 305*a(n-2) + 250*a(n-3) for n > 2.

A328353 a(n)*S is the sum of all positive integers whose decimal expansion is up to n digits and uses six distinct nonzero digits d1,d2,d3,d4,d5,d6 such that d1+d2+d3+d4+d5+d6=S.

Original entry on oeis.org

0, 1, 67, 4063, 244039, 14643895, 878643031, 52718637847, 3163118606743, 189787118420119, 11387227117300375, 683233627110581911, 40994017627070271127, 2459641057626828406423, 147578463457625377218199, 8854707807457616670088855, 531282468447457564427312791, 31876948106847457250970656407

Offset: 0

Pierre-Alain Sallard, Nov 26 2019

This sequence is the building block for the calculation of the sums of positive integers whose decimal notation only uses six distinct, nonzero digits: see the attached pdf document.

			For n=2, the sum of all positive integers whose decimal notation is only made of, let's say, the 4,5,6,7,8,9 digits with at most n=2 such digits, i.e. the sum 4+5+6+7+8+9+44+45+46+47+48+49+54+55+56+57+58+59+64+65+66+67+68+69+74+75+76+77+78+79+84+85+86+87+88+89+94+95+96+97+98+99 is equal to a(2)*(4+5+6+7+8+9) = 67*39 = 2613.

Pierre-Alain Sallard, Table of n, a(n) for n = 0..50
Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (67,-426,360).

Cf. A328348, A328350, A328351, A328352, A328354, A328355, A328356.

LinearRecurrence[{67,-426,360},{0,1,67},20] (* Harvey P. Dale, Feb 11 2022 *)

[(50*60**n-59*6**n+9)//2655 for n in range(20)]

a(n) = (50*60^n - 59*6^n + 9) / 2655.
a(n) = 61*a(n-1) - 60*a(n-2) + 6^(n-1) for n > 1.
G.f.: x / (1 - 67*x + 426*x^2 -360*x^3).
a(n) = 67*a(n-1) - 426*a(n-2) + 360*a(n-3) for n > 2.

A328354 a(n)*S is the sum of all positive integers whose decimal expansion is up to n digits and uses seven distinct nonzero digits d1,d2,d3,d4,d5,d6,d7 such that d1+d2+d3+d4+d5+d6+d7=S.

Original entry on oeis.org

0, 1, 78, 5517, 386590, 27064101, 1894506678, 132615604717, 9283093290990, 649816537094901, 45487157643722278, 3184101035390113917, 222887072479614855390, 15602095073589188045701, 1092146655151356200377878, 76450265860595725286703117, 5351518610241706308890979790, 374606302716919480394120916501

Offset: 0

Pierre-Alain Sallard, Nov 26 2019

This sequence is the building block for the calculation of the sums of positive integers whose decimal notation uses seven distinct, nonzero digits: see the attached pdf document.

			For n=2, the sum of all positive integers whose decimal notation is made of, let's say, the 3,4,5,6,7,8 and 9 digits with at most n=2 such digits, i.e., the sum 3+4+5+6+7+8+9+33+34+35+36+37+38+39+43+44+45+46+47+48+49+53+54+55+56+57+58+59+63+64+65+66+67+68+69+73+74+75+76+78+79+83+84+85+86+87+88+89+93+94+95+96+97+98+99, is equal to a(2)*(3+4+5+6+7+8+9) = 78*42 = 3276.

Pierre-Alain Sallard, Table of n, a(n) for n = 0..50
Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (78,-567,490).

Cf. A328348, A328350, A328351, A328352, A328353, A328355, A328356.

LinearRecurrence[{78,-567,490},{0,1,78},20] (* Harvey P. Dale, Jan 30 2024 *)

concat(0, Vec(x / ((1 - x)*(1 - 7*x)*(1 - 70*x)) + O(x^17))) \\ Colin Barker, Dec 06 2019

[(60*70**n-69*7**n+9)//3726 for n in range(20)]

a(n) = (60*70^n-69*7^n+9) / 3726.
a(n) = 71*a(n-1) - 70*a(n-2) + 7^(n-1) for n > 1.
G.f.: x / (1 - 78*x + 567*x^2 - 490*x^3).
a(n) = 78*a(n-1) - 567*a(n-2) + 490*a(n-3) for n > 2.

A328355 Let S be any integer in the range 36 <= S <= 44. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and uses eight distinct nonzero digits d1,d2,d3,d4,d5,d6,d7,d8 such that d1+d2+d3+d4+d5+d6+d7+d8=S.

Original entry on oeis.org

0, 1, 89, 7193, 576025, 46086681, 3686971929, 294958053913, 23596646709785, 1887731755956761, 151018540629932569, 12081483251621739033, 966518660139556190745, 77321492811243031804441, 6185719424900070836714009, 494857553992010693275990553, 39588604319360895672790202905

Offset: 0

Pierre-Alain Sallard, Dec 10 2019

This sequence is the building block for the calculation of the sums of positive integers whose decimal expansion uses exactly eight distinct, nonzero digits: see the attached pdf documents.

			For n=2, the sum of all positive integers whose decimal notation is made of any digit different from 0 and, let's say, 9 with at most n=2 such digits, i.e., the sum 1+2+3+4+5+6+7+8+11+12+13+14+15+16+17+18+21+...+28+31+...+38+41+...+48+51+...+58+61+...+68+71+...+78+81+...+88, is equal to a(2)*(1+2+3+4+5+6+7+8) = 89*36 = 3204.
Similarly, and always with n=2, the sum of all positive integers whose decimal notation is made of any digit different from 0 and, let's say, 8, i.e., the sum 1+2+3+4+5+6+7+9+11+..+17+19+21+...+27+29+31+...+37+39+41+...+47+49+51+...+57+59+61+...+67+69+71+...+77+79+91+...+97+99 is equal to a(2)*(1+2+3+4+5+6+7+9) = 89*37 = 3293.

Pierre-Alain Sallard, Integers sequences A324348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (89, -728, 640).

Cf. A328348, A328350, A328351, A328352, A328353, A328354, A328356.

CoefficientList[Series[x/(1 - 89 x + 728 x^2 - 640 x^3), {x, 0, 16}], x] (* Michael De Vlieger, Dec 10 2019 *)

[(70*80**n-79*8**n+9)//4977 for n in range(20)]

a(n) = (70*80^n - 79*8^n + 9) / 4977.
a(n) = 81 a(n-1) - 80 a(n-2) + 8^(n-1) for n > 1.
G.f.: x / (1 - 89*x + 728*x^2 - 640*x^3).
a(n) = 89*a(n-1) - 728*a(n-2) + 640*a(n-3) for n > 2.
E.g.f.: (9*exp(x) - 79*exp(8*x) + 70*exp(80*x))/4977. - Stefano Spezia, Dec 11 2019

A328356 a(n) is the sum of all positive integers whose decimal expansion is up to n digits and does not contain the 0 digit.

Original entry on oeis.org

0, 45, 4500, 409095, 36855450, 3317322645, 298562027400, 26870609370195, 2418355085455350, 217651959870221745, 19588676407933119300, 1762980876890499197295, 158668278921733593899250, 14280145102970321446216845, 1285213059267457612117075200

Offset: 0

Pierre-Alain Sallard, Dec 10 2019

			For n=2, the sum of all integers from 1 to 99 except those containing a zero (i.e., except multiples of 10: 10, 20, ..., 90) is equal to a(2) = 4500.
For n=3, the sum of all integers from 1 to 999 except those containing a zero is equal to a(3) = 409095.

Colin Barker, Table of n, a(n) for n = 0..500
Pierre-Alain Sallard, Integer sequences A328348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (100,-909,810).

Cf. A328348, A328350, A328351, A328352, A328353, A328354, A328355.

concat(0, Vec(45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)) + O(x^15))) \\ Colin Barker, Dec 10 2019

[(80*90**n-89*9**n+9)*5//712 for n in range(20)]

a(n) = (80*90^n - 89*9^n + 9) * 5 / 712.
a(n) = 91*a(n-1) - 90*a(n-2) + 45*9^(n-1) for n > 1.
G.f.: 45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)). - Colin Barker, Dec 10 2019

cp's OEIS Frontend

A328350 Let S be any integer in the range 6 <= S <= 24. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and contains at most three distinct nonzero digits d1, d2, d3 such that d1+d2+d3 = S.

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A328351 Let S be any integer in the range 10 <= S <= 30. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and contains at most four distinct nonzero digits d1, d2, d3 and d4 such that d1+d2+d3+d4=S.

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A328352 Similar to A328350, but for 5 digits rather then 3.

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A328353 a(n)*S is the sum of all positive integers whose decimal expansion is up to n digits and uses six distinct nonzero digits d1,d2,d3,d4,d5,d6 such that d1+d2+d3+d4+d5+d6=S.

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A328354 a(n)*S is the sum of all positive integers whose decimal expansion is up to n digits and uses seven distinct nonzero digits d1,d2,d3,d4,d5,d6,d7 such that d1+d2+d3+d4+d5+d6+d7=S.

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A328355 Let S be any integer in the range 36 <= S <= 44. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and uses eight distinct nonzero digits d1,d2,d3,d4,d5,d6,d7,d8 such that d1+d2+d3+d4+d5+d6+d7+d8=S.

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A328356 a(n) is the sum of all positive integers whose decimal expansion is up to n digits and does not contain the 0 digit.

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