A071817 -id:A071817 - cp's OEIS frontend

A090579 Number of numbers with 4 decimal digits and sum of digits = n.

1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 279, 342, 405, 465, 519, 564, 597, 615, 615, 597, 564, 519, 465, 405, 342, 279, 219, 165, 120, 84, 56, 35, 20, 10, 4, 1

Offset: 1

Hugo Pfoertner, Jan 12 2004

There are 9000 numbers with 4 decimal digits, the smallest being 1000 and the largest 9999.

			a(2)=4: 1001, 1010, 1100, 2000.

Cf. A071817 3-digit numbers, A090580 5-digit numbers, A090581 6-digit numbers.

nn=36;Drop[CoefficientList[Series[(x-x^10)/(1-x)(1-x^10)^3/(1-x)^3,{x,0,nn}],x],1] (* Geoffrey Critzer, Feb 09 2014 *)

b=vector(36,i,0);for(n=1000,9999,a=eval(Vec(Str(n)));b[sum(j=1,4,a[j])]++);for(n=1,36,print1(b[n],",")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

G.f.: (x-x^10)/(1-x)*((1-x^10)/(1-x))^3. - Geoffrey Critzer, Feb 09 2014

A090580 Number of numbers with 5 decimal digits and sum of digits = n.

Original entry on oeis.org

1, 5, 15, 35, 70, 126, 210, 330, 495, 714, 992, 1330, 1725, 2170, 2654, 3162, 3675, 4170, 4620, 4998, 5283, 5460, 5520, 5460, 5283, 4998, 4620, 4170, 3675, 3162, 2654, 2170, 1725, 1330, 992, 714, 495, 330, 210, 126, 70, 35, 15, 5, 1

Offset: 1

Hugo Pfoertner, Jan 12 2004

There are 90000 numbers with 5 decimal digits, the smallest being 10000 and the largest 99999.

			a(2)=5: 10001, 10010, 10100, 11000, 20000.

Cf. A071817 3-digit numbers, A090579 4-digit numbers, A090581 6-digit numbers.

Rest@ CoefficientList[Series[(x - x^10)/(1 - x) ((1 - x^10)/(1 - x))^4, {x, 0, 45}], x] (* or *)
Function[w, Count[w, #] & /@ Range[Max@ w]]@ Map[Total@ IntegerDigits@ # &, Range[10^#, 10^(# + 1) - 1]] &@ 4 (* Michael De Vlieger, Dec 07 2016 *)
Tally[Total[IntegerDigits[#]]&/@Range[10000,99999]][[All,2]] (* Harvey P. Dale, Jul 08 2019 *)

b=vector(45,i,0);for(n=10000,99999,a=eval(Vec(Str(n)));b[sum(j=1,5,a[j])]++);for(n=1,45,print1(b[n],",")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

G.f.: (x - x^10)/(1 - x)*((1 - x^10)/(1 - x))^4. - Michael De Vlieger, Dec 07 2016

A090581 Number of numbers with 6 decimal digits and sum of digits = n.

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 462, 792, 1287, 2001, 2992, 4317, 6027, 8162, 10746, 13782, 17247, 21087, 25212, 29496, 33787, 37917, 41712, 45002, 47631, 49467, 50412, 50412, 49467, 47631, 45002, 41712, 37917, 33787, 29496, 25212, 21087, 17247, 13782, 10746, 8162, 6027, 4317, 2992, 2001, 1287, 792, 462, 252, 126, 56, 21, 6, 1

Offset: 1

Hugo Pfoertner, Jan 12 2004

There are 900000 6-digit numbers, the smallest being 100000 and the largest 999999.

			a(2)=6: 100001, 100010, 100100, 101000, 110000, 200000.

Cf. A071817 3-digit numbers, A090579 4-digit numbers, A090580 5-digit numbers.

Rest@ CoefficientList[Series[(x - x^10)/(1 - x) ((1 - x^10)/(1 - x))^#, {x, 0, 9 (# + 1)}], x] &@ 5 (* or *)
Function[w, Count[w, #] & /@ Range[Max@ w]]@ Map[Total@ IntegerDigits@ # &, Range[10^#, 10^(# + 1) - 1]] &@ 5 (* Michael De Vlieger, Dec 07 2016 *)

b=vector(54,i,0);for(n=100000,999999,a=eval(Vec(Str(n)));b[sum(j=1,6,a[j])]++);for(n=1,54,print1(b[n],",")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

G.f.: (x - x^10)/(1 - x)*((1 - x^10)/(1 - x))^5. - Michael De Vlieger, Dec 07 2016

A071816 Number of ordered solutions to x+y+z = u+v+w, 0 <= x, y, z, u, v, w < n.

Original entry on oeis.org

1, 20, 141, 580, 1751, 4332, 9331, 18152, 32661, 55252, 88913, 137292, 204763, 296492, 418503, 577744, 782153, 1040724, 1363573, 1762004, 2248575, 2837164, 3543035, 4382904, 5375005, 6539156, 7896825, 9471196, 11287235, 13371756

Offset: 1

Graeme McRae, Jun 07 2002

Number of 6-digit numbers in base n (with leading zeros allowed) such that the sum of the first three digits equals the sum of the last three digits.

a(n) = largest coefficient of (1+...+x^(n-1))^6. - R. H. Hardin, Jul 23 2009

			For n = 2 there are 20 ordered solutions (x,y,z,u,v,w) to x+y+z = u+v+w: (0,0,0,0,0,0), (0,0,1,0,0,1), (0,0,1,0,1,0), (0,0,1,1,0,0), (0,1,0,0,0,1), (0,1,0,0,1,0), (0,1,0,1,0,0), (0,1,1,0,1,1), (0,1,1,1,0,1), (0,1,1,1,1,0), (1,0,0,0,0,1), (1,0,0,0,1,0), (1,0,0,1,0,0), (1,0,1,0,1,1), (1,0,1,1,0,1), (1,0,1,1,1,0), (1,1,0,0,1,1), (1,1,0,1,0,1), (1,1,0,1,1,0), (1,1,1,1,1,1).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. B. Nathanson, Growth polynomials for additive quadruples and (h, k)-tuples, arXiv preprint arXiv:1305.7172 [math.NT], 2013.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A008528, A071817, A077042.

First differences are in A070302.

[n*(11*n^4+5*n^2+4)/20: n in [1..30]]; // Vincenzo Librandi, Sep 05 2011

A071816 := proc(n) n*(11*n^4+5*n^2+4)/20 ; end proc: # R. J. Mathar, Sep 04 2011

The sum of the squares of the number of different 3-digit numbers that add up to k (summed over all possible k's) - cf. A071817.
a(n) = A077042(n,6).
a(n) = n*(11*n^4+5*n^2+4)/20. Recurrence: a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). G.f.: x*(1+14*x+36*x^2+14*x^3+x^4)/(1-x)^6. - Vladeta Jovovic, Jun 09 2002

New definition from Vladeta Jovovic, Jun 09 2002
Comment revised by Franklin T. Adams-Watters, Jul 27 2009
Edited by N. J. A. Sloane, Jul 28 2009

A278969 Number of 7-digit numbers whose sum of digits is n.

Original entry on oeis.org

1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5004, 7995, 12306, 18312, 26418, 37038, 50568, 67353, 87648, 111573, 139068, 169863, 203463, 239148, 275988, 312873, 348558, 381723, 411048, 435303, 453438, 464653, 468448, 464653, 453438, 435303, 411048, 381723, 348558, 312873, 275988, 239148, 203463, 169863, 139068, 111573, 87648, 67353, 50568, 37038, 26418, 18312, 12306, 7995, 5004, 3003, 1716, 924, 462, 210, 84, 28, 7, 1

Offset: 1

Daniel Mondot, Dec 02 2016

There are 9000000 numbers with 7 decimal digits, the smallest being 1000000 and the largest 9999999.

Differs for n >= 10 (5004 vs 5005) from A000579(n+5) = binomial(n+5,6). - M. F. Hasler, Mar 05 2017

			a(2)=7: 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000.

A071817 (3-digit numbers), A090579 (4-digit numbers), A090580 (5-digit numbers), A090581 (6-digit numbers), A278971 (8-digit numbers).

Rest@ CoefficientList[Series[(x - x^10)/(1 - x) ((1 - x^10)/(1 - x))^#, {x, 0, 9 (# + 1)}], x] &@ 6 (* or *)
Function[w, Count[w, #] & /@ Range[Max@ w]]@ Map[Total@ IntegerDigits@ # &, Range[10^#, 10^(# + 1) - 1]] &@ 6 (* Michael De Vlieger, Dec 07 2016 *)

b=vector(63, i, 0); for(n=1000000, 9999999, a=eval(Vec(Str(n))); b[sum(j=1, 7, a[j])]++); for(n=1, 63, print1(b[n], ", "))

Vec((1-x^9)*(1-x^10)^6/(1-x)^7) \\ shorter than (1-x^9)/(1-x)*((1-x^10)/(1-x))^6, but not better. - M. F. Hasler, Mar 05 2017

G.f.: (x - x^10)/(1 - x)*((1 - x^10)/(1 - x))^6. - Michael De Vlieger, Dec 07 2016

a(64-n) = a(n), 1 <= n <= 63. - M. F. Hasler, Mar 05 2017

A278971 Number of 8-digit numbers whose sum of digits is n.

Original entry on oeis.org

1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11439, 19433, 31732, 50016, 76350, 113178, 163284, 229713, 315645, 424215, 558279, 720147, 911304, 1132140, 1381710, 1657545, 1955535, 2269905, 2593305, 2917035, 3231405, 3526195, 3791180, 4016685, 4194135, 4316565, 4379055, 4379055, 4316565, 4194135, 4016685, 3791180, 3526195, 3231405, 2917035, 2593305, 2269905, 1955535, 1657545, 1381710, 1132140, 911304, 720147, 558279, 424215, 315645, 229713, 163284, 113178, 76350, 50016, 31732, 19433, 11439, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1

Offset: 1

Daniel Mondot, Dec 02 2016

There are 90000000 numbers with 8 decimal digits, the smallest being 10000000 and the largest 99999999.

			a(2)=8: 10000001, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000.

Cf. A071817 (3-digit numbers), A090579 (4-digit numbers), A090580 (5-digit numbers), A090581 (6-digit numbers), A278969 (7-digit numbers).

Rest@ CoefficientList[Series[(x - x^10)/(1 - x) ((1 - x^10)/(1 - x))^#, {x, 0, 9 (# + 1)}], x] &@ 7 (* Michael De Vlieger, Dec 07 2016 *)

b=vector(72, i, 0); for(n=10000000, 99999999, a=eval(Vec(Str(n))); b[sum(j=1, 8, a[j])]++); for(n=1, 72, print1(b[n], ", "))

G.f.: (x - x^10)/(1 - x)*((1 - x^10)/(1 - x))^7. - Michael De Vlieger, Dec 07 2016

A289354 Number of 9-digit numbers whose sum of digits is n.

Original entry on oeis.org

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24309, 43741, 75465, 125445, 201675, 314523, 477015, 705012, 1017225, 1435005, 1981845, 2682559, 3562131, 4644255, 5949615, 7493982, 9286233, 11326425, 13604085, 16096905, 18770031, 21576079, 24455955, 27340500, 30152925, 32811945, 35235465, 37344615, 39067875, 40344975, 41130255, 41395240, 41130255, 40344975, 39067875, 37344615, 35235465, 32811945, 30152925, 27340500, 24455955, 21576079, 18770031, 16096905, 13604085, 11326425, 9286233, 7493982, 5949615, 4644255, 3562131, 2682559, 1981845, 1435005, 1017225, 705012, 477015, 314523, 201675, 125445, 75465, 43741, 24309, 12870, 6432, 3003, 1287, 495, 165, 45, 9, 1

Offset: 1

Miquel Cerda, Jul 03 2017

There are 900000000 numbers with 9 decimal digits, the smallest being 100000000 and the largest 999999999.

			a(2)=9: 100000001, 100000010, 100000100, 100001000, 100010000, 100100000, 101000000, 11000000, 20000000.

Miquel Cerda, Table of n, a(n) for n = 1..81

Cf. A071817 (3-digit numbers), A090579 (4-digit numbers), A090580 (5-digit numbers), A090581 (6-digit numbers), A278969 (7-digit numbers), A278971 (8-digit numbers).

With[{d = 9}, Rest@ CoefficientList[Series[(x - x^10)/(1 - x) ((1 - x^10)/(1 - x))^(d - 1), {x, 0, 9 d}], x]] (* Michael De Vlieger, Jul 04 2017 *)

a(n) = a(82 - n). - David A. Corneth, Jul 03 2017

G.f.: (x - x^10)/(1 - x)*((1 - x^10)/(1 - x))^8 - Michael De Vlieger, Jul 04 2017

A289410 Irregular triangular array T(m,k) with m (row) >= 1 and k (column) >= 1 read by rows: number of m-digit numbers whose digit sum is k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 54, 61, 66, 69, 70, 69, 66, 61, 54, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 279, 342, 405, 465, 519, 564, 597, 615, 615, 597, 564, 519, 465, 405, 342, 279, 219, 165, 120, 84

Offset: 1

Miquel Cerda, Jul 05 2017

The m-th row is palindromic; T(m,k) = T(m,9*m+1-k).

			The irregular triangle T(m,k) begins:
m\k  1  2  3  4  5   6   7   8   9   10   11  12   13   14  15  16  17  18  19
1    1  1  1  1  1   1   1   1   1;
2    1  2  3  4  5   6   7   8   9    9    8   7    6    5   4   3   2   1;
3    1  3  6  10 15  21  28  36  45   54   61  66   69   70  69  66  61  54 45,...;
4    1  4  10 20 35  56  84  120 165  219  279 342  405  465,...;
5    1  5  15 35 70  126 210 330 495  714  992 1330 1725,...;
6    1  6  21 56 126 252 462 792 1287 2001 2992,...;
etc.
Row m(2), column k(4) there are 4 numbers of 2-digits whose digits sum = 4: 13, 22, 31, 40.

Miquel Cerda, Rows n=1..10 of triangle, flattened

The row sums = 9*10^(m-1) = A052268(n). The row lengths = 9*m = A008591(n). The middle diagonal = A071976. (row m=3) = A071817, (row m=4) = A090579, (row m=5) = A090580, (row m=6) = A090581, (row m=7) = A278969, (row m=8) = A278971, (row m=9) = A289354, (column k=3) = A000217, (column k=4) = A000292, (column k=5) = A000332, (column k=6) = A000389, (column k=7) = A000579, (column k=8) = A000580, (column k=9) = A000581, (column k=10) = A035927.

row:= proc(m) local g; g:= normal((1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m);
seq(coeff(g,x,j),j=1..9*m) end proc:
seq(row(k),k=1..5); # Robert Israel, Jul 19 2017

G.f. of row m: (1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m.

G.f. as array: (1+x+x^2)*(1+x^3+x^6)*x*y/(1-y*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)). - Robert Israel, Jul 19 2017

Edited by Robert Israel, Jul 19 2017

A289380 Number of 10-digit numbers whose sum of digits is n.

Original entry on oeis.org

1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48619, 92359, 167815, 293215, 494725, 808753, 1284481, 1986490, 2997280, 4419415, 6376951, 9015769, 12502435, 17021245, 22769185, 29948644, 38757862, 49379275, 61966135, 76628035, 93416221, 112309741, 133203565, 155899810, 180103120

Offset: 1

Miquel Cerda, Jul 04 2017

The 10-digit numbers distributed according to the sum of their digits n.

The sequence is symmetrical; a(n) = a(91 - n), 1 <= n <= 91.

			a(2)=10: 1000000001, 1000000010, 1000000100, 1000001000, 1000010000, 1000100000, 1001000000, 1010000000, 110000000, 200000000.

Miquel Cerda, Table of n, a(n) for n = 1..90

Cf. A071817 (3-digit numbers), A090579 (4-digit numbers), A090580 (5-digit numbers), A090581 (6-digit numbers), A278969 (7-digit numbers), A278971 (8-digit numbers), A289354 (9-digit numbers).

CoefficientList[Series[(1 - x^10)^9*(1 - x^9)/(1 - x)^10, {x, 0, 40}],
x] (* Wesley Ivan Hurt, Jul 09 2017 *)

G.f.: (1 - x^10)^9*(x - x^10)/(1 - x)^10.

A382531 Number of n-digit base-10 numbers whose digit sum is equal to ceiling(9*n/2).

Original entry on oeis.org

1, 9, 70, 615, 5520, 50412, 468448, 4379055, 41395240, 392406145, 3748943890, 35866068766, 345143007910, 3323483518810, 32150758083580, 311088525668335, 3021445494584902, 29344719005694973, 285904843977651598, 2785022004925340460, 27203012941819689340

Offset: 1

Miquel Cerda, Mar 30 2025

Digit sum ceiling(9*n/2) = A130877(n+1) has highest frequency among all n-digit base-10 numbers.

The count excludes numbers with leading zeros.

			a(2) = 9, the 2-digit numbers with digit sum 9 are: 18, 27, 36, 45, 54, 63, 72, 81, 90.

Alois P. Heinz, Table of n, a(n) for n = 1..1002

Cf. A210736 (analogous for base-2 digits).
Cf. A025015 (maximal coefficient of (1+...+x^9)^n).
Cf. A007953, A071817, A071976, A130877.

b:= proc(n, s, t) option remember; `if`(9*n b(n, ceil(9*n/2), 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Apr 12 2025

a(n) = [x^ceiling(9*n/2)] (f^n - f^(n-1)) with f = (x^10-1)/(x-1). - Alois P. Heinz, Apr 12 2025

cp's OEIS Frontend

A090579 Number of numbers with 4 decimal digits and sum of digits = n.

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A090580 Number of numbers with 5 decimal digits and sum of digits = n.

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A090581 Number of numbers with 6 decimal digits and sum of digits = n.

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A071816 Number of ordered solutions to x+y+z = u+v+w, 0 <= x, y, z, u, v, w < n.

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A278969 Number of 7-digit numbers whose sum of digits is n.

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A278971 Number of 8-digit numbers whose sum of digits is n.

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A289354 Number of 9-digit numbers whose sum of digits is n.

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