A026121 -id:A026121 - cp's OEIS frontend

A185282 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of n-element subsets that can be chosen from {1,2,...,2*n^k} having element sum n^(k+1).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 3, 0, 1, 1, 7, 36, 7, 0, 1, 1, 15, 351, 785, 18, 0, 1, 1, 31, 3240, 56217, 26404, 51, 0, 1, 1, 63, 29403, 3695545, 18878418, 1235580, 155, 0, 1, 1, 127, 265356, 238085177, 12107973904, 11163952389, 74394425, 486, 0

Offset: 0

Alois P. Heinz, Jan 25 2012

A(n,k) is the number of partitions of n^(k+1) into n distinct parts <= 2*n^k.

			A(0,0) = 1: {}.
A(1,1) = 1: {1}.
A(2,2) = 3: {1,7}, {2,6}, {3,5}.
A(3,1) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
A(4,1) = 7: {1,2,5,8}, {1,2,6,7}, {1,3,4,8}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}.
A(2,3) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.
Square array A(n,k) begins:
  1,   1,      1,         1,            1, ...
  1,   1,      1,         1,            1, ...
  0,   1,      3,         7,           15, ...
  0,   3,     36,       351,         3240, ...
  0,   7,    785,     56217,      3695545, ...
  0,  18,  26404,  18878418,  12107973904, ...

Rows n=1-3 give: A000012, A000225, A026121.

Columns k=1-3 give: A202261, A186730, A185062.

b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n^(k+1), 2*n^k, n):
seq(seq(A(n, d-n), n=0..d), d=0..8);

$RecursionLimit = 10000; b[n_, i_, t_] := b[n, i, t] = If [i < t || n < t*(t+1)/2 || n > t*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]]; A[0, ] = A[1, ] = 1; A[n_ /; n > 1, 0] = 0; A[n_, k_] := b[n^(k+1), 2*n^k, n]; Table[Print[ta = Table [A[n, d-n], {n, 0, d}]]; ta, {d, 0, 9}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A121544 Sum of all proper base 4 numbers with n digits (those not beginning with 0).

Original entry on oeis.org

6, 114, 1896, 30624, 491136, 7862784, 125822976, 2013241344, 32212156416, 515395682304, 8246335635456, 131941389041664, 2111062300164096, 33776997104615424, 540431954881806336, 8646911282940739584, 138350580546379186176, 2213609288819376390144

Offset: 1

Jonathan Vos Post, Sep 08 2006

Sum of the first 3 * 4^(n-1) integers starting with 4^(n-1).
Sum of the integers from 4^(n-1) to 4^n -1.
First differences of A026337.

			a(1) = 6 = 1 + 2 + 3.
a(2) = 114 = 10_4 + 11_4 + 12_4 + 13_4 + 20_4 + 21_4 + 22_4 + 23_4 + 30_4 + 31_4 + 32_4 + 33_4 = (4+5+6+7+8+9+10+11+12+13+14+15)_10.

G. C. Greubel, Table of n, a(n) for n = 1..820
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A007090, A010036, A026121, A026337, A101291.

[3*Binomial(5*4^(n-1), 2)/5: n in [1..20]]; // G. C. Greubel, Nov 07 2024

Table[3*4^(n-1)*(5*4^(n-1) - 1)/2, {n,20}] (* James C. McMahon, Oct 19 2024 *)

def A121544(n): return 3*binomial(5*4^(n-1), 2)//5
[A121544(n) for n in range(1,21)] # G. C. Greubel, Nov 07 2024

a(n) = 3 * 4^(n-1) * (4^(n-1) + 4^n - 1)/2.
G.f.: 6*x*(1-x) / ((1-4*x)*(1-16*x)). - Colin Barker, Apr 30 2013
From G. C. Greubel, Nov 07 2024: (Start)
a(n) = (3/5)*binomial(5*4^(n-1), 2).
E.g.f.: (3/32)*(-1 - 4*exp(4*x) + 5*exp(16*x)). (End)

More terms from Colin Barker, Apr 30 2013

Edited by Michel Marcus, Apr 15 2024

A226508 a(n) = Sum_{i=3^n..3^(n+1)-1} i.

Original entry on oeis.org

3, 33, 315, 2889, 26163, 235953, 2125035, 19129689, 172180323, 1549662273, 13947078555, 125524061289, 1129717614483, 10167461718993, 91507165036875, 823564514029689, 7412080712360643, 66708726669526113, 600378540800575995, 5403406869529706889

Offset: 0

Michel Marcus, Jun 10 2013

Partial sums give 3, 36, 351, 3240, 29403,...: A026121.

a(n) is the sum of all integers having n+1 digits in their ternary expansion (without leading zeros). - Jonathan Vos Post, Sep 07 2006

			a(0) = 1+2 = 3,
a(1) = 3+4+5+6+7+8 = 33.

Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A010035, A010036 (base 2), A026121, A101291 (base 10).

Cf. A007089 (numbers in base 3).

Table[3^(n - 1) (4 3^(n + 1) - 3), {n, 0, 20}] (* Bruno Berselli, Jun 11 2013 *)
LinearRecurrence[{12,-27},{3,33},30] (* Harvey P. Dale, Jun 19 2013 *)

a(n) = sum(i=3^n, 3^(n+1)-1, i) \\ Michel Marcus, Jun 11 2013

G.f.: 3*(1-x)/(1-12*x+27*x^2). [Bruno Berselli, Jun 11 2013]
a(n) = 3^(n-1)*(4*3^(n+1)-3). [Bruno Berselli, Jun 11 2013]
a(0)=3, a(1)=33, a(n)=12*a(n-1)-27*a(n-2). - Harvey P. Dale, Jun 19 2013

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A185282 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of n-element subsets that can be chosen from {1,2,...,2*n^k} having element sum n^(k+1).

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A121544 Sum of all proper base 4 numbers with n digits (those not beginning with 0).

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A226508 a(n) = Sum_{i=3^n..3^(n+1)-1} i.

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