A136602 -id:A136602 - cp's OEIS frontend

A136601 Complement of A136602.

211, 219, 221, 227, 229, 233, 235, 239, 241, 247, 277, 284, 286, 287, 300, 302, 310, 377, 388, 390, 395, 397, 400, 403, 404, 406, 413, 469, 485, 487, 493, 494, 499, 505, 507, 508, 512, 518, 526, 559, 575, 593, 598, 607, 608, 611, 613, 616, 632, 635, 665

Offset: 1

Reinhard Zumkeller, Jan 12 2008

A066957(a(n)) = 0.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A066957 a(n) = number of ways of placing '+' and '-' among the digits 123456789 so that the result of the expression is n, '-' before 1 IS allowed.

Original entry on oeis.org

22, 43, 18, 41, 18, 40, 24, 39, 18, 46, 17, 38, 27, 38, 24, 43, 18, 39, 23, 37, 23, 43, 25, 36, 32, 36, 25, 44, 25, 35, 34, 31, 26, 37, 24, 35, 32, 32, 27, 37, 31, 26, 34, 34, 29, 42, 27, 27, 35, 26, 28, 35, 29, 33, 30, 26, 23, 29, 32, 25, 33, 30, 24, 34, 22, 26, 32, 30, 25

Offset: 0

Miklos SZABO (mike(AT)ludens.elte.hu), Feb 01 2002

A066957(9)=46 is the largest term; A066957(456789)=2 is the last term greater than 1; a(n)=0 for n>123456789; 1715 terms are greater than 0; a(A136601(n))=0; a(A136602(n))>0; a(A136603(n))=1; partial sums are A136604. - Reinhard Zumkeller, Jan 12 2008

R. Zumkeller, Table of n, a(n) for n = 0..10000
R. Zumkeller, Illustration of initial terms

Cf. A066956.

A136603 Positive numbers having a unique representation by inserting minus and plus-signs in 123456789.

Original entry on oeis.org

176, 191, 194, 196, 209, 213, 215, 217, 223, 237, 253, 259, 263, 272, 274, 275, 289, 290, 292, 293, 295, 305, 307, 308, 353, 361, 367, 379, 380, 381, 382, 386, 392, 399, 401, 405, 415, 416, 418, 419, 427, 443, 461, 463, 467, 475, 477, 479, 481, 490, 491

Offset: 1

Reinhard Zumkeller, Jan 12 2008

A066957(a(n)) = 1; subsequence of A136601;

the sequence is finite with 869 terms.

			a(1) = 176 = 1 - 2 + 34 + 56 + 78 + 9;
a(2) = 191 = 123 + 4 + 56 + 7 - 8 + 9;
a(3) = 194 = 1 - 2 + 34 + 5 + 67 + 89;
a(4) = 196 = -1 + 2 + 34 + 5 + 67 + 89;
a(5) = 209 = 1 + 234 + 56 + 7 - 89;
a(6) = 213 = 123 + 4 + 5 - 6 + 78 + 9;
a(7) = 215 = 123 + 4 - 5 + 6 + 78 + 9;
a(8) = 217 = 123 - 4 + 5 + 6 + 78 + 9;
a(9) = 223 = -1 + 23 + 45 + 67 + 89;
a(10) = 237 = -123 + 456 - 7 - 89;
a(11) = 253 = -1 + 2 + 345 - 6 - 78 - 9;
a(12) = 259 = -1 + 234 - 56 - 7 + 89;
a(869) = 123456789 = A136602(1715).

R. Zumkeller, Table of n, a(n) for n = 1..869 (full sequence)

A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

0, 3, 12, 40, 98, 253, 540, 1199, 2415, 4893, 9268, 17864, 32421, 59265, 104632, 184338, 315414, 540155, 901845, 1504173, 2461932, 4013511, 6443170, 10314675, 16281749, 25608450, 39838855, 61716941, 94682665, 144726102

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the nonnegative integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A001477 to have the same row sums for at least 8 terms: A089867, A089868, A089869, A089870, A118760, A123719, A130696, A136602, A254109, A258069, A258070, A258071, A266279, A272813, A273885, A273886, A273887, A273888.

			Illustration of the first few terms
.
0   | 0
3   | 1,  2
12  | 3,  4,  5
40  | 6,  7,  8,  9,  10
98  | 11, 12, 13, 14, 15, 16, 17
253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43
.

Cf. A373300, original version, with positive integers A000027.
Cf. A001477, the nonnegative integers.
Cf. A027480, the sequence of row sums for a regular triangle.

Module[{s = -1},
 Table[s +=
   PartitionsP[
    n - 1]; (s + PartitionsP[n]) (s + PartitionsP[n] - 1)/2 -
   s (s - 1)/2, {n, 1, 30}]]

cp's OEIS Frontend

A136601 Complement of A136602.

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A066957 a(n) = number of ways of placing '+' and '-' among the digits 123456789 so that the result of the expression is n, '-' before 1 IS allowed.

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A136603 Positive numbers having a unique representation by inserting minus and plus-signs in 123456789.

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A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

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Mathematica