A077340 -id:A077340 - cp's OEIS frontend

A077339 Triangle in which n-th row contains the first n numbers beginning with n.

1, 2, 20, 3, 30, 31, 4, 40, 41, 42, 5, 50, 51, 52, 53, 6, 60, 61, 62, 63, 64, 7, 70, 71, 72, 73, 74, 75, 8, 80, 81, 82, 83, 84, 85, 86, 9, 90, 91, 92, 93, 94, 95, 96, 97, 10, 100, 101, 102, 103, 104, 105, 106, 107, 108, 11, 110, 111, 112, 113, 114, 115, 116, 117, 118

Offset: 1

Amarnath Murthy, Nov 05 2002

Repetitions are allowed.

The first term in row m is T(m,1)=m, followed by T(m,2)=10m, 10m+1,..., 10m+9; then, T(m,12)=100m, 100m+1, ..., 100m+99, then T(m,112)=1000m,...,1000m+999, etc. (as long as the second index does not exceed the first, given the definition of this triangular sequence. But in principle one can consider the n-th number "starting with m" without that restriction). The first repetitions to occur seem to be T(20,1)=20=T(2,2), T(30,1)=30=T(3,2), T(31,1)=31=T(3,3), ... These cause the differences with A077341, and also the differences between A077340 and A077343. - M. F. Hasler, Jan 07 2013

			The triangle starts:
1;
2,  20;
3,  30,  31;
4,  40,  41,  42;
5,  50,  51,  52,  53;
6,  60,  61,  62,  63,  64;
7,  70,  71,  72,  73,  74,  75;
8,  80,  81,  82,  83,  84,  85,  86;
9,  90,  91,  92,  93,  94,  95,  96,  97;
10, 100, 101, 102, 103, 104, 105, 106, 107, 108;
11, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119;
12, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 1200; ...

Cf. A077340. Different from A077341.

A077339(m,n,p=1)={!n & n=m+(1-m=(sqrtint(8*m-7)+1)\2)*m\2; while(n>p,n-=p;p*=10);p*m+n-1} \\ returns the n-th number starting with m, allowing also n>m (useful for A077341 ff), or the m-th term of the sequence, if no 2nd arg is given. \\ - M. F. Hasler, Jan 07 2013

Edited by N. J. A. Sloane, Jun 12 2007

A077341 Triangle in which n-th row contains the first n numbers, not already present, that begin with n.

Original entry on oeis.org

1, 2, 20, 3, 30, 31, 4, 40, 41, 42, 5, 50, 51, 52, 53, 6, 60, 61, 62, 63, 64, 7, 70, 71, 72, 73, 74, 75, 8, 80, 81, 82, 83, 84, 85, 86, 9, 90, 91, 92, 93, 94, 95, 96, 97, 10, 100, 101, 102, 103, 104, 105, 106, 107, 108, 11, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 12

Offset: 1

Amarnath Murthy, Nov 05 2002

Every positive number appears exactly once. Proof: N will appear as the first term of row N, unless it has already appeared. And no term is repeated, by definition.

The first term of the 20th row is 200 unlike the 20th row of A077339 where it is 20.

			Triangle begins:
1
2 20
3 30 31
4 40 41 42
5 50 51 52 53
...

M. F. Hasler, Table of n, a(n) for n = 1..5350

Cf. A077339, A077340 (see there for differences), A077342, A077343.

{u=[];for(m=1,19,o=0;for(n=1,m,while(setsearch(u,a=A077339(m,n+o)),o++);u=setunion(u,Set(a));print1(a",")))} \\ M. F. Hasler, Jan 08 2013

Edited by N. J. A. Sloane, Jun 12 2007

A077343 Final terms of rows in A077341.

Original entry on oeis.org

1, 20, 31, 42, 53, 64, 75, 86, 97, 108, 119, 1200, 1301, 1402, 1503, 1604, 1705, 1806, 1907, 2009, 2109, 2210, 2311, 2412, 2513, 2614, 2715, 2816, 2917, 3019, 3120, 3220, 3321, 3422, 3523, 3624, 3725, 3826, 3927, 4029, 4130, 4231, 4331, 4432, 4533, 4634, 4735, 4836, 4937, 5039

Offset: 1

Amarnath Murthy, Nov 05 2002

Differs from A077340 in a(20), a(30), a(31), a(40..42), a(50..53), ... - M. F. Hasler, Jan 07 2013

M. F. Hasler and Hans Havermann, Table of n, a(n) for n = 1..10000 (terms 1..199 from _M. F. Hasler_)

Cf. A077339, A077340, A077341, A077342.

{u=[];for(m=1,199,o=0;for(n=1,m,while(setsearch(u,a=A077339(m,n+o)),o++);u=setunion(u,Set(a)));print1(a","))} \\ - M. F. Hasler, Jan 07 2013

More terms from M. F. Hasler, Jan 07 2013

A077342 First column of A077341.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 200, 21, 22, 23, 24, 25, 26, 27, 28, 29, 300, 310, 32, 33, 34, 35, 36, 37, 38, 39, 400, 410, 420, 43, 44, 45, 46, 47, 48, 49, 500, 510, 520, 530, 54, 55, 56, 57, 58, 59, 600, 610, 620, 630, 640, 65, 66, 67

Offset: 1

Amarnath Murthy, Nov 05 2002

a(20) = 200 and not 20. Can anyone derive a formula for a(n)?

a(n) = n when n > 10 * floor(n/10) + floor(n/10) - 2. Otherwise it is guaranteed that n has appeared in a previous row of the triangle. - Alonso del Arte, Jan 07 2013

Hans Havermann, Table of n, a(n) for n = 1..10000

Cf. A077339, A077340, A077341, A077343.

A081929 Sum of the n-th row of A077339.

Original entry on oeis.org

1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, 2467, 3979, 5692, 7606, 9721, 12037, 14554, 17272, 20191, 23311, 26632, 30154, 33877, 37801, 41926, 46252, 50779, 55507, 60436, 65566, 70897, 76429, 82162, 88096, 94231, 100567, 107104

Offset: 1

Amarnath Murthy, Apr 01 2003

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Not the same as A069178 (A069178(12)=1387 <> 2467=a(12) ). Cf. A077339, A077340.

For n>10, a(n) = 1/2 * (201n^2 - 2001n + 2).
From Chai Wah Wu, Jan 12 2023: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 13.
G.f.: x*(900*x^12 - 1080*x^11 - x^2 - 19*x - 1)/(x - 1)^3. (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

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A077339 Triangle in which n-th row contains the first n numbers beginning with n.

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A077341 Triangle in which n-th row contains the first n numbers, not already present, that begin with n.

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A077343 Final terms of rows in A077341.

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A077342 First column of A077341.

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A081929 Sum of the n-th row of A077339.

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