cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 24 results. Next

A195836 Triangle read by rows which arises from A195826 in the same way as A175003 arises from A195310. Column k starts at row A000217(k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 5, -2, 13, 7, -3, -1, 16, 10, -4, -1, 21, 13, -5, -1, 28, 16, -7, -2, 35, 21, -10, -3, 43, 28, -13, -4, 1, 55, 35, -16, -5, 1, 70, 43, -21, -7, 1, 86, 55, -28, -10, 2, 105, 70, -35, -13, 3, 130, 86, -43, -16, 4
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. It appears that this sequence is related to the generalized hexagonal numbers (A000217), A195826 and A006950 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. It appears that row sums give A006950. See comments in A195825.

Examples

			Written as a triangle:
.  1,
.  1,
.  1,   1,
.  2,   1,
.  3,   1,
.  4,   2,  -1,
.  5,   3,  -1,
.  7,   4,  -1,
. 10,   5,  -2,
. 13,   7,  -3,   -1,
. 16,  10,  -4,   -1,
. 21,  13,  -5,   -1,
. 28,  16,  -7,   -2,
. 35,  21,  -10,  -3,
. 43,  28,  -13,  -4,   1,
. 55,  35,  -16,  -5,   1,
. 70,  43,  -21,  -7,   1,
. 86,  55,  -28, -10,   2,

Crossrefs

Cf. A000217, A006950, A175003, A195825, A195826, A195837, A195838, A195939, A195840, A195841, A195842, A195843.

A195837 Triangle read by rows which arises from A195827, in the same way as A175003 arises from A195310. Column k starts at row A085787(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 5, -3, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 14, -7, -1, 33, 16, -10, -2, 37, 21, -12, -3, 1, 44, 27, -14, -4, 1, 54, 33, -16, -4, 1, 68, 37, -21, -5, 1, 80, 44, -27, -7, 2
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. It appears that this sequence is related to the generalized heptagonal numbers A085787, A195827 and A036820 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. It appears that row sums give A036820. See comments in A195825.

Examples

			Written as a triangle:
.  1;
.  1;
.  1;
.  1,   1;
.  2,   1;
.  3,   1;
.  4,   1,  -1;
.  4,   2,  -1;
.  5,   3,  -1;
.  7,   4,  -1;
. 10,   4,  -2;
. 12,   5,  -3;
. 14,   7,  -4,  -1;
. 16,  10,  -4,  -1;
. 21,  12,  -5,  -1;
. 27,  14,  -7,  -1;
. 33,  16, -10,  -2;
. 37,  21, -12,  -3,  1;
. 44,  27, -14,  -4,  1;
. 54,  33, -16,  -4,  1;

Crossrefs

Cf. A085787, A036820, A175003, A195825, A195826, A195827, A195836, A195838, A195839, A195840, A195841, A195842, A195843.

A195838 Triangle read by rows which arises from A195828, in the same way as A175003 arises from A195310. Column k starts at row A001082(k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 14, -10, -1, 35, 16, -12, -2, 1, 38, 21, -13, -3, 1, 44, 32, -14, -4, 1
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized octagonal numbers A001082, A195828 and A195848 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1,   1;
.  2,   1;
.  3,   1;
.  4,   1,  -1;
.  4,   1,  -1;
.  4,   2,  -1;
.  5,   3,  -1;
.  7,   4,  -1;
. 10,   4,  -2;
. 12,   4,  -3;
. 13,   5,  -4;
. 14,   7,  -4,  -1;
. 16,  10,  -4,  -1;
. 21,  12,  -5,  -1;
. 27,  13,  -7,  -1;
. 32,  14, -10,  -1;
. 35,  16, -12,  -2,   1;
. 38,  21, -13,  -3,   1;

Crossrefs

Row sums give A195848.
Cf. A001082, A175003, A195828, A195836, A195837.

A195839 Triangle read by rows which arises from A195829, in the same way as A175003 arises from A195310. Column k starts at row A118277(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 13, -10, -1, 34, 14, -12, -1, 36, 16, -13, -2, 1
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized enneagonal numbers A118277, A195829 and A195849 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1;
.  1,  1;
.  2,  1;
.  3,  1;
.  4,  1, -1;
.  4,  1, -1;
.  4,  1, -1;
.  4,  2, -1;
.  5,  3, -1;
.  7,  4, -1;
. 10,  4, -2;
. 12,  4, -3;
. 13,  4, -4;
. 13,  5, -4;
. 14,  7, -4, -1;
. 16, 10, -4, -1;
. 21, 12, -5, -1;

Crossrefs

Row sums give A195849.
Cf. A001082, A175003, A195828, A195836, A195837, A195838.

A195840 Triangle read by rows which arises from A195830, in the same way as A175003 arises from A195310. Column k starts at row A074377(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 13, -10, -1, 34
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized decagonal numbers A074377, A195830 and A195850 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1;
.  1;
.  1,  1;
.  2,  1;
.  3,  1;
.  4,  1,  -1;
.  4,  1,  -1;
.  4,  1,  -1;
.  4,  1,  -1;
.  4,  2,  -1;
.  5,  3,  -1;
.  7,  4,  -1;
. 10,  4,  -2;
. 12,  4,  -3;
. 13,  4,  -4;
. 13,  4,  -4;
. 13,  5,  -4;
. 14,  7,  -4,  -1;
. 16, 10,  -4,  -1;
. 21, 12,  -5,  -1;
. 27, 13,  -7,  -1;
. 32, 13, -10,  -1;
. 34, 13, -12,  -1,  1;

Crossrefs

Row sums give A195850.
Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839.

A195841 Triangle read by rows which arises from A195831, in the same way as A175003 arises from A195310. Column k starts at row A195160(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized hendecagonal numbers A195160, A195831 and A195851 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Crossrefs

Row sums give A195851.
Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840.

A195842 Triangle read by rows which arises from A195832, in the same way as A175003 arises from A195310. Column k starts at row A195162(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized dodecagonal numbers A195162, A195832 and A195852 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Crossrefs

Row sums give A195852.
Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840, A195841.

A195843 Triangle read by rows which arises from A195833, in the same way as A175003 arises from A195310. Column k starts at row A195313(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized tridecagonal numbers A195313, A195833 and A196933 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Crossrefs

Row sums give A196933.
Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840, A195841, A195842.

A210954 Triangle read by rows which arises from A210944 in the same way as A175003 arises from A195310. Column k starts at row A195818(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1
Offset: 1

Views

Author

Omar E. Pol, Jun 16 2012

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. Also 1 together with the row sums give A210964. This sequence is related to the generalized 14-gonal numbers A195818, A210954 and A210964 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as an irregular triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1;
1;
1,  1;
2,  1;
3,  1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  2, -1;
5,  3, -1;
7,  4, -1;
10, 4, -2;
12, 4, -3;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 5, -4;
14, 7, -4, -1;

Crossrefs

Cf. A175003, A195818, A195825, A195836, A195837, A195838, A195839, A195840, A195841, A195842, A195843, A210944, A210964.

A195311 Row sums of A195310.

Original entry on oeis.org

0, 1, 3, 5, 7, 10, 13, 17, 21, 25, 29, 33, 38, 43, 48, 54, 60, 66, 72, 78, 84, 90, 97, 104, 111, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 199, 208, 217, 226, 235, 245, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 356, 367, 378, 389, 400
Offset: 1

Views

Author

Omar E. Pol, Sep 21 2011

Keywords

Examples

			If written as an irregular triangle in which row n has length A026741(n+1) then the first differences in row n are always n (see below).
Triangle begins:
0,
1,3,5,
7,10,
13,17,21,25,29,
33,38,43,
48,54,60,66,72,78,84,
90,97,104,111,
118,126,134,142,150,158,166,174,182,
190,199,208,217,226,
235,245,255,265,275,285,295,305,315,325,335,
345,356,367,378,389,400

Crossrefs

Cf. A001318, A026741, A195309, A195310.

Programs

  • Mathematica
    a1318[n_] := If[EvenQ[n], n (3 n/2 + 1)/4, (n + 1) (3 n + 1)/8];
a[n_] := DeleteCases[Table[n - a1318[k], {k, 1, n}], _?Negative] // Total;
Array[a, 56] (* Jean-François Alcover, Jun 26 2019 *)
  • Sage
    def A195311(n):
    return add(max(0,n-k*(3*k-1)/2)+max(0,n-k*(3*k+1)/2) for k in (1..n))
[A195311(n) for n in (1..56)]  # Peter Luschny, Oct 12 2012
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