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.

Previous Showing 31-37 of 37 results.

A246336 Partial sums of A151548.

Original entry on oeis.org

1, 4, 9, 16, 21, 32, 49, 64, 69, 80, 97, 116, 137, 176, 225, 256, 261, 272, 289, 308, 329, 368, 417, 452, 473, 512, 565, 624, 705, 832, 961, 1024, 1029, 1040, 1057, 1076, 1097, 1136, 1185, 1220, 1241, 1280, 1333, 1392, 1473, 1600, 1729, 1796, 1817, 1856, 1909, 1968, 2049, 2176
Offset: 0

Views

Author

N. J. A. Sloane, Aug 30 2014

Keywords

Comments

Arises in the analysis of a certain 2-D cellular automaton (see A169707).
a(46) = 1729 is also the Hardy-Ramanujan number. - Omar E. Pol, Feb 17 2015
It appears that sums of two successive terms give the numbers greater than 1 in A194811. - Omar E. Pol, Mar 05 2015

Links

Crossrefs

Cf. A169707, A151548, A160552, A170903.

Formula

G.f.: 1/(1-x^2) + (4*x/(1-x))*mul(1+x^(2^k-1)+2*x^(2^k),k=1..oo).
From Omar E. Pol, Feb 18 2015: (Start)
It appears that:
a(2^k-2) = (2^k-1)^2, if k >= 1.
a(2^k-1) = 4^k, if k >= 1.
a(2^k) = 4^k + 5, if k >= 1.
(End)

A266530 Partial sums of A266529.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 10, 11, 12, 15, 18, 23, 28, 35, 42, 43, 44, 47, 50, 55, 60, 67, 74, 79, 84, 95, 106, 123, 140, 155, 170, 171, 172, 175, 178, 183, 188, 195, 202, 207, 212, 223, 234, 251, 268, 283, 298, 303, 308, 319, 330, 347, 364, 383, 402, 423, 444, 483, 522, 571, 620, 651, 682, 683, 684, 687, 690, 695, 700
Offset: 1

Views

Author

Omar E. Pol, Jan 02 2016

Keywords

Comments

Also the toothpick sequence A139250 and twice the terms of A255747 interleaved.
It appears that this sequence has a fractal (or fractal-like) behavior.
First differs from A266510 at a(55), with which it shares infinitely many terms.
First differs from A266540 at a(25), with which it shares infinitely many terms.

Links

Crossrefs

Cf. A139250, A160552, A255747, A266510, A266529, A266540.

Programs

  • Mathematica
    Accumulate[Riffle[#, #] &@ Nest[Join[#, 2 # + Append[Rest@#, 1]] &, {0}, 6]] (* Ivan Neretin, Feb 09 2017 *)

Formula

a(2n-1) = A139250(n).
a(2n) = 2 * A255747(n-1).
a(n) = (a(n-1) + a(n+1))/2, if n is an odd number greater than 1.

A163311 Triangle read by rows in which the diagonals give the infinite set of Toothpick sequences.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 7, 5, 1, 5, 10, 11, 7, 1, 6, 13, 19, 15, 10, 1, 7, 16, 29, 25, 23, 13, 1, 8, 19, 41, 37, 40, 35, 14, 1, 9, 22, 55, 51, 61, 67, 43, 16, 1, 10, 25, 71, 67, 86, 109, 94, 47, 19, 1, 11, 28, 89, 85, 115, 161, 173, 100, 55, 22, 1, 12, 31, 109, 105, 148, 223, 286, 181
Offset: 1

Views

Author

Gary W. Adamson, Jul 24 2009

Keywords

Comments

Apart from the second diagonal (which gives the toothpick sequence A139250), the rest of the diagonals cannot be represented with toothpick structures. - Omar E. Pol, Dec 14 2016

Examples

			Triangle begins:
1;
1, 2;
1, 3,  4;
1, 4,  7,  5;
1, 5,  10, 11,  7;
1, 6,  13, 19,  15,  10;
1, 7,  16, 29,  25,  23,  13;
1, 8,  19, 41,  37,  40,  35,  14;
1, 9,  22, 55   51,  61,  67,  43,  16;
1, 10, 25, 71,  67,  86,  109, 94,  47,  19;
1, 11, 28, 89,  85,  115, 161, 173, 100, 55,  22;
1, 12, 31, 109, 105, 148, 223, 286, 181, 115, 67,  25;
1, 13, 34, 131, 127, 185, 295, 439, 296, 205, 142, 79,  30;
1, 14, 37, 155, 151, 226, 377, 638, 451, 331, 253, 175, 95, 36;
...

Links

Crossrefs

Row sums = A163312: (1, 3, 8, 17, 34, 64,...).
Right border = A163267, toothpick sequence for N=1.
Next diagonal going to the left = A139250, toothpick sequence for N=2.
Then 1, 4, 10, 19,... = A162958, toothpick sequence for N=3.
Cf. A160552, A162958, A163311, A163312.

Formula

See A162958 for rules governing the generation of N-th Toothpick sequences. By way of example, (N+2), A139250. The generator is A160552, which uses the multiplier "2". Then A160552 convolved with (1, 2, 2, 2,...) = A139250 the Toothpick sequence for N=2. Similarly, we create an array for Toothpick sequences N=1, 2, 3,...etc = A163267, A139250, A162958,...; then take the antidiagonals, creating triangle A163311.

A162957 A162956 convolved with (1, 3, 0, 0, 0, ...).

Original entry on oeis.org

1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 364, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 370, 40, 91, 148, 178, 211, 415, 553, 421, 211, 421, 622, 745, 1048, 1792, 2005, 1093
Offset: 1

Views

Author

Gary W. Adamson, Jul 18 2009

Keywords

Comments

Rows of A162956 tend to this sequence. Analogous to rows of A160552 tending to A151548.

Crossrefs

Cf. A162956, A162958.

Formula

Equals (1, 3, 0, 0, 0, ...) * (1, 1, 4, 1, 4, 7, 13, 1, 4, 7, 13, 7, 19, 34, 40, ...).

Extensions

a(20), a(21) corrected and more terms from Georg Fischer, Jul 17 2025

A151549 a(n) = (A151548(n)-1)/2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 8, 7, 2, 5, 8, 9, 10, 19, 24, 15, 2, 5, 8, 9, 10, 19, 24, 17, 10, 19, 26, 29, 40, 63, 64, 31, 2, 5, 8, 9, 10, 19, 24, 17, 10, 19, 26, 29, 40, 63, 64, 33, 10, 19, 26, 29, 40, 63, 66, 45, 40, 65, 82, 99, 144, 191, 160, 63, 2, 5, 8, 9, 10, 19, 24, 17, 10, 19, 26, 29, 40, 63
Offset: 0

Views

Author

N. J. A. Sloane, May 18 2009

Keywords

Examples

			From _Omar E. Pol_, Nov 30 2013: (Start)
Written as an irregular triangle in which row lengths is A011782 the right border gives A000225.
Triangle begins:
0;
1;
2,3;
2,5,8,7;
2,5,8,9,10,19,24,15;
2,5,8,9,10,19,24,17,10,19,26,29,40,63,64,31;
2,5,8,9,10,19,24,17,10,19,26,29,40,63,64,33,10,19,26,29,40,63,66,45,40,65,82,99,144,191,160,63;
(End)

Links

Crossrefs

Cf. A160552.

A160704 Jacobsthal sequence A001045 convolved with A139251 (first differences of toothpick numbers).

Original entry on oeis.org

1, 3, 9, 19, 41, 87, 181, 363, 729, 1463, 2933, 5871, 11753, 23523, 47061, 94123, 188249, 376503, 753013, 1506031, 3012073
Offset: 1

Views

Author

Gary W. Adamson, May 24 2009

Keywords

Examples

			a(4) = 19 = (1, 1, 3, 5) dot (4, 4, 2, 1) = (4 + 4 + 6 + 5).
a(4) = 19 = 2*9 + 1, where "1" = A160552(4), A160552 = (1, 1, 3, 1, 3, 5, 7,...)
Using the latter method, we create a heading = row 1:
1,...1,...3,...1,....3,...5,....7,....1,....3,.....5, = A160552 with offset 1.
1...3,....9,...19,..41,..87,..181,..363,..729,..1463,...
...such that a(n) = 2*a(n-1) = term above a(n), or a(n) = 2*a(n-1) + A160552(n). Example: a(6) = 87 = 2*41 + 5, where 5 = A160552(6).

Links

Crossrefs

Cf. A139251, A160552, A139250, A001045

Formula

a(n) = 2*a(n-1) + A160552, where A160552 begins with offset 1: (1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5,...).

A255046 a(n) = (1 + A151548(n))/2.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 9, 8, 3, 6, 9, 10, 11, 20, 25, 16, 3, 6, 9, 10, 11, 20, 25, 18, 11, 20, 27, 30, 41, 64, 65, 32, 3, 6, 9, 10, 11, 20, 25, 18, 11, 20, 27, 30, 41, 64, 65, 34, 11, 20, 27, 30, 41, 64, 67, 46, 41, 66, 83, 100, 145, 192, 161, 64, 3, 6, 9, 10, 11, 20
Offset: 0

Views

Author

Omar E. Pol, Feb 15 2015

Keywords

Comments

It appears that when A255045 is regarded as a triangle with rows of lengths 1, 2, 4, 8, 16, ..., this is what the rows converge to.

Examples

			Written as an irregular triangle in which the row lengths are the terms of A011782:
1;
2;
3,4;
3,6,9,8;
3,6,9,10,11,20,25,16;
3,6,9,10,11,20,25,18,11,20,27,30,41,64,65,32;
3,6,9,10,11,20,25,18,11,20,27,30,41,64,65,34,11,20,27,30,41,64,67,46,41,66,83,100,145,192,161,64;
...
Right border gives A000079.

Links

Crossrefs

Cf. A000079, A011782, A139251, A151548, A160552, A255045.
Previous Showing 31-37 of 37 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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