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.

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A153004 First differences of toothpick numbers A153003.

Original entry on oeis.org

1, 3, 3, 3, 6, 9, 6, 3, 6, 9, 9, 12, 21, 24, 12, 3, 6, 9, 9, 12, 21, 24, 15, 12, 21, 27, 30, 45, 66, 60, 24, 3, 6, 9, 9, 12, 21, 24, 15, 12, 21, 27, 30, 45, 66, 60, 27, 12, 21, 27, 30, 45, 66, 63, 42, 45, 69, 84, 105, 156, 192, 144, 48, 3
Offset: 1

Views

Author

Omar E. Pol, Jan 02 2009

Keywords

Examples

			Triangle begins:
1;
3,3;
3,6,9,6;
3,6,9,9,12,21,24,12;
3,6,9,9,12,21,24,15,12,21,27,30,45,66,60,24;

Links

Crossrefs

Cf. A139251, A152968, A152978, A152998, A153000, A153003.
Toothpick sequence: A139250.

Programs

  • Mathematica
    A139250[n_] := A139250[n] = Module[{m, k}, If[n == 0, Return[0]]; m = 2^(Length[IntegerDigits[n, 2]] - 1); k = (2 m^2 + 1)/3; If[n == m, k, k + 2 A139250[n - m] + A139250[n - m + 1] - 1]];
b[n_] := If[n == 0, 0, (3/4) (A139250[n + 1] - 3) + 1];
b /@ Range[0, 64] // Differences (* Jean-François Alcover, Apr 06 2020 *)

A160413 a(n) = A160411(n+1)/4.

Original entry on oeis.org

2, 1, 7, 2, 13, 3, 19, 4, 25, 5, 31, 6, 37, 7, 43, 8, 49, 9, 55, 10, 61, 11, 67, 12, 73, 13, 79, 14, 85, 15, 91, 16, 97, 17, 103, 18, 109, 19, 115, 20, 121, 21, 127, 22, 133, 23, 139, 24, 145, 25, 151, 26, 157, 27, 163, 28, 169, 29, 175, 30, 181, 31, 187, 32, 193
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Links

Crossrefs

Cf. A152978, A160117 A160411, A160417.

Programs

  • Mathematica
    LinearRecurrence[{0, 2, 0, -1}, {2, 1, 7, 2, 13}, 100] (* Amiram Eldar, Feb 02 2024 *)

Formula

G.f.: x*(x^4 + 3*x^2 + x+2) / ((x-1)^2*(x+1)^2). - Colin Barker, Mar 04 2013
From Colin Barker, Apr 06 2013: (Start)
a(n) = -1 + (-1)^n - (1/4)*(-7 + 5*(-1)^n)*n for n > 1.
a(n) = 2*a(n-2) - a(n-4) for n > 5. (End)

Extensions

More terms from Colin Barker, Apr 06 2013
More terms from Amiram Eldar, Feb 02 2024

A160417 a(n) = A160415(n+1)/4.

Original entry on oeis.org

2, 1, 7, 1, 7, 3, 21, 1, 7, 3, 21, 3, 21, 9, 63, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21, 9, 63, 9, 63, 27, 189, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21, 9, 63, 9, 63, 27, 189, 3, 21, 9, 63, 9, 63, 27, 189, 9, 63, 27, 189, 27, 189, 81, 567, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Crossrefs

Cf. A152978, A160118, A160413, A160415.

Programs

  • Mathematica
    With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; f[n_] := If[OddQ[n], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; f[0] = 0; f[1] = 1; Differences[Array[f, 100]]/4] (* Amiram Eldar, Feb 02 2024 *)

Extensions

More terms from Max Alekseyev, Dec 12 2011
More terms from Amiram Eldar, Feb 02 2024

A172309 Number of L-toothpicks added to the L-toothpick structure of A172308 (First quadrant) at the n-th stage.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 4, 2, 4, 6, 6, 8, 6, 12, 10, 6, 4, 6, 10, 12, 10, 14, 14, 16, 16, 18, 14, 16, 12, 20, 22, 8, 8, 16, 18, 16, 10, 16, 20, 24, 26, 36, 24, 32, 30, 40, 28, 24
Offset: 0

Views

Author

Omar E. Pol, Feb 06 2010

Keywords

Links

Crossrefs

Cf. First differences of A172308.
Cf. A139250, A139251, A152978, A172304, A172305, A172306, A172307, A172308, A172310, A172311, A172312, A172313.

Formula

a(0) = 0, a(n) = A172305(n+1)/4, for n>=1.

Extensions

More terms from Colin Barker, Apr 19 2015

A160727 a(n) = A161415(n+1)/4.

Original entry on oeis.org

2, 3, 7, 3, 9, 9, 23, 3, 9, 9, 27, 9, 27, 27, 73, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 227, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 697, 3, 9, 9, 27, 9
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Examples

			From _Omar E. Pol_, Jan 01 2014: (Start)
Written as an irregular triangle in which row lengths is A000079 the sequence begins:
2;
3,7;
3,9,9,23;
3,9,9,27,9,27,27,73;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,227;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27, 81,81,243,27,81,81,243,81,243,243,697;
(End)

Links

Crossrefs

Cf. A152978, A153000, A160414, A161415.

Programs

  • Mathematica
    A160727[n_]:=3^DigitCount[n,2,1]-If[IntegerQ[Log2[n+1]],(n+1)/2,0];Array[A160727,100] (* Paolo Xausa, Sep 01 2023 *)

Formula

a(n) = A048883(n), except a(n) = A048883(n) - (n+1)/2 if n is a power of 2 minus 1. - Omar E. Pol, Jan 06 2014

Extensions

a(11)-a(58) from M. F. Hasler, Dec 03 2012
a(59)-a(68) from Omar E. Pol, Jan 06 2014

A152973 Records in A152968.

Original entry on oeis.org

1, 2, 4, 6, 8, 14, 16, 18, 20, 30, 44, 46, 56, 70, 104, 128, 130, 148, 182, 244, 336, 352, 354, 372, 478, 608, 824, 1024, 1026, 1048, 1054, 1198, 1564, 2040, 2672, 2976, 2978, 3000, 3150, 3960, 5168, 6752, 8320, 8322, 8344, 8348, 8494, 8550, 9306
Offset: 1

Views

Author

Omar E. Pol, Dec 16 2008

Keywords

Comments

Excluding the initial 1, this sequence also gives the records in A151688. - Nathaniel Johnston, Apr 10 2011

Links

Crossrefs

Toothpick structure: A139250.
Cf. A139251, A139252, A152968.
Cf. A152978.

Extensions

More terms from Omar E. Pol, Dec 20 2008
a(17)-a(49) from Nathaniel Johnston, Apr 10 2011

A168113 First differences of A168112.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 6, 7, 6, 3, 6, 7, 8, 9, 16, 17, 10, 3, 6, 7, 8, 9, 16, 17, 12, 9, 16, 19, 22, 31, 46, 41, 18, 3, 6, 7, 8, 9, 16, 17, 12, 9, 16, 19, 22, 31, 46, 41, 20, 9, 16, 19, 22, 31, 46, 43, 30, 31, 48
Offset: 1

Views

Author

Omar E. Pol, Dec 07 2009

Keywords

Comments

a(n) is the number of toothpicks added at the n-th round in the toothpick structure of A168112.

Examples

			If written as a triangle, begins:
1;
1;
2;
3,3;
3,6,7,6;
3,6,7,8,9,16,17,10;
3,6,7,8,9,16,17,12,9,16,19,22,31,46,41,18;
Rows converge to A168114.

Links

Crossrefs

Cf. A139250, A139251, A152978, A152980, A160407, A168112, A168114.

Formula

Observation: It appears that a(2^i+2) = 3, for i>0.
a(n) = A160731(n)/2. [From Nathaniel Johnston, Mar 28 2011]

Extensions

Terms after a(34) from Nathaniel Johnston, Mar 28 2011

A222173 Number of pentagons added at n-th stage to the structure of A222172.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 1, 2, 4, 4, 3, 5, 7, 4, 1, 2, 4, 6, 8
Offset: 0

Views

Author

Omar E. Pol, Mar 21 2013

Keywords

Comments

Essentially the first differences of A222172.
Also number of P-toothpicks added at n-th stage to the P-toothpick structure of A222172.
First differs from A152978 at a(9).

Examples

			Apparently this is an irregular triangle:
0;
1, 1;
1, 2, 3, 2;
1, 2, 4, 4, 3, 5, 7, 4;
1, 2, 4, 6, 8,...

Links

Crossrefs

Cf. A139251, A152978, A222172, A222177, A222180, A222181.

Formula

a(n) = A222181(n+2)/10 = A222177(n+1)/2, n >= 1.

A152999 Primes in toothpick sequence A152998.

Original entry on oeis.org

3, 5, 7, 11, 17, 23, 47, 61, 97, 103, 151, 173, 191, 211, 241, 347, 353, 359, 367, 397, 467, 541, 599, 607, 659, 733, 1109, 1237, 1367, 1439, 1453, 1471, 1663, 2029, 2357, 2399, 2671, 2797, 3373, 3607, 3719, 3911, 4241, 5479, 5501, 5527, 5701, 5741, 5779, 5923
Offset: 1

Views

Author

Omar E. Pol, Dec 23 2008

Keywords

Links

Crossrefs

Cf. A139250, A139251, A139253, A152768, A152968, A152978, A152998, A153000, A153002, A153006, A153009.

Extensions

More terms from Jinyuan Wang, Mar 15 2020

A159785 a(n) = A152980(n)*3.

Original entry on oeis.org

3, 6, 9, 9, 12, 21, 24, 15, 12, 21, 27, 30, 45, 66, 60, 27, 12, 21, 27, 30, 45, 66, 63, 42, 45, 69, 84, 105, 156, 192, 144, 51, 12, 21, 27, 30, 45, 66, 63, 42, 45, 69, 84, 105, 156, 192, 147, 66, 45, 69, 84, 105, 156, 195, 168, 129, 159, 222, 273, 366, 504, 528
Offset: 1

Views

Author

Omar E. Pol, May 02 2009

Keywords

Links

Crossrefs

Toothpick sequence: A139250.
Cf. A139251, A147646, A152968, A152978, A152980, A153004, A153006.

Extensions

More terms from 3*A152980(n) by Jinyuan Wang, Mar 14 2020
Previous Showing 11-20 of 23 results. Next

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