A131062 -id:A131062 - cp's OEIS frontend

A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 21, 13, 1, 1, 17, 37, 37, 17, 1, 1, 21, 57, 77, 57, 21, 1, 1, 25, 81, 137, 137, 81, 25, 1, 1, 29, 109, 221, 277, 221, 109, 29, 1, 1, 33, 141, 333, 501, 501, 333, 141, 33, 1, 1, 37, 177, 477, 837, 1005, 837, 477, 177, 37, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131062: (1, 2, 7, 20, 49, 110, 235, ...); the binomial transform of (1, 1, 4, 4, 4, ...).

Triangle equals 4*A007318 - 3*A000012 as infinite lower triangular matrices. - Emeric Deutsch, Jun 21 2007

			First few rows of the triangle are
  1;
  1,  1;
  1,  5,  1;
  1,  9,  9,  1;
  1, 13, 21, 13,  1;
  1, 17, 37, 37, 17,  1;
  1, 21, 57, 77, 57, 21, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A109128, A123203, A131060, A131063, A131064, A131065, A131066, A131067, A131068.

[4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 4*binomial(n, k)-3 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007

Table[4*Binomial[n, k] -3, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 21 2007

More terms from Emeric Deutsch, Jun 21 2007

A123203 a(n) = 2^(n+1) - 3*n.

Original entry on oeis.org

1, 2, 7, 20, 49, 110, 235, 488, 997, 2018, 4063, 8156, 16345, 32726, 65491, 131024, 262093, 524234, 1048519, 2097092, 4194241, 8388542, 16777147, 33554360, 67108789, 134217650, 268435375, 536870828, 1073741737, 2147483558

Offset: 1

Gary W. Adamson, Jun 13 2007

An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 186, leads to this sequence. For the central square this vector leads to the companion sequence A036563. - Johannes W. Meijer, Aug 15 2010

			a(4) = 20, row sums of 4th row of triangle A131062: (1, 9, 9, 1).
a(4) = 20 = (1, 3, 3, 1) dot (1, 1, 4, 4) = (1 + 3 + 12 + 4).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1, 4.
Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.

[2^(n+1) -3*n: n in [1..40]]; // G. C. Greubel, Sep 14 2024

Table[2^(n+1) - 3*n, {n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
LinearRecurrence[{4,-5,2},{1,2,7},40] (* Harvey P. Dale, Mar 30 2024 *)

def A123203(n): return 2^(n+1) -3*n
[A123203(n) for n in range(1,41)] # G. C. Greubel, Sep 14 2024

Binomial transform of [1, 1, 4, 4, 4, ...].
Equals row sums of triangle A131061.
From Johannes W. Meijer, Aug 15 2010; corrected by Colin Barker, Jul 28 2012: (Start)
a(n) = 2^(1+n) - 3*n.
a(n) = 3*A000295(n-1) + A000079(n-1).
(End)
G.f.: x*(1 - 2*x + 4*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Jul 28 2012
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Colin Barker, Jul 29 2012
E.g.f.: 2*exp(2*x) - 3*x*exp(x) - 2. - G. C. Greubel, Sep 14 2024

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008

Title changed by G. C. Greubel, Sep 14 2024

A214832 Integer part of A440 piano key frequencies, start with A0 = the 1st key.

Original entry on oeis.org

27, 29, 30, 32, 34, 36, 38, 41, 43, 46, 48, 51, 55, 58, 61, 65, 69, 73, 77, 82, 87, 92, 97, 103, 110, 116, 123, 130, 138, 146, 155, 164, 174, 184, 195, 207, 220, 233, 246, 261, 277, 293, 311, 329, 349, 369, 391, 415, 440, 466, 493, 523, 554, 587, 622, 659, 698, 739, 783, 830, 880, 932, 987, 1046, 1108, 1174, 1244, 1318, 1396, 1479, 1567, 1661, 1760, 1864, 1975, 2093, 2217, 2349, 2489, 2637, 2793, 2959, 3135, 3322, 3520, 3729, 3951, 4186

Offset: 1

Jon Perry, Mar 07 2013

A254531(a(k)) = k, k = 1..88. - Reinhard Zumkeller, Feb 04 2015

			Middle C is 261.626 Hz so a(40) = 261.

BGfL, A Virtual Keyboard
Wikipedia, Piano Key Frequencies
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Cf. A059620, A079731, A131062, A101285, A131071.

Cf. A254531, A010774.

a214832 = floor . (* 440) . (2 **) . (/ 12) . fromIntegral . subtract 49
-- Reinhard Zumkeller, Nov 23 2014

for (i=1;i<=88;i++) document.write(Math.floor(Math.pow(2,(i-49)/12)*440)+", ");

PARI
```
a(n)=floor(440*2^((n-49)/12));
```

a(n) = floor[2^((n-49)/12)*440] (Hz) for 1 <= n <= 88.

A131071 12-note scale in Hertz (rounded to integers).

Original entry on oeis.org

261, 275, 293, 309, 330, 348, 366, 391, 412, 440, 464, 495, 521

Offset: 1

Hans Isdahl, Sep 24 2007

nonn

Wikipedia, Pythagorean tuning.
Index entries for sequences related to music.

Cf. A131062 for the corresponding C major scale. [M. F. Hasler, Oct 07 2011]

Cf. A214832.

The scale involves 9/8 and 256/243 as fractions and the start is A = 440 Hz.

The initial term (rounded frequency of the C) is calculated as 16/27 * 440 Hz = 260.74 Hz, cf. the Wikipedia page on Pythagorean tuning for the ratios of the frequencies. - M. F. Hasler, Oct 07 2011

A319727 Rounded frequencies of notes in the shruti scale of Indian classical music, starting with 260.7 Hertz for C-equivalent note.

Original entry on oeis.org

261, 275, 278, 290, 293, 309, 313, 326, 330, 348, 352, 367, 371, 391, 412, 417, 435, 440, 464, 469, 489, 495, 521, 549, 556, 579, 587, 618, 626, 652, 660, 695, 704, 733, 743, 782, 824, 834, 869, 880, 927, 939, 978, 990

Offset: 1

Jim Singh, Sep 26 2018

nonn

A shruti can be interpreted as the smallest interval of pitch the ear can detect and a singer or musical instrument can produce, and accordingly the 'Grama' system divides an octave into 22 parts.
The scale involves 256/243, 25/24 and 81/80 as fractions.
Note that ((81/80)^10) * ((256/243)^7) * ((25/24)^5) = 2.
The frequencies correspond to the ratios [1/1, 256/243, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 81/64, 4/3, 27/20, 45/32, 729/512, 3/2, 128/81, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 243/128, 2/1].
The start is A-equivalent note = 440 Hz. The initial term (rounded frequency of C-equivalent note) is calculated as (16/27) * 440 Hz = 260.7 Hz.

Wikipedia, Shruti (music)

Cf. A131071, A131062.

Ratios={[1/1, 256/243, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 81/64, 4/3, 27/20, 45/32, 729/512, 3/2, 128/81, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 243/128];}
a(n)={n--; round(440*16/27*2^(n\22)*Ratios[n%22+1])} \\ Andrew Howroyd, Sep 27 2018

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A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

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A123203 a(n) = 2^(n+1) - 3*n.

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A214832 Integer part of A440 piano key frequencies, start with A0 = the 1st key.

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A131071 12-note scale in Hertz (rounded to integers).

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A319727 Rounded frequencies of notes in the shruti scale of Indian classical music, starting with 260.7 Hertz for C-equivalent note.

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