login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a131062 -id:a131062
Displaying 1-5 of 5 results found.
page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A131061
Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.
+10
13
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
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
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
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 21 2007
EXAMPLE
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;
...
MAPLE
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
MATHEMATICA
Table[4*Binomial[n, k] -3, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
PROG
(Magma) [4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
(Sage) [[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020
CROSSREFS
Cf. A109128, A123203, A131060, A131063, A131064, A131065, A131066, A131067, A131068.
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jun 13 2007
EXTENSIONS
More terms from Emeric Deutsch, Jun 21 2007
STATUS
approved
A123203
a(n) = 2^(n+1) - 3*n.
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
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).
FORMULA
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
EXAMPLE
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).
MATHEMATICA
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 *)
PROG
(Magma) [2^(n+1) -3*n: n in [1..40]]; // G. C. Greubel, Sep 14 2024
(SageMath)
def A123203(n): return 2^(n+1) -3*n
[A123203(n) for n in range(1, 41)] # G. C. Greubel, Sep 14 2024
CROSSREFS
Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jun 13 2007
EXTENSIONS
More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008
Title changed by G. C. Greubel, Sep 14 2024
STATUS
approved
A214832
Integer part of A440 piano key frequencies, start with A0 = the 1st key.
+10
5
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A254531(a(k)) = k, k = 1..88. - Reinhard Zumkeller, Feb 04 2015
LINKS
Table of n, a(n) for n=1..88.
BGfL, A Virtual Keyboard
Wikipedia, Piano Key Frequencies
Wikipedia, Twelfth root of two
Index entries for sequences based on music
FORMULA
a(n) = floor[2^((n-49)/12)*440] (Hz) for 1 <= n <= 88.
EXAMPLE
Middle C is 261.626 Hz so a(40) = 261.
PROG
(JavaScript)
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));
(Haskell)
a214832 = floor . (* 440) . (2 **) . (/ 12) . fromIntegral . subtract 49
-- Reinhard Zumkeller, Nov 23 2014
CROSSREFS
Cf. A059620, A079731, A131062, A101285, A131071.
Cf. A254531, A010774.
KEYWORD
nonn,fini,full
AUTHOR
Jon Perry, Mar 07 2013
STATUS
approved
A131071
12-note scale in Hertz (rounded to integers).
+10
4
261, 275, 293, 309, 330, 348, 366, 391, 412, 440, 464, 495, 521
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..13.
Wikipedia, Pythagorean tuning.
Index entries for sequences related to music.
FORMULA
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
CROSSREFS
Cf. A131062 for the corresponding C major scale. [M. F. Hasler, Oct 07 2011]
Cf. A214832.
KEYWORD
nonn
AUTHOR
Hans Isdahl, Sep 24 2007
STATUS
approved
A319727
Rounded frequencies of notes in the shruti scale of Indian classical music, starting with 260.7 Hertz for C-equivalent note.
+10
0
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..44.
Wikipedia, Shruti (music)
PROG
(PARI)
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
CROSSREFS
Cf. A131071, A131062.
KEYWORD
nonn
AUTHOR
Jim Singh, Sep 26 2018
STATUS
approved
page 1

Search completed in 0.006 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified October 26 00:42 EDT 2024. Contains 377104 sequences.
License Agreements, Terms of Use, Privacy Policy