A000210 -id:A000210 - cp's OEIS frontend

A082977 Numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12.

0, 1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 102, 104, 106, 108, 109, 111

Offset: 1

N. J. A. Sloane, May 31 2003

James Ingram suggests that this (with the initial 0 omitted) is the correct version of Fludd's sequence, rather than A047329. See also A083026.

Key-numbers of the pitches of a Hypophrygian mode scale on a standard chromatic keyboard, with root = 0. A Hypophrygian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone B. - James Ingram (j.ingram(AT)t-online.de), Jun 01 2003

Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Robert Fludd, Page 158 of "Utriusque Cosmi" in Beinecke Rare Book and Manuscript Library Photonegatives Collection.
Robert Fludd, Larger version of the same image
Robert Fludd, Utriusque Cosmi, Maioris scilicet et Minoris, metaphysica, physica, atque technica Historia, available as ZIP or PDF download.
Wikipedia, Robert Fludd
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A047329. Different from A000210.
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): this sequence
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032

a082977 n = a082977_list !! (n-1)
a082977_list = [0, 1, 3, 5, 6, 8, 10] ++ map (+ 12) a082977_list
-- Reinhard Zumkeller, Jan 07 2014

[n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 6, 8, 10]]; // Wesley Ivan Hurt, Jul 19 2016

A082977:=n->12*floor(n/7)+[0, 1, 3, 5, 6, 8, 10][(n mod 7)+1]: seq(A082977(n), n=0..100); # Wesley Ivan Hurt, Jul 19 2016

CoefficientList[Series[x(1 + x + 2*x^4)(1 + x + x^2)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2013 *)
fQ[n_] := MemberQ[{0, 1, 3, 5, 6, 8, 10}, Mod[n, 12]]; Select[ Range[0, 111], fQ] (* Robert G. Wilson v, Jan 07 2014 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 5, 6, 8, 10, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Floor@Range[0,2^8,12/7] (* Federico Provvedi, Oct 18 2018 *)

x='x+O('x^99); concat(0, Vec(x*(1+x+2*x^4)*(1+x+x^2)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)))) \\ Jianing Song, Sep 22 2018

G.f.: x*(1 + x + 2*x^4)*(1 + x + x^2)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - R. J. Mathar, Sep 17 2008
From Wesley Ivan Hurt, Jul 19 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
a(n) = (84*n - 105 - 2*(n mod 7) - 2*((n + 1) mod 7) + 5*((n + 2) mod 7) - 2*((n + 3) mod 7) - 2*((n + 4) mod 7) + 5*((n + 5) mod 7) - 2*((n + 6) mod 7))/49.
a(7k) = 12k - 2, a(7k-1) = 12k - 4, a(7k-2) = 12k - 6, a(7k-3) = 12k - 7, a(7k-4) = 12k - 9, a(7k-5) = 12k - 11, a(7k-6) = 12k - 12. (End)
a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018
a(n) = floor(12*(n-1)/7). - Federico Provvedi, Oct 18 2018

A185616 Lower Wythoff values for sequence A185615(n).

Original entry on oeis.org

1, 6, 12, 40, 80, 174, 273, 393, 414, 546, 786, 828, 1179, 1242, 1572, 1656, 1870, 1965, 3740, 5460, 8090, 12816, 13254, 14154, 16180, 20710, 24270, 25632, 32360, 40450, 81186, 87841, 137830, 142725, 162372, 169854, 175682, 212078, 285450

Offset: 1

Paul D. Hanna and Sean A. Irvine, Jan 31 2011

nonn

Lower Wythoff values for sequence A185615(n)

Cf. A000210, A185616, A185617.

A000210(A185615(n))

A187472 Rank transform of the sequence floor((e-1)n); complement of A187473.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 14, 17, 18, 20, 22, 24, 26, 28, 30, 32, 33, 35, 38, 40, 41, 43, 45, 47, 49, 51, 53, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 115, 117, 119, 121, 123, 125

Offset: 1

Clark Kimberling, Mar 10 2011

nonn

See A187224.

Cf. A187224, A187473.

m = E-1;
seqA = Table[Floor[m*n], {n, 1, 180}]  (* A000210 *)
seqB = Table[n, {n, 1, 80}];           (* A000027 *)
jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {_, 1}],
Flatten@Position[#1, {_, 2}]} &[Sort@Flatten[{{#1, 1} & /@ seqA,
{#1, 2} & /@ seqB}, 1]];
limseqU = FixedPoint[jointRank[{seqA, #1[[1]]}] &, jointRank[{seqA, seqB}]][[1]]                                     (* A187472 *)
Complement[Range[Length[seqA]], limseqU]  (* A187473 *)
(* by Peter J. C. Moses, Mar 10 2011 *)

A279590 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = e - 1.

Original entry on oeis.org

1, -3, 4, -3, -1, 8, -15, 18, -13, -3, 30, -63, 89, -86, 29, 97, -278, 453, -511, 314, 245, -1151, 2170, -2795, 2305, -6, -4331, 9921, -14534, 14549, -5887, -13958, 43029, -72127, 83898, -55979, -30079, 174330, -342124, 454087, -393943, 45299, 638945

Offset: 0

Clark Kimberling, Dec 16 2016

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000210.

z = 30; r = E - 1;
f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}]; f[x]
CoefficientList[Series[1/f[x], {x, 0, 2*z}], x]

G.f.: 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = e - 1.

A279632 Coefficients in the expansion of ([s] + [2s]x + [3s]x^2 + ...)/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = e - 1, s = r/(1-r).

Original entry on oeis.org

2, -2, 3, -2, -2, 8, -14, 17, -12, -5, 34, -68, 91, -80, 11, 126, -308, 467, -488, 235, 382, -1316, 2291, -2760, 1995, 638, -5220, 10738, -14725, 13447, -3007, -18467, 47914, -74806, 80821, -43890, -51936, 201548, -363193, 450980, -347117, -55972, 782359

Offset: 0

Clark Kimberling, Dec 18 2016

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000210, A054385.

z = 100;
r = E - 1; f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];
s = r/(r - 1); g[x_] := g[x] = Sum[Floor[s*(k + 1)] x^k, {k, 0, z}]
CoefficientList[Series[g[x]/f[x], {x, 0, z}], x]

G.f.: ([s] + [2s]x + [3s]x^2 + ...)/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = sqrt(2), s = r/(1-r).

A332502 Rectangular array read by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 4, 4, 3, 3, 6, 5, 5, 4, 4, 8, 7, 6, 6, 5, 5, 9, 9, 8, 7, 7, 6, 6, 11, 10, 10, 9, 8, 8, 7, 7, 12, 12, 11, 11, 10, 9, 9, 8, 8, 14, 13, 13, 12, 12, 11, 10, 10, 9, 9, 16, 15, 14, 14, 13, 13, 12, 11, 11, 10, 10, 17, 17, 16, 15, 15, 14, 14, 13

Offset: 0

Clark Kimberling, May 08 2020

Every nonnegative integer occurs exactly once in the union of row 0 and the main diagonal.
Column 0: Nonnegative integers, A001477.
Row 0: Lower Wythoff sequence, A000201.
Row 1: A026351.
Row 2: A026355 (and essentially A099267).
Main Diagonal: Upper Wythoff sequence, A001950.
Diagonal (1,4,6,9,...) = A003622;
Diagonal (3,5,8,11,...) = A026274;
Diagonal (1,3,6,8,...) = A026352;
Diagonal (2,4,7,9,...) = A026356.

			Northwest corner:
  0   1   3   4   6   8    9    11   12   14   16
  1   2   4   5   7   9    10   12   13   15   17
  2   3   5   6   8   10   11   13   14   16   18
  3   4   6   7   9   11   12   14   15   17   19
  4   5   7   8   10  12   13   15   16   18   20
  5   6   8   9   11  13   14   16   17   19   21
As a triangle (antidiagonals):
  0
  1   1
  2   2   3
  3   3   4   4
  4   4   5   5   6
  5   5   6   6   7   8
  6   6   7   7   8   9   9

Cf. A000210, A001950, A001622, A084531, A167267, A019446.

t[n_, k_] := Floor[n + k*GoldenRatio];
Grid[Table[t[n, k], {n, 0, 10}, {k, 0, 10}]] (* array *)
u = Table[t[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten  (* sequence *)

T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

A340534 a(n) is the least product of n consecutive primes that is divisible by the sum of those primes, or 0 if there is no such product.

Original entry on oeis.org

2, 0, 30, 0, 15015, 0, 37182145, 9699690, 33426748355, 0, 3710369067405, 0, 304250263527210, 0, 37420578814667938361329, 0, 18598027670889965365580513, 0, 107254825578022430263302818471, 0, 44510752614879308559270669665465, 0, 267064515689275851355624017992790, 0, 116431182179248680450031658440253681535, 0

Offset: 1

J. M. Bergot and Robert Israel, Jan 10 2021

nonn

a(27) > 10^225 if it is not 0.
If n is even, a(n) is either A002110(n) or 0.
a(n) = A002110(n) for n in A051838.

			a(5) = 15015 = 3*5*7*11*13 is the product of 5 consecutive primes and is divisible by 3+5+7+11+13 = 39.

Cf. A000210, A051838, A086487.

f:= proc(n) local L,i,p;
   L:= [seq(ithprime(i),i=1..n)]:
   p:= convert(L,`*`);
   if n::even then
     if p mod convert(L,`+`) = 0 then return p else return 0 fi
   else
     do
       p:= convert(L,`*`);
       if p mod convert(L,`+`) = 0 then return p fi;
       if p > 10^225 then return FAIL fi;
       L:= [op(L[2..-1]),nextprime(L[-1])];
     od
   fi;
end proc:
map(f, [$1..26]);

cp's OEIS Frontend

A082977 Numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12.

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A185616 Lower Wythoff values for sequence A185615(n).

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A187472 Rank transform of the sequence floor((e-1)n); complement of A187473.

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A279590 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = e - 1.

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A279632 Coefficients in the expansion of ([s] + [2s]x + [3s]x^2 + ...)/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = e - 1, s = r/(1-r).

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A332502 Rectangular array read by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

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A340534 a(n) is the least product of n consecutive primes that is divisible by the sum of those primes, or 0 if there is no such product.

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Maple