_Georg Fischer - cp's OEIS frontend

A378234 From higher-order arithmetic progressions: Corrected version of A259461.

40, 5000, 472500, 43218000, 4148928000, 432081216000, 49509306000000, 6275893932000000, 881135508052800000, 136878615942868800000, 23474682634201999200000, 4432282735129048800000000, 918537831584839065600000000, 208281986149676045967360000000, 51516317681413623440962560000000

Offset: 0

Georg Fischer, Dec 16 2024

nonn

Only the first 5 terms of A259461 are correct. - R. J. Mathar, Jul 14 2015

"2 over n!" on page 13 in the Dienger article is A006472; A_3 is A001303.

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Cf. A001303, A006472, A259459, A259460, A259461.

rV := proc(n,a,d)
    n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259461 := proc(n)
    mul(rV(i,a,d),i=1..n+3) ;
    coeftayl(%,d=0,3) ;
    coeftayl(%,a=0,n) ;
end proc:
seq(A259461(n),n=1..5) ; # R. J. Mathar, Jul 14 2015

D-finite with recurrence: -2*n*(n+2)*a(n) + (n+4)^3*(n+5)*a(n-1) = 0.

a(n) = (n+5)!*(n+4)!^3 / (1296*2^(n+4)*n!^2*(n+2)*(n+1)).

A373101 Triangle read by rows, T(n,k) = (binomial(n,k)^3 - binomial(n,k))/6 for k=1..n-1 and n >= 2.

Original entry on oeis.org

1, 4, 4, 10, 35, 10, 20, 165, 165, 20, 35, 560, 1330, 560, 35, 56, 1540, 7140, 7140, 1540, 56, 84, 3654, 29260, 57155, 29260, 3654, 84, 120, 7770, 98770, 333375, 333375, 98770, 7770, 120, 165, 15180, 287980, 1543465, 2667126, 1543465, 287980, 15180, 165

Offset: 2

Georg Fischer, May 23 2024

This triangle was mentioned in A143420 with the wrong A-number A143419.

			T(n,k) for n=2..7:
   1;
   4,    4;
  10,   35,   10;
  20,  165,  165,   20;
  35,  560, 1330,  560,   35;
  56, 1540, 7140, 7140, 1540, 56;

Cf. A143418, A143420.

seq(print(n,seq((binomial(n,k)^3 - binomial(n,k))/6,k=1..n-1)),n=2..10);

A363839 Numbers in the witch's multiplication table (German: "Hexeneinmaleins") in Goethe's Faust.

Original entry on oeis.org

1, 10, 2, 3, 4, 5, 6, 7, 8, 9, 1, 10, 0, 1

Offset: 1

Georg Fischer, Jun 23 2023

In German, the witch declaims the following magic spell to Faust:
  Du mußt verstehn!
  Aus Eins[1] mach' Zehn[10],
  Und Zwey[2] laß gehn,
  Und Drey[3] mach' gleich,
  So bist du reich.
  Verlier' die Vier[4]!
  Aus Fünf[5] und Sechs[6],
  So sagt die Hex',
  Mach' Sieben[7] und Acht[8],
  So ist's vollbracht:
  Und Neun[9] ist Eins[1],
  Und Zehn[10] ist keins[0].
  Das ist das Hexen-Einmal-Eins[1]!
(The numbers in "[...]" do not apppear in the original.)
Literal translation into English:
  You must understand!
  From one make ten,
  And two let go,
  And three make equal,
  So you are rich.
  Lose the four!
  From five and six,
  So says the witch,
  Make seven and eight,
  So it is finished:
  And nine is one,
  And ten is none.
  This is the Witch's one-times-one!
.
From Peter Luschny, Jun 23 2023: (Start)
Sometimes the witch's one-times-one is interpreted as a construction guide for a magic square, which describes a transformation like this:
 1  2  3        4  9  2
 4  5  6   ->   3  5  7
 7  8  9        8  1  6
(End)

Johann Wolfgang von Goethe: Faust - Der Tragödie erster Teil. Page 161 and page 162. Tübingen: Cotta, 1808. Digital full-text edition at Wikisource, version of Aug 18 2016.
Poetry in Translation, Faust Scene VI, translated by A. S. Kline, 2003.
Wikipedia (German), Hexeneinmaleins.

Cf. A080992, A108579.

A356686 Decimal expansion of the constant p^*_0 related to Shallit's constant (A086276).

Original entry on oeis.org

1, 4, 4, 7, 0, 5, 4, 3, 5, 0, 0, 1, 6, 2, 7, 9, 4, 0, 6, 5, 6, 4, 3, 6, 5, 3, 2, 0, 2, 2, 3, 2, 2, 1, 5, 0, 1, 3, 4, 5, 1, 1, 4, 7, 7, 6, 6, 0, 9, 9, 6, 3, 3, 5, 4, 1, 9, 1, 1, 6, 0, 4, 2, 6, 0, 9, 2, 8, 8, 8, 4, 5, 9, 4, 9, 5, 5, 3, 8, 1, 5

Offset: 1

Georg Fischer, Aug 23 2022

The constant is defined in Sadov's paper (equation 13).

			1.44705435001627940656436532022322150134511477660996...

Georg Fischer, Table of n, a(n) for n = 1..401 [All 410 digits from Sadov's paper]
Sergey Sadov, On Shallit's minimization problem, arXiv:1806.03651 [math.CA], Jun 10 2018 (see p. 25).

Cf. A086276.

A351524 Number of powers of 11 modulo n.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 2, 2, 12, 3, 2, 4, 16, 6, 3, 2, 6, 2, 22, 2, 5, 12, 18, 6, 28, 2, 30, 8, 3, 16, 3, 6, 6, 3, 12, 2, 40, 6, 7, 3, 6, 22, 46, 4, 21, 5, 16, 12, 26, 18, 2, 6, 6, 28, 58, 2, 4, 30, 6, 16, 12, 3, 66, 16, 22, 3, 70, 6, 72, 6, 10, 6, 4, 12, 39, 4, 54, 40, 41, 6, 16, 7, 28

Offset: 1

Georg Fischer, Feb 13 2022

This is the original version of A054711 rev. #1 as defined by Henry Bottomley, Apr 20 2000.

A054711 is now different from the sequence here.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A054711.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, 11]}, e + MultiplicativeOrder[11, n/11^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

A341862 a(n) is the even term in the linear recurrence signature for numerators and denominators of continued fraction convergents to sqrt(n), or 0 if n is a square.

Original entry on oeis.org

0, 0, 2, 4, 0, 4, 10, 16, 6, 0, 6, 20, 14, 36, 30, 8, 0, 8, 34, 340, 18, 110, 394, 48, 10, 0, 10, 52, 254, 140, 22, 3040, 34, 46, 70, 12, 0, 12, 74, 50, 38, 64, 26, 6964, 398, 322, 48670, 96, 14, 0, 14, 100, 1298, 364, 970, 178, 30, 302, 198, 1060, 62, 59436

Offset: 0

Georg Fischer, Feb 22 2021

nonn

The Everest et al. link states that "the continued fraction expansion of a quadratic irrational is eventually periodic, which implies that the numerators px and denominators qx of its convergents satisfy linear recurrence relations".
Let k be the period length minus one of the continued fraction of sqrt(n). Then the linear recurrence signatures with constant coefficients have the form (0, 0, ..., 0, a(n), 0, 0, ..., 0, (-1)^(n+1)), with k zeroes before and behind a(n).
a(n) is twice the numerator of the convergent to sqrt(n) with index k (starting with 0).
These properties result from the mirrored structure of the period of such continued fractions.
The sequence has remarkably many terms in common with A180495 and with 2*A033313.

			The numerators for sqrt(13) begin with 3, 4, 7, 11, 18, 119, ... (A041018) and have the signature (0,0,0,0,36,0,0,0,0,1). The continued fraction has period [1,1,1,1,6], so k=4 and a(13) = 2*A041018(4) = 2*18 = 36. The signature ends with (-1)^4.
The numerators for sqrt(19) begin with 4, 9, 13, 48, 61, 170, 1421, ... (A041028) and have the signature (0,0,0,0,0,340,0,0,0,0,0,-1). The continued fraction has period [2,1,3,1,2,8], so k=5 and a(19) = 2*A041028(5) = 2*170 = 340. The signature ends with (-1)^5.

Jean-François Alcover, Table of n, a(n) for n = 0..10000
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, AMS Mathematical Surveys and Monographs, Volume 104 (2003) p. 8, 5th paragraph.

Cf. A006702, A033313, A180495.

a(n) = 2*A006702(n) if n is not square, otherwise 0.

A338207 Inverse permutation to A307048.

Original entry on oeis.org

2, 1, 14, 6, 4, 3, 8, 12, 20, 5, 122, 18, 10, 7, 50, 24, 38, 9, 32, 30, 16, 11, 26, 36, 56, 13, 68, 42, 22, 15, 104, 48, 74, 17, 176, 54, 28, 19, 44, 60, 92, 21, 1094, 66, 34, 23, 158, 72, 110, 25, 86, 78, 40, 27, 62, 84, 128, 29, 446, 90, 46, 31, 212, 96, 146, 33, 338, 102, 52, 35, 80, 108, 164, 37, 284, 114, 58, 39, 266

Offset: 1

Georg Fischer, Oct 16 2020

nonn

Permutation of the positive integers.

Index entries for sequences that are permutations of the natural numbers

Cf. A307048.

A338208 Inverse permutation to A322469.

Original entry on oeis.org

2, 3, 1, 16, 7, 14, 4, 17, 12, 8, 5, 15, 21, 33, 6, 126, 26, 13, 9, 69, 31, 50, 10, 124, 38, 22, 11, 34, 43, 67, 18, 127, 48, 27, 19, 122, 55, 86, 20, 70, 60, 32, 23, 51, 65, 103, 24, 125, 74, 39, 25, 179, 79, 120, 28, 287, 84, 44, 29, 68, 91, 143, 30, 1100, 96, 49, 35, 232, 101, 160, 36, 123, 108, 56, 37, 87, 113, 177, 40, 611

Offset: 1

Georg Fischer, Oct 16 2020

nonn

Permutation of the positive integers.
There is a hierarchy of such permutations derived by selecting and mapping the terms of the form 6*k - 2 to k:
  Level 0: A307407
  Level 1: A322469, inverse is A338208 (this sequence)
  Level 2: A307048             A338207
  Level 3: A160016             A338206
  Level 4: A000027  (the positive integers)

Cf. A160016, A307048, A307407, A322469, A338206, A338207.

A338206 Inverse of permutation in A160016.

Original entry on oeis.org

0, 2, 1, 6, 3, 10, 4, 14, 5, 18, 7, 22, 8, 26, 9, 30, 11, 34, 12, 38, 13, 42, 15, 46, 16, 50, 17, 54, 19, 58, 20, 62, 21, 66, 23, 70, 24, 74, 25, 78, 27, 82, 28, 86, 29, 90, 31, 94, 32, 98, 33, 102, 35, 106, 36, 110, 37, 114, 39, 118, 40, 122, 41, 126, 43, 130, 44, 134, 45, 138, 47, 142, 48, 146, 49, 150, 51, 154, 52, 158

Offset: 0

Georg Fischer, Oct 16 2020

Permutation of the nonnegative integers.

Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1,0,-1).

Cf. A160016.

gf := (x*(1 + x^2)*(2 + x + 2*x^2 + x^3 + 2*x^4))/((-1 + x)^2*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)): ser := series(gf, x, 82):
seq(coeff(ser, x, n), n=0..79); # Peter Luschny, Oct 16 2020

LinearRecurrence[{0,1,0,0,0,1,0,-1}, {0,2,1,6,3,10,4,14},80]

my(x='x+O('x^80)); Vec((x*(2+x+4*x^2+2*x^3+4*x^4+x^5+2*x^6))/((1-x^2)^2*(1+x^2+x^4)))

Blocks of 6 numbers: a(6*k+0 .. 6*k+5) = (4*k+0, 12*k+2, 4*k+1, 12*k+6, 4*k+3, 12*k+10) for k >= 0.
O.g.f.: x*(1 + x^2)*(2 + x + 2*x^2 + x^3 + 2*x^4)/((1 - x^2)^2*(1 + x^2 + x^4)).
If n is odd, then a(n) = 2*n; otherwise, a(n) = nearest integer to 2*n/3. - Philippe Deléham, Nov 09 2023

A338186 Expansion of (2-6x-12x^2)/((1-x)^2(1-9x)).

Original entry on oeis.org

2, 16, 126, 1100, 9850, 88584, 797174, 7174468, 64570098, 581130752, 5230176622, 47071589436, 423644304746, 3812798742520, 34315188682470, 308836698142004, 2779530283277794, 25015772549499888, 225141952945498718, 2026277576509488172, 18236498188585393242, 164128483697268538856

Offset: 0

Georg Fischer, Oct 15 2020

The locally small terms 4^k in A322469 occur at the positions a(k) (for k = 0..9, and probably in general; cf. conjectures in A322469).

			A322469(a(4)) = A322469(9850) = 256 = 4^4.

Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. A322469.

f:= gfun:-rectoproc({a(n)=11*a(n-1)-19*a(n-2)+9*a(n-3), a(0)=2, a(1)=16, a(2)=126}, a(n), remember): map(f, [$0..21]);

CoefficientList[Series[(2-6*x-12*x^2)/((1-x)^2*(1-9*x)), {x,0,21}], x]

my(x='x+O('x^22)); Vec((2-6*x-12*x^2)/((1-x)^2*(1-9*x)))

a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3) for n >= 3.

cp's OEIS Frontend

User: _Georg Fischer

A378234 From higher-order arithmetic progressions: Corrected version of A259461.

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A373101 Triangle read by rows, T(n,k) = (binomial(n,k)^3 - binomial(n,k))/6 for k=1..n-1 and n >= 2.

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A363839 Numbers in the witch's multiplication table (German: "Hexeneinmaleins") in Goethe's Faust.

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A356686 Decimal expansion of the constant p^*_0 related to Shallit's constant (A086276).

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A351524 Number of powers of 11 modulo n.

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A341862 a(n) is the even term in the linear recurrence signature for numerators and denominators of continued fraction convergents to sqrt(n), or 0 if n is a square.

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A338207 Inverse permutation to A307048.

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A338208 Inverse permutation to A322469.

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A338206 Inverse of permutation in A160016.

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A338186 Expansion of (2-6*x-12*x^2)/((1-x)^2*(1-9*x)).

A338186 Expansion of (2-6x-12x^2)/((1-x)^2(1-9x)).