Raul Prisacariu - cp's OEIS frontend

A383873 a(n) = 3a(n-1) - 2a(n-2) + 5*a(n-3) starting with 1, 2, 3.

1, 2, 3, 10, 34, 97, 273, 795, 2324, 6747, 19568, 56830, 165089, 479447, 1392313, 4043490, 11743079, 34103822, 99042758, 287636025, 835341669, 2425966747, 7045397028, 20460965935, 59421937484, 172570865722, 501173551873, 1455488611595, 4226973059649

Offset: 0

Raul Prisacariu, May 18 2025

The sequence appears as an example of recurring series in "A Mathematical and Philosophical Dictionary" by Charles Hutton.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Charles Hutton, Recurring Series, A mathematical and philosophical dictionary Vol. 2 (1795), 440.
Index entries for linear recurrences with constant coefficients, signature (3,-2,5).

I:=[ 1, 2, 3]; [n le 3 select I[n] else 3*Self(n-1)-2*Self(n-2)+5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 21 2025

LinearRecurrence[{3,-2,5},{1,2,3},30] (* Vincenzo Librandi, May 21 2025 *)

G.f.: (x^2+x-1)/(5*x^3-2*x^2+3*x-1). - Alois P. Heinz, May 19 2025

A371201 a(n) = Sum_{k=prime(n)..prime(n+1)-1} k, with a(0) = 1.

Original entry on oeis.org

1, 2, 7, 11, 34, 23, 58, 35, 82, 153, 59, 201, 154, 83, 178, 297, 333, 119, 381, 274, 143, 453, 322, 513, 740, 394, 203, 418, 215, 442, 1673, 514, 801, 275, 1435, 299, 921, 957, 658, 1017, 1053, 359, 1855, 383, 778, 395, 2454, 2598, 898, 455, 922, 1413, 479, 2455, 1521, 1557, 1593

Offset: 0

Raul Prisacariu, Mar 15 2024

The sequence can be obtained graphically using the following grid walk rules. From an origin the first movement iteration consists of moving 1 unit in any direction. The n-th movement iteration consists of moving in the same direction n units. If n is a prime number, the movement iteration consists of first changing the movement direction by 90 degrees and then moving n units in the new direction. If n is a nonprime number, the movement iteration consists of moving n units in the same direction as the previous movement iteration. The sequence is obtained by measuring the length of each 90-degree turn.

a(0) is the length of the grid segment before doing any 90-degree turns and a(1) is the length of the first 90-degree turn.

			a(0) = 1.
a(1) = 2.
a(2) = 3 + 4 = 7.
a(3) = 5 + 6 = 11.
a(4) = 7 + 8 + 9 + 10 = 34.
a(5) = 11 + 12 = 23.
a(6) = 13 + 14 + 15 + 16 = 58.
a(7) = 17 + 18 = 35.
The natural numbers are summed in groups where each prime begins a new group,
  primes     v   v       v       v
         1   2   3   4   5   6   7   8   9  10  ...
        \-/ \-/ \-----/ \-----/ \-------------/
  a(n) = 1   2     7       11          34
    n  = 0   1     2       3           4

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000040, A054265, A138383.

Cf. A008837 (partial sums).

ithprime(0):=1:
a:= n-> ((j, k)-> (k-1+j)*(k-j)/2)(map(ithprime, [n, n+1])[]):
seq(a(n), n=0..56);  # Alois P. Heinz, Mar 16 2024

Join[{1},Table[Prime[n]+(Prime[n+1]+Prime[n])*(Prime[n+1]-Prime[n]-1)/2,{n,56}]] (* James C. McMahon, Apr 20 2024 *)

first(n) = {
	my(res = primes(n), t = 0);
	for(i = 1, n,
		res[i] = binomial(res[i],2) - t;
		t+=res[i];
	);
	res	
} \\ David A. Corneth, Mar 16 2024

from sympy import nextprime, prime
def A371201(n):
    if n == 0: return 1
    q = nextprime(p:=prime(n))
    return (q-p)*(p+q-1)>>1 # Chai Wah Wu, Jun 01 2024

For n > 0, a(n) = A138383(n) - (prime(n+1) - prime(n)).
a(n) = binomial(prime(n+1), 2) - Sum_{k=0..n-1} a(k). - David A. Corneth, Mar 15 2024
a(n) = prime(n) + A054265(n), for n >= 1. - Michel Marcus, Mar 15 2024
a(n) = (prime(n+1)-prime(n))*(prime(n+1)+prime(n)-1)/2 for n>=1. - Chai Wah Wu, Jun 01 2024

More terms from Michel Marcus, Mar 15 2024

A370491 The numerators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

Original entry on oeis.org

1, 1, -1, -5, 19, -3, -10187, 146847, 3268961, -211632497, 393324007, 5402916117, -3884618921299, -774402304798329, 148294948981707557, -3311395903665985169, -43463254022673425965, 14469962812566878696039, 6554498075974546253080309, -3074689522272735111427973673

Offset: 1

Raul Prisacariu, Feb 19 2024

sign

Whittaker's root series formula is applied to 1 - 2x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - ..., which is the Taylor expansion of -x + e^(-x). We obtain the following infinite series that converges to the Omega constant (LambertW(1)): LambertW(1) = 1/2 + 1/14 - 1/259 - 5/9657 + 19/200187 - 3/18671081 ... . The sequence is formed by the numerators of the infinite series.

			a(1) is the numerator of -1/-2 = 1/2.
a(2) is the numerator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!))) = -(1/2)/((-2)*(7/2)) = 1/14.
a(3) is the numerator of -det ToeplitzMatrix((1/2!,-2),(1/2!,-1/3!))/(det ToeplitzMatrix((-2,1),(-2,1/2!))*det ToeplitzMatrix((-2,1,0),(-2,1/2!,-1/3!))) = -(-1/12)/((7/2)*(-37/6)) = -1/259.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A030178, A370490 (denominator).

For n > 1, a(n) is the numerator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n-1)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))), where c(0)=1, c(1)=-2, c(n) = (-1)^n/n!.

a(9)-a(20) from Chai Wah Wu, Mar 23 2024

A370490 The denominators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

Original entry on oeis.org

2, 14, 259, 9657, 200187, 18671081, 7313976065, 1273374259615, 285038137030769, 79755360301275363, 9091712937155442435, 149243024021521700285, 1085736156475373087072485, 3071709182054627484879798019, 2005459027715242401528647218817, 1496371535371115486607560677791759

Offset: 1

Raul Prisacariu, Feb 19 2024

nonn

Whittaker's root series formula is applied to 1 - 2x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - ..., which is the Taylor expansion of -x + e^(-x). We obtain the following infinite series that converges to the Omega constant (LambertW(1)): LambertW(1) = 1/2 + 1/14 - 1/259 - 5/9657 + 19/200187 - 3/18671081 ... . The sequence is formed by the denominators of the infinite series.

			a(1) is the denominator of -1/-2 = 1/2.
a(2) is the denominator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!))) = -(1/2)/((-2)*(7/2)) = 1/14.
a(3) is the denominator of -det ToeplitzMatrix((1/2!,-2),(1/2!,-1/3!))/(det ToeplitzMatrix((-2,1),(-2,1/2!))*det ToeplitzMatrix((-2,1,0),(-2,1/2!,-1/3!))) = -(-1/12)/((7/2)*(-37/6)) = -1/259.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A030178, A370491 (numerator).

for n>1, a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n-1)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))), where c(0)=1, c(1)=-2, c(n) = (-1)^n/n!.

a(9)-a(16) from Chai Wah Wu, Mar 23 2024

A369187 The numerators of a series that converges to the Dottie Number (A003957).

Original entry on oeis.org

1, -1, 1, -3, 1, 205, -4439, 111021, -1724351, 2074717, 2567577481, -246042951203, 14444487376705, -726562139423955, 1473171168838825, 1178164765176836393, -204468301714665778099, 138848947223110087743421, -11701779801284441802592247, 7774256876827576332115737

Offset: 1

Raul Prisacariu, Jan 15 2024

sign

Whittaker's root series formula is applied to 1 - x - x^2/2! + x^4/4! - x^6/6! + ..., which is the Taylor expansion of cos(x) - x. The following infinite series for the Dottie number (D) is obtained: D = 1/1 - 1/3 + 1/12 - 3/260 + 1/5720 + 205/314248 - 4439/17255072 ... . The sequence is formed by the numerators of the series.

			a(1) is the numerator of -1/-1 = 1/1.
a(2) is the numerator of simplified -(-1/2!)/(-1* det ToeplitzMatrix((-1,1),(-1,-1/2!))) = (1/2)/(-3/2) = -1/3.
a(3) is the numerator of the simplified -det ToeplitzMatrix((-1/2!,-1),(-1/2!,0))/(det ToeplitzMatrix((-1,1),(-1,-1/2!))*det ToeplitzMatrix((-1,1,0),(-1,-1/2!,0))) = -(1/4)/((3/2)*-2) = 1/12.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A003957.

a(1) = 1; for n > 1, a(n) is the numerator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=-1, c(2)=-1/2!, c(3)=0, c(4)=1/4!, c(5)=0, c(6)=-1/6!, and c(n) is the coefficient of x^n in the Taylor expansion of cos(x)-x.

a(8)-a(20) from Chai Wah Wu, Feb 10 2024

A369186 The denominators of a series that converges to the Dottie Number (A003957).

Original entry on oeis.org

1, 3, 12, 260, 5720, 314248, 17255072, 1769058016, 181357735680, 29880655637760, 4923158441956352, 1189676108826729472, 287484053261423565824, 95784714773484796761088, 31913810779214031287095296, 2804341960426298188743438336, 1232120770958699233546743119872

Offset: 1

Raul Prisacariu, Jan 15 2024

nonn

Whittaker's root series formula is applied to 1 - x - x^2/2! + x^4/4! - x^6/6! + ..., which is the Taylor expansion of cos(x) - x. The following infinite series for the Dottie number (D) is obtained: D = 1/1 - 1/3 + 1/12 - 3/260 + 1/5720 + 205/314248 - 4439/17255072 ... . The sequence is formed by the denominators of the series.

			a(1) is the denominator of -1/-1 = 1/1.
a(2) is the denominator of simplified -(-1/2!)/(-1* det ToeplitzMatrix((-1,1),(-1,-1/2!))) = (1/2)/(-3/2) = -1/3.
a(3) is the denominator of the simplified -det ToeplitzMatrix((-1/2!,-1),(-1/2!,0))/(det ToeplitzMatrix((-1,1),(-1,-1/2!))*det ToeplitzMatrix((-1,1,0),(-1,-1/2!,0))) = -(1/4)/((3/2)*-2) = 1/12.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A003957.

a(1)=1;

for n > 1, a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=-1, c(2)=-1/2!, c(3)=0, c(4)=1/4!, c(5)=0, c(6)=-1/6!, and c(n) is the coefficient of x^n in the Taylor expansion of cos(x)-x.

a(8)-a(17) from Chai Wah Wu, Feb 10 2024

A368862 Numerators of an infinite series that converges to the negative inverse of Backhouse's constant (A088751).

Original entry on oeis.org

-1, -3, 1, 1, -1, 5, -19, -9, 41, -103, 17, 289, -169, 331, -689, -4991, 3999, 7833, -6509, 21827, -22165, -87637, 119441, -190981, -152513, 1482023, -425985, -1045091, 1071237, -14108791, 5845271, 39852203, -35832801, 54451699, 44061359, -435442725, 261309855, -22217917

Offset: 1

Raul Prisacariu, Jan 08 2024

sign

Whittaker's root series formula is applied to 1 + Sum_{k>=1} prime(k) x^k. The following infinite series that converges to the negative inverse of Backhouse's constant (-x) is obtained:
x = -1/(1*2) - 3/(2*1) + 1/(1*1) + 1/(1*2) - 1/(2*3) + 5/(3*7) - 19/(7*10) - 9/(10*13) + 41/(13*21) - 103/(21*26) + 17/(26*33) + 289/(33*53) ...
The denominators of the infinite series are obtained by multiplying the absolute values of 2 consecutive terms from the sequence A030018.

			a(1) = -1;
a(2) = -3;
a(3) = -det ToeplitzMatrix((3,2),(3,5)) = 1;
a(4) = -det ToeplitzMatrix((3,2,1),(3,5,7)) = 1;
a(5) = -det ToeplitzMatrix((3,2,1,0),(3,5,7,11)) = -1;
a(6) = -det ToeplitzMatrix((3,2,1,0,0),(3,5,7,11,13)) = 5;
a(7) = -det ToeplitzMatrix((3,2,1,0,0,0),(3,5,7,11,13,17)) = -19.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A088751, A030018, A072508.

a(1) = -1.

For n > 1, a(n) = -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n))), where c(0)=1 and c(n) is the n-th prime number.

a(21)-a(38) from Stefano Spezia, Jan 09 2024

A368205 a(n) = 3a(n-1) - 2a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.

Original entry on oeis.org

1, 3, 7, 14, 25, 40, 56, 63, 37, -71, -350, -945, -2064, -3952, -6783, -10381, -13625, -13330, -2359, 33208, 117672, 288959, 598325, 1099385, 1812546, 2640543, 3197152, 2497824, -1541375, -12816925, -37865849, -86422322, -170718343, -301444536, -476474600, -655816385, -713055419, -351058887, 1028750562, 4501424879, 11797832400, 25361896880, 47988600961

Offset: 0

Raul Prisacariu, Dec 18 2023

sign

Whittaker's Root Series Formula is applied to the polynomial equation -1+2x+3x^2+x^3. The following infinite series involving the Plastic Ratio (rho) is obtained: rho - 1 = 1/2 - 3/(2*7) + 7/(7*21) - 14/(21*65) + 25/(65*200) - 40/(200*616) + 56/(616*1897) - ...

The terms of the sequence appear in the numerators of the infinite sequence (with alternating signs). The denominators of the sequence are formed by multiplying consecutive terms from the sequence A218836.

			a(0) = 1,
a(1) = 3*a(0) = 3*1 = 3,
a(2) = 3*a(1) - 2*a(0) = 3*3 - 2*1 = 7,
a(3) = 3*a(2) - 2*a(1) - a(0) = 3*7 - 2*3 - 1 = 14.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A218836 (denominator), A060006.

a:=proc(n) local c1,c2,c3;
 option remember;
c1:=3; c2:=2; c3:=1;
if n=0 then 1
elif n=1 then 3
elif n=2 then 7
else c1*a(n-1)-c2*a(n-2)-c3*a(n-3); fi;
end; # N. J. A. Sloane, Dec 31 2023
[seq(a(n),n=0..30)];

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3).

a(n) = determinant of the n X n Toeplitz Matrix((3,2,-1,0,0,...,0),(3,1,0,0,0,...,0)).

A367597 The numerators of a series that converges to log(2) obtained using Whittaker's Root Series Formula.

Original entry on oeis.org

1, -1, 1, 1, -7, -13, 115, 5657, -50759, -2129735, 14303129, 567824371, -668626477, -2536766233351, -22685437440167, 6386546752479769, 164094712558603577, -8391397652598907629, -441809873725219477581, 29040435128150468206405, 2814195975064870847100773

Offset: 1

Raul Prisacariu, Nov 24 2023

The Whittaker's root series formula is applied to -1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + ..., which is the Taylor expansion of e^x with the first coefficient having a negative sign (-1 instead of 1). We obtain log(2) = 1 - 1/3 + 1/39 + 1/975 - 7/40575 - 13/844501 + 115/73824373 + 5657/25814174655 .... The sequence is formed by the numerators of the series.

			a(1) is the numerator of -(-1)/1=1/1
a(2) is the numerator of -(-1)^2*(1/2!)/(1*det((1,1/2!),(-1,1)))=-(1/2)/(1*(3/2))=-1/3
a(3) is the numerator of -(-1)^3*det((1/2!,1/3!),(1,1/2!))/(det((1,1/2!),(-1,1))*det((1,1/2!,1/3!),(-1,1,1/2!),(0,-1,1)))=(1/12)/((3/2)*(13/6))=1/39

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A002162, A365595, A367596 (denominators).

c[k_] := If[k < 0, 0, SeriesCoefficient[Exp[x] - 2, {x, 0, k}]]; Join[{1}, Table[(-1)^n*Det[ToeplitzMatrix[Table[c[3 - j], {j, 1, n}], Table[c[j + 1], {j, 1, n}]]] / (Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n}], Table[c[j], {j, 1, n}]]] * Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n + 1}], Table[c[j], {j, 1, n + 1}]]]), {n, 1, 20}] // Numerator] (* Vaclav Kotesovec, Nov 26 2023 *)

a(n) is the numerator of the simplified fraction -(-1)^n*det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=-1, c(1)=1, c(2)=1/2!, c(3)=1/3!, c(4)=1/4!,c(n)=1/n!.

More terms from Vaclav Kotesovec, Nov 26 2023

A367596 The denominators of a series that converges to log(2) obtained using Whittaker's root series formula.

Original entry on oeis.org

1, 3, 39, 975, 40575, 844501, 73824373, 25814174655, 3868475107935, 724655165594943, 165910226233669599, 15194097535426090645, 4933425635511640104565, 5606480381963363479902783, 2450522415523358900846598879, 1224105922303030827661963930815, 693005978151926719613680243125855

Offset: 1

Raul Prisacariu, Nov 24 2023

The Whittaker's root series formula is applied to -1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + ..., which is the Taylor expansion of e^x with the first coefficient having a negative sign (-1 instead of 1). We obtain log(2) = 1 - 1/3 + 1/39 + 1/975 - 7/40575 - 13/844501 + 115/73824373 + 5657/25814174655 .... The sequence is formed by the denominators of the series.

			a(1) is the denominator of -(-1)/1 = 1/1.
a(2) is the denominator of -(-1)^2*(1/2!)/(1*det((1,1/2!),(-1,1))) = -(1/2)/(1*(3/2)) = -1/3.
a(3) is the denominator of -(-1)^3*det((1/2!,1/3!),(1,1/2!))/(det((1,1/2!),(-1,1))*det((1,1/2!,1/3!),(-1,1,1/2!),(0,-1,1))) = (1/12)/((3/2)*(13/6)) = 1/39.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A002162, A365594, A367597 (numerators).

c[k_] := If[k < 0, 0, SeriesCoefficient[Exp[x] - 2, {x, 0, k}]]; Join[{1}, Table[(-1)^n*Det[ToeplitzMatrix[Table[c[3 - j], {j, 1, n}], Table[c[j + 1], {j, 1, n}]]] / (Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n}], Table[c[j], {j, 1, n}]]] * Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n + 1}], Table[c[j], {j, 1, n + 1}]]]), {n, 1, 20}] // Denominator] (* Vaclav Kotesovec, Nov 26 2023 *)

a(n) is the denominator of the simplified fraction -(-1)^n*det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=-1, c(1)=1, c(2)=1/2!, c(3)=1/3!, c(4)=1/4!, c(n)=1/n!.

More terms from Vaclav Kotesovec, Nov 26 2023

cp's OEIS Frontend

User: Raul Prisacariu

A383873 a(n) = 3*a(n-1) - 2*a(n-2) + 5*a(n-3) starting with 1, 2, 3.

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A371201 a(n) = Sum_{k=prime(n)..prime(n+1)-1} k, with a(0) = 1.

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A370491 The numerators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

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A370490 The denominators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

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A369187 The numerators of a series that converges to the Dottie Number (A003957).

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A369186 The denominators of a series that converges to the Dottie Number (A003957).

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A368862 Numerators of an infinite series that converges to the negative inverse of Backhouse's constant (A088751).

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A368205 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.

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A383873 a(n) = 3a(n-1) - 2a(n-2) + 5*a(n-3) starting with 1, 2, 3.

A368205 a(n) = 3a(n-1) - 2a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.