A061278 -id:A061278 - cp's OEIS frontend

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16803, 49059, 143235, 418195, 1220979, 3564819, 10407987, 30387571, 88720755, 259032627, 756281907, 2208070579, 6446770227, 18822245427, 54954172467, 160446376243, 468445588275, 1367692273971, 3993168476979, 11658612678451

Offset: 0

N. J. A. Sloane, Nov 17 2002, Jun 05 2007

Index entries for linear recurrences with constant coefficients, signature (3,0,0,-2).

Partial sums of S(n, x), for x=1...11, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826.

CoefficientList[Series[(1-x)^(-1)/(1-2x-2x^2-2x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 27 2011 *)

Vec((1-x)^(-1)/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

Original entry on oeis.org

1, 3, 11, 42, 157, 588, 2204, 8259, 30949, 115977, 434606, 1628619, 6103000, 22870056, 85702025, 321155187, 1203479779, 4509855786, 16899992477, 63330128340, 237319937332, 889320046107, 3332590398005, 12488371098225, 46798254229006, 175369276077483

Offset: 0

N. J. A. Sloane, Nov 17 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,3).

Partial sums of S(n, x), for x=1...15, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829.

CoefficientList[Series[1/(1-3x-2x^2-3x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,3},{1,3,11},40] (* Harvey P. Dale, Nov 05 2021 *)

A102207 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(0) = 4, a(1) = 17, a(2) = 65.

Original entry on oeis.org

4, 17, 65, 244, 912, 3405, 12709, 47432, 177020, 660649, 2465577, 9201660, 34341064, 128162597, 478309325, 1785074704, 6661989492, 24862883265, 92789543569, 346295291012, 1292391620480, 4823271190909, 18000693143157

Offset: 0

Creighton Dement, Dec 30 2004

nonn

Index entries for linear recurrences with constant coefficients, signature (5, -5, 1).

Cf. A061278, A092184, A001834, A001353, A102206.

a[0] = 4; a[1] = 17; a[2] = 65; a[n_] := a[n] = 5a[n - 1] - 5a[n - 2] + a[n - 3]; Table[ a[n], {n, 0, 22}] (* Or *)
CoefficientList[ Series[(3x - 4)/((x - 1)(x^2 - 4x + 1)), {x, 0, 22}], x] (* Robert G. Wilson v, Jan 12 2005 *)
LinearRecurrence[{5,-5,1},{4,17,65},30] (* or *) With[{c=Sqrt[3]},Table[ Simplify[ ((3-7c)(2-c)^x+(2+c)^x (3+7c)-6)/12],{x,30}]] (* Harvey P. Dale, Mar 15 2013 *)

G.f.: (3x-4)/((x-1)(x^2-4x+1))
(1/2) [A001353(n+1) + 5*A001353(n) - 1 ]. - Ralf Stephan, May 17 2007
a(n)=1/12*((3-7*Sqrt[3])*(2-Sqrt[3])^n+(3+7*Sqrt[3])*(2+Sqrt[3])^n-6). - Harvey P. Dale, Mar 15 2013

More terms from Robert G. Wilson v, Jan 12 2005

A228025 a(1) = least k such that 1/2+1/3+1/4+1/5 < H(k) - H(5); a(2) = least k such that H(a(1)) - H(5) < H(k) -H(a(1)), and for n > 2, a(n) = least k such that H(a(n-1)) - H(a(n-2)) > H(k) - H(a(n-1)), where H = harmonic number.

Original entry on oeis.org

20, 76, 285, 1065, 3976, 14840, 55385, 206701, 771420, 2878980, 10744501, 40099025, 149651600, 558507376, 2084377905, 7779004245, 29031639076, 108347552060, 404358569165, 1509086724601, 5631988329240, 21018866592360, 78443478040201, 292755045568445

Offset: 1

Clark Kimberling, Aug 03 2013

Suppose that x and y are positive integers and that x <=y. Let a(1) = least k such that H(y) - H(x-1) < H(k) - H(y); let a(2) = least k such that H(a(1)) - H(y) < H(k) - H(a(1)); and for n > 2, let a(n) = least k such that greatest such H(a(n-1)) - H(a(n-2)) < H(k) - H(a(n-1)). The increasing sequences H(a(n)) - H(a(n-1)) and a(n)/a(n-1) converge. For what choices of (x,y) is the sequence a(n) linearly recurrent?

For A228025, (x,y) = (2,5); it appears that H(a(n)) - H(a(n-1)) approaches log(2 + sqrt(3)) and that and a(n)/a(n-1) approaches sqrt(3).

			The first two values (a(1),a(2)) = (20,76) match the beginning of the following inequality chain:  1/2+1/3+1/4+1/5 < 1/6+...+1/20 < 1/21+...+1/76 < ...

Clark Kimberling, Table of n, a(n) for n = 1..100

Cf. A224820, A224965, A228016.

z = 100; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 2; y = 5;
a[1] = Ceiling[w /. FindRoot[h[w] == 2 h[y] - h[x - 1], {w, 1}, WorkingPrecision -> 400]]; a[2] = Ceiling[w /. FindRoot[h[w] == 2 h[a[1]] - h[y], {w, a[1]},      WorkingPrecision -> 400]]; Do[s = 0; a[t] = Ceiling[w /. FindRoot[h[w] == 2 h[a[t - 1]] - h[a[t - 2]], {w, a[t - 1]}, WorkingPrecision -> 400]], {t, 3, z}];
m = Map[a, Range[z]] (* A227653, Peter J. C. Moses, Jul 12 2013 *)

a(n) = A061278(n+1) (conjectured).
a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) (conjectured).
G.f.: (-20 + 24 x - 5 x^2)/(-1 + 5 x - 5 x^2 + x^3) (conjectured).

A341894 For square n > 0, a(n) = 0; for nonsquare n > 0, a(n) is the rank r such that t(r) + t(r-1) = u(r) - u(r-1) - 1, where u(r) and t(r) are indices of some triangular numbers in the Diophantine relation T(u(r)) = n*T(t(r)).

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 2, 2, 0, 3, 2, 2, 4, 2, 2, 0, 2, 2, 3, 2, 4, 4, 2, 2, 0, 3, 2, 4, 4, 2, 4, 2, 2, 2, 2, 0, 2, 2, 2, 4, 4, 2, 4, 2, 4, 6, 2, 2, 0, 3, 3, 4, 4, 2, 4, 2, 4, 4, 2, 2, 8, 2, 2, 0, 2, 4, 4, 2, 4, 4, 4, 2, 6, 2, 2, 6, 4, 4, 2, 2, 0, 3, 2, 2, 8, 4, 2, 4, 4, 2, 6, 4, 4, 4, 2, 4, 4, 2, 2, 0, 2, 2, 4, 2, 4, 8, 2, 2, 8, 2

Offset: 1

Vladimir Pletser, Mar 06 2021

nonn

Let t(i) and u(j) be the indices of triangular numbers that satisfy the Diophantine relation T(u(j)) = n*T(t(i)) for some integers i and j. The number of solutions (t(i), u(j)) of T(u(j)) = n*T(t(i)) is 0 or 1 for square n, and an infinity for nonsquare n.
For square n, a(n) is arbitrarily set to 0.
For nonsquare n, a(n) is the index r in the sequence of t(i) and u(j) such that t(r) + t(r-1) = u(r) - u(r-1) - 1.
Alternatively, for nonsquare n, a(n) is the index r such that the ratio t(i)/t(i-r) is decreasing monotonically without jumps for increasing values of i.
Alternatively, for n > 4, a(n) is the index r such that the ratio t(r)/t(r-1) varies between (s+1)/(s-1) and (s+2)/s, with s = [sqrt(n)], where [x] = floor(x).
Alternatively, for nonsquare n, a(n) is the number of fundamental solutions (X_f, Y_f) of the generalized Pell equation X^2 - n*Y^2 = 1 - n providing odd solutions, i.e., with X_f odd and Y_f odd (or Y_f even if y_f is odd, where y_f is the fundamental solution of the associated simple Pell equation x^2 - n*y^2 = 1).

			The following table gives the first values of nonsquare n and a(n) and the sequences yielding the values of t, u, T(t) and T(u) such that T(u) = n*T(t).
n       2       3       5       6       7       8      10
a(n)    1       1       2       2       2       2       3
t    A053141 A061278 A077259 A077288 A077398 A336623  A341893*
u    A001652 A001571 A077262 A077291 A077401 A336625* A341895*
T(t) A075528 A076139 A077260 A077289 A077399 A336624  A068085*
T(u) A029549 A076140 A077261 A077290 A077400 A336626*   -
With a(n) = r, the definition t(r) + t(r-1) = u(r) - u(r-1) - 1 yields:
- For n = 2, a(n) = 1: A053141(1) + A053141(0) = A001652(1) - A001652(0) - 1, i.e., 2 + 0 = 3 + 0 - 1 = 2.
- For n = 5, a(n) = 2: A077259(2) + A077259(1) = A077262(2) - A077262(1) - 1, i.e., 6 + 2 = 14 - 5 - 1 = 8.
- For n = 10, a(n) = 3: A341893(3+1*) + A341893(2+1*) = A341895(3+1*) - A341895(2+1*) - 1, i.e., 12 + 6 = 39 - 20 - 1 = 18.
Note that for those sequences marked with an *, the first term 0 appears for n = 1, contrary to all the other sequences, where the first term 0 appears for n = 0; the numbering must therefore be adapted and 1 must be added to compensate for this shift in indices.
The monotonic decrease of t(i)/t(i-r) can be seen also as:
- For n = 2, a(n) = 1: for 1 <= i <= 6, A053141(i)/A053141(i-1) decreases monotonically from 7 to 5.829.
- For n = 5, a(n) = 2: for 3 <= i <= 8, A077259(i)/A077259(i-2) decreases monotonically from 22 to 17.948, while A077259(i)/A077259(i-1) takes values alternatively varying between 3 and 2.618 and between 7.333 and 6.855.
- For n = 10, a(n) = 3: for 4 <= i <= 10, A341893(i)/A341893(i-3) decreases monotonically from 55 to 38, while A077259(i) / A077259(i-1) takes values alternatively varying between 6 and 4.44 and between 2 and 1.925.
For n > 4, the relation (s+1)/(s-1) <=  t(r)/t(r-1) <= (s+2)/s, with s = [sqrt(n)], yields:
- For n = 5, a(n) = 2: A077259(2)/A077259(1) = 6/2 = 3, and s = [sqrt(5)] = 2, (s+1)/(s-1) = 3 and (s+2)/s = 2.
- For n = 10, a(n) = 3: A077259(3+1*)/A077259(2+1*) = 12/6 = 2, and s = [sqrt(10)] = 3, (s+1)/(s-1) = 2 and (s+2)/s = 5/3 = 1.667.
Finally, the number of fundamental solutions of the generalized Pell equation is as follows.
- For n = 2, X^2 - 2*Y^2 = -1 has a single fundamental solution, (X_f, Y_f) = (1, 1), and the rank a(n) is 1.
- For n = 5, X^2 - 5*Y^2 = -4 has two fundamental solutions, (X_f, Y_f) = (1, 1) and (-1, 1), and the rank a(n) is 2.
- For n = 10, X^2 - 10*Y^2 = -9 has three fundamental solutions, (X_f, Y_f) = (1, 1), (-1, 1), and (9, 3), and the rank a(n) is 3.

J. S. Chahal and H. D'Souza, "Some remarks on triangular numbers", in A.D. Pollington and W. Mean, eds., Number Theory with an Emphasis on the Markov Spectrum, Lecture Notes in Pure Math, Dekker, New York, 1993, 61-67.

Vladimir Pletser, Table of n, a(n) for n = 1..256
T. Breiteig, Quotients of triangular numbers, The Mathematical Gazette, 99, 2015, 243-255.
Keith Matthews, The Diophantine Equation x^2 - Dy^2 = N, D > 0, in integers, Expositiones Mathematicae, 18, 2000, 323-331.
Keith Matthews, Quadratic Diophantine equations BCMATH programs, 2020.
Vladimir Pletser, Searching for Multiple of Triangular Numbers being Triangular Numbers, ResearchGate, DOI: 10.13140/RG.2.2.35428.91527, 2021.
Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv: 2102.13494 [math.NT], 2021.
Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv: 2101.00998 [math.NT], 2021.

Cf. A068085, A053141, A061278, A077259, A077288, A077398, A336623, A341893, A001652, A001571, A077262, A077291, A077401, A336625, A341895, A075528, A076139, A077260, A077289, A077399, A336624, A068085, A029549, A076140, A077261, A077290, A077400, A336626, A000217, A341895, A341896.

A145856 Least number k>1 such that centered n-gonal number n*k(k-1)/2+1 is a perfect square, or 0 if no such k exists.

Original entry on oeis.org

3, 0, 2, 4, 3, 8, 16, 2, 17, 9, 15, 5, 6, 16, 2, 3, 6, 0, 7, 4, 3, 40, 7, 2, 22, 8, 111, 4, 16, 8, 16, 0, 3, 9, 2, 5, 990, 9, 15, 3, 46, 16, 10, 5, 6, 336, 10, 2, 30, 0, 31, 16, 11, 416, 7, 3, 11, 33, 55, 4, 78, 56, 2, 6, 3, 8, 47751, 12, 16, 24, 48, 0, 49, 25, 17, 13, 6, 9, 2640, 2, 6721

Offset: 1

Alexander Adamchuk, Oct 22 2008

nonn

Jonathan Vos Post, When Centered Polygonal Numbers are Perfect Squares, submitted to Mathematics Magazine, 4 May 2004, manuscript no. 04-1165, unpublished, available upon request. - Jonathan Vos Post, Oct 25 2008

Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers

Cf. A120744, A001542, A166259, A006451, A001921, A129444, A001570, A001652, A129556, A053606, A105038, A105040, A053141, A061278.

a(n) = 0 for n in A166259.

a(n) = A120744(n) + 1. - Alexander Adamchuk, Oct 10 2009

Edited by Max Alekseyev, Jan 23 2010

cp's OEIS Frontend

A077827 Expansion of (1-x)^(-1)/(1-2*x-2*x^2-2*x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A077830 Expansion of 1/(1-3*x-2*x^2-3*x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A102207 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(0) = 4, a(1) = 17, a(2) = 65.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A228025 a(1) = least k such that 1/2+1/3+1/4+1/5 < H(k) - H(5); a(2) = least k such that H(a(1)) - H(5) < H(k) -H(a(1)), and for n > 2, a(n) = least k such that H(a(n-1)) - H(a(n-2)) > H(k) - H(a(n-1)), where H = harmonic number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A341894 For square n > 0, a(n) = 0; for nonsquare n > 0, a(n) is the rank r such that t(r) + t(r-1) = u(r) - u(r-1) - 1, where u(r) and t(r) are indices of some triangular numbers in the Diophantine relation T(u(r)) = n*T(t(r)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A145856 Least number k>1 such that centered n-gonal number n*k(k-1)/2+1 is a perfect square, or 0 if no such k exists.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).