Miklos Kristof - cp's OEIS frontend

A190818 Expansion of e.g.f.: 1/(1-2*tanh(x)).

1, 2, 8, 44, 320, 2912, 31808, 405344, 5903360, 96722432, 1760811008, 35260703744, 770296217600, 18229999665152, 464622502289408, 12687528814751744, 369557965317079040, 11437129322496131072, 374778854976227115008, 12963259774166774841344, 471986702056014668103680

Offset: 0

Miklos Kristof, May 21 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A011782 (e.g.f. of 1/(1-tanh(x))).

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 1/(1-2*Tanh(x)) ))); // G. C. Greubel, Dec 03 2023

E(x):=1/(1-2*tanh(x)):
a[0]:=E(x):
for n from 1 to 30 do a[n]:=diff(a[n-1],x) od:
x:=0:
seq(a[n],n=0..30);

CoefficientList[Series[1/(1-2*Tanh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

x='x+O('x^66);
Vec(serlaplace(1/(1-2*tanh(x)))) /* Joerg Arndt, May 21 2011 */

def A190818_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/(1-2*tanh(x)) ).egf_to_ogf().list()
A190818_list(40) # G. C. Greubel, Dec 03 2023

E.g.f: 1/(1-2*tanh(x)).

a(n) ~ n! * 2^(n+2)/(3*(log(3))^(n+1)). - Vaclav Kotesovec, Jun 26 2013

A146301 a(n) = (8n+3)(8*n+7).

Original entry on oeis.org

21, 165, 437, 837, 1365, 2021, 2805, 3717, 4757, 5925, 7221, 8645, 10197, 11877, 13685, 15621, 17685, 19877, 22197, 24645, 27221, 29925, 32757, 35717, 38805, 42021, 45365, 48837, 52437, 56165, 60021, 64005, 68117, 72357, 76725, 81221

Offset: 0

Miklos Kristof, Oct 29 2008

Sum_{n>=0} 1/((8*n+3)*(8*n+7)) = (1/16)*sqrt(2)*(log(sqrt(2)-1) + Pi/2) = 0.60936936799920131042...

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Maple
```
seq((8*n+3)*(8*n+7),n=0..40);
```

Table[(8n+3)(8n+7),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{21,165,437},40] (* Harvey P. Dale, Aug 16 2015 *)

a(n)=(8*n+3)*(8*n+7) \\ Charles R Greathouse IV, Jun 17 2017

G.f: (21 + 102*x + 5*x^2)/(1-x)^3.

E.g.f.: (21 + 144*x + 64*x^2)*exp(x).

A146302 a(n) = (8n+5)(8*n+9).

Original entry on oeis.org

45, 221, 525, 957, 1517, 2205, 3021, 3965, 5037, 6237, 7565, 9021, 10605, 12317, 14157, 16125, 18221, 20445, 22797, 25277, 27885, 30621, 33485, 36477, 39597, 42845, 46221, 49725, 53357, 57117, 61005, 65021, 69165, 73437, 77837, 82365

Offset: 0

Miklos Kristof, Oct 29 2008

From Miklos Kristof, Nov 03 2008: (Start)
f(y) = y^4*(1 + y^4) = y^4 - y^8 + y^12 - y^16 + y^20 - y^24 + ...
Integral_{y} f(y) dy = y^5/5 - y^9/9 + y^13/13 - y^17/17 + y^21/21 - y^25/25 + ...
Integral_{y=0..1} f(y) dy = 1/5 - 1/9 + 1/13 - 1/17 + 1/21 - 1/25 + ...
                  = (9 - 5)/(5*9) + (17 - 13)/(13*17) + (25 - 21)/(21*25) + ...
                  = 4/(5*9) + 4/(13*17) + 4/(21*25) + ...
Integral_{y=0..1} f(y) dy = Sum_{m>=0} 4/((8*m+5)*(8*m+9))
                  = -(1/8)*sqrt(2)*Pi + 1 - (1/4)*sqrt(2)*log(1+sqrt(2))
                  = 0.13302701266008896241... (End)

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

seq((8*m+5)*(8*m+9),m=0..40); # Miklos Kristof, Nov 03 2008

Table[(8n+5)(8n+9),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{45,221,525},40] (* Harvey P. Dale, Oct 10 2015 *)

a(n)=(8*n+5)*(8*n+9) \\ Charles R Greathouse IV, Jun 17 2017

G.f: (45 + 86*x - 3*x^2)/(1-x)^3.
E.g.f.: (45 + 176*x + 64*x^2)*exp(x).
a(n) = A004770(n) * A004768(n). - Reinhard Zumkeller, Oct 30 2008

A141759 a(n) = 16n^2 + 32n + 15.

Original entry on oeis.org

15, 63, 143, 255, 399, 575, 783, 1023, 1295, 1599, 1935, 2303, 2703, 3135, 3599, 4095, 4623, 5183, 5775, 6399, 7055, 7743, 8463, 9215, 9999, 10815, 11663, 12543, 13455, 14399, 15375, 16383, 17423, 18495, 19599, 20735, 21903, 23103, 24335, 25599

Offset: 0

Miklos Kristof, Sep 15 2008

Via the partial fraction decomposition 1/((4n+3)*(4n+5)) = (1/2) *(1/(4n+3) -1/(4n+5)) we find 2*Sum_{n>=0} (-1)^n/a(n) = 2*Sum_{n>=0} (-1)^n/( (4*n+3)*(4*n+5) ) = 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ... = (1/1 + 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ..)-1 = Sum_{n>=0} (-1)^n/A016813(n) + Sum_{n>=0} (-1)^n/A004767(n) -1 = -1 + Sum_{n>=0} b(n)/n^1 where b(n) = 1, 0, 1, 0, -1, 0, -1, 0 is a sequence with period length 8, one of the Dirichlet L-series modulo 8. The alternating sum becomes -1 +L(m=8,r=4,s=1) = Pi*sqrt(2)/4-1 = A093954 - 1.
Pi = 4 - 8*Sum(1/a(n)) noted by Bronstein-Semendjajew for the variant a(n) = (4n-1)*(4n+1) starting at n=1. - Frank Ellermann, Sep 18 2011
The identity (16*n^2-1)^2 - (64*n^2-8)*(2*n)^2 = 1 can be written as a(n)^2 - A158487(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 09 2012
Sequence found by reading the line from 15, in the direction 15, 63,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
Essentially the least common multiple of 4*n+1 and 4*n-1. - Colin Barker, Feb 11 2017

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed., 1965, ch. 4.1.8.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A074377, A093954, A133818, A158487.

[(4*n+3)*(4*n+5): n in [0..50]]; // Vincenzo Librandi, Sep 22 2011

A141759:=n->16*n^2 + 32*n + 15: seq(A141759(n), n=0..60);

LinearRecurrence[{3, -3, 1}, {15, 63, 143}, 50] (* Vincenzo Librandi, Feb 09 2012 *)

a(n)=n*(n+2)<<4+15 \\ Charles R Greathouse IV, Oct 27 2011

G.f.: (15+18*x-x^2)/(1-x)^3.
E.g.f.: (15+48*x+16*x^2)*exp(x).
a(n) = a(-n-2) = A016802(n+1) - 1. - Bruno Berselli, Sep 22 2011
From Amiram Eldar, Feb 04 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = Pi/(2*sqrt(2)) (A093954).
Product_{n>=0} (1 - 1/a(n)) = sin(Pi/(2*sqrt(2))). (End)

Formula indices corrected by R. J. Mathar, Jul 07 2009

A135399 a(n) = (-1)^n + (-2)^n + 3^n (-1, -2 and 3 are the roots of the equation x^3 = 7*x + 6).

Original entry on oeis.org

3, 0, 14, 18, 98, 210, 794, 2058, 6818, 19170, 60074, 175098, 535538, 1586130, 4799354, 14316138, 43112258, 129009090, 387682634, 1161737178, 3487832978, 10458256050, 31385253914, 94134790218, 282446313698, 847255055010, 2541932937194, 7625463267258

Offset: 0

Miklos Kristof, Dec 11 2007

seq(a(3*n+2)/14, n=0..9) = 1, 15, 487, 12507, 342811, 9214935, 249130927, 6723913587, 181566638371, 4902131463855
seq(a(3*n+1)/98, n=0..9) = 0, 1, 21, 613, 16185, 439921, 11854461, 320257693, 8645459745, 233439396841
seq(a(2*n+1)/6, n=0..14) = 0, 3, 35, 343, 3195, 29183, 264355, 2386023, 21501515, 193622863, 1743042675, 15689131703, 141209175835, 1270910544543, 11438306748995
seq(a(6*n+1)/294, n=0..4) = 0, 7, 5395, 3951487, 2881819915

			a(3) = (-1)^3 + (-2)^3 + 3^3 = -1 - 8 + 27 = 18.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,7,6).

Table[(-1)^n+(-2)^n+3^n,{n,0,30}] (* or *) LinearRecurrence[{0,7,6},{3,0,14},30] (* Harvey P. Dale, Oct 18 2015 *)

a(n)=(-1)^n + (-2)^n + 3^n \\ Charles R Greathouse IV, Oct 12 2016

G.f.: (3 - 7*x^2)/(1-7*x^2-6*x^3).
E.g.f.: exp(-x) + exp(-2*x) + exp(3*x)
a(0)=3, a(1)=0, a(2)=14, a(n) = 7*a(n-2) + 6*a(n-3). - Harvey P. Dale, Oct 18 2015

A133818 a(n) = (8n+3)(8n+5)(8n+7)(8*n+9).

Original entry on oeis.org

945, 36465, 229425, 801009, 2070705, 4456305, 8473905, 14737905, 23961009, 36954225, 54626865, 77986545, 108139185, 146289009, 193738545, 251888625, 322238385, 406385265, 506025009, 622951665, 759057585, 916333425, 1096868145

Offset: 0

Miklos Kristof, Jan 06 2008, Sep 15 2008

Also 1/3-1/5-1/7+1/9+1/11-1/13-1/15+1/17+1/19--++... = Pi*sqrt(2)/4-1 - Miklos Kristof, Sep 15 2008

Also sum(2*(-1)^n/((4*n+3)*(4*n+5)), n=0..infinity) = Pi*sqrt(2)/4-1 - Miklos Kristof, Sep 15 2008

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

seq((8*n+3)*(8*n+5)*(8*n+7)*(8*n+9), n=0..30);
sum(32*(4*n+3)/((8*n+3)*(8*n+5)*(8*n+7)*(8*n+9)), n=0..infinity) = Pi*sqrt(2)/4-1. Maple: evalf(Pi*sqrt(2)/4-1, 30); gives 0.11072073453959156175397024752... - Miklos Kristof, Sep 15 2008

Times@@@(#+{3,5,7,9}&/@(8Range[0,25])) (* Harvey P. Dale, Mar 14 2011 *)

G.f.: 3*(315 + 10580*x + 18850*x^2 + 3028*x^3 - 5*x^4)/(1-x)^5.
E.g.f: (945 + 35520*x + 78720*x^2 + 36864*x^3 + 4096*x^4)*exp(x).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Apr 26 2021

A135400 a(n) = (4n^4 - 4n^3 - n^2 + 3*n)/2.

Original entry on oeis.org

1, 17, 108, 382, 995, 2151, 4102, 7148, 11637, 17965, 26576, 37962, 52663, 71267, 94410, 122776, 157097, 198153, 246772, 303830, 370251, 447007, 535118, 635652, 749725, 878501, 1023192, 1185058, 1365407, 1565595

Offset: 1

Miklos Kristof, Dec 11 2007

Form the infinite matrix:
    1   2   4   7  11 ...
    3   5   8  12  17 ...
    6   9  13  18  24 ...
   10  14  19  25  32 ...
   15  20  26  33  41 ...
   ...
The diagonal elements are b(n) = 1, 5, 13, 25, 41, ... = 2*n*(n-1)+1 = A001844(n-1).
M(n,m) = ((n+m)^2-n-3*m+2)/2.
a(n) = M(n,b(n)) = M(1,1), M(2,5), M(3,13), M(4,25), M(5,41), ...
Let us define the PHI algebra as follows:
The basis of the PHI algebra is the PHI(1), PHI(2), PHI(3), ... elements, and the production rules are:
PHI(M(n,m))*PHI(n) = PHI(m) and every other production is zero.
An element of the PHI algebra is X = Sum_{n>=1} c(n)*PHI(n), where c(n) are real or complex constants.
UNIT = Sum_{n>=1} PHI(b(n)) = PHI(1) + PHI(5) + PHI(13) + PHI(25)+ ...
For every X elements: UNIT*X = X.
OMEGA = Sum_{n>=1} PHI(n) = PHI(1) + PHI(2) + PHI(3) + ...
ULTRA = Sum_{n>=1} PHI(a(n)) = PHI(1) + PHI(17) + PHI(108) + PHI(382) + ...
ULTRA*OMEGA = UNIT.
The PHI algebra is a nonassociative, but universal algebra; every finite or countable algebra can be modeled in the PHI algebra.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001844.

seq(2*n^4-2*n^3-1/2*n^2+3/2*n,n=1..30); for n from 1 to 30 do b[n]:=2*n*(n-1)+1 od: seq(((n+b[n])^2-n-3*b[n]+2)/2,n=1..30);

Table[2n^4-2n^3-n^2/2+(3n)/2,{n,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,17,108,382,995},30] (* Harvey P. Dale, May 25 2012 *)

a(n)=n*(4*n^3-4*n^2-n+3)/2 \\ Charles R Greathouse IV, Oct 12 2016

G.f.: (2*x^4 + 33*x^3 + 12*x^2 + x)/(1-x)^5.
E.g.f.: (1/2)*(4*x^4 + 20*x^3 + 15*x^2 + 2*x)*exp(x).
a(1)=1, a(2)=17, a(3)=108, a(4)=382, a(5)=995, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, May 25 2012

A133766 a(n) = (4n+1)(4n+3)(4*n+5).

Original entry on oeis.org

15, 315, 1287, 3315, 6783, 12075, 19575, 29667, 42735, 59163, 79335, 103635, 132447, 166155, 205143, 249795, 300495, 357627, 421575, 492723, 571455, 658155, 753207, 856995, 969903, 1092315, 1224615, 1367187, 1520415, 1684683, 1860375, 2047875, 2247567, 2459835

Offset: 0

Miklos Kristof, Jan 02 2008

L. B. W. Jolley, Summation of Series, Dover, 1961.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001539, A154633.

Maple
```
seq((4*n+1)*(4*n+3)*(4*n+5),n=0..40);
```

Table[c=4n;(c+1)(c+3)(c+5),{n,0,30}] (* or *) LinearRecurrence[{4,-6,4,-1},{15,315,1287,3315},30] (* Harvey P. Dale, May 06 2012 *)

a(n)=(4*n+1)*(4*n+3)*(4*n+5) \\ Charles R Greathouse IV, Oct 16 2015

G.f.: 3*(5 + 85*x + 39*x^2 - x^3)/(1-x)^4 .
E.g.f: (15 + 300*x + 336*x^2 + 64*x^3)*exp(x) .
Sum_{n>=0} 4/a(n) = (Pi-2)/4. [Jolley, eq. 238]
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Harvey P. Dale, May 06 2012
Sum_{n>=0} (-1)^n/a(n) = 1/8 + (log(2*sqrt(2)+3) - Pi)/(16*sqrt(2)). - Amiram Eldar, Feb 27 2022

A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

Original entry on oeis.org

0, 2, 0, 3, 2, 7, 10, 23, 42, 87, 170, 343, 682, 1367, 2730, 5463, 10922, 21847, 43690, 87383, 174762, 349527, 699050, 1398103, 2796202, 5592407, 11184810, 22369623, 44739242, 89478487, 178956970, 357913943, 715827882, 1431655767, 2863311530, 5726623063, 11453246122, 22906492247, 45812984490

Offset: 0

Miklos Kristof, Dec 07 2007

Partial sums of A155980 for n > 2. - Klaus Purath, Jan 30 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A005578, A099754.
Cf. A001045, A327767, A328284.
Cf. A007583, A062092, A087289, A020988 (even bisection), A163834 (odd bisection), A078008, A084247, A181565.

List([0..40], n-> (2^n+3-7*(-1)^n+3*0^n)/6); # G. C. Greubel, Sep 02 2019

a135351:=func< n | (2^n+3-7*(-1)^n+3*0^n)/6 >; [ a135351(n): n in [0..32] ]; // Klaus Brockhaus, Dec 05 2009

G(x):=x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)): f[0]:=G(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/n!,n=0..30);

Join[{0}, Table[(2^n +3 -7*(-1)^n)/6, {n,40}]] (* G. C. Greubel, Oct 11 2016 *)
LinearRecurrence[{2,1,-2},{0,2,0,3},40] (* Harvey P. Dale, Feb 13 2024 *)

a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; \\ Michel Marcus, Oct 11 2016

[(2^n+3-7*(-1)^n+3*0^n)/6 for n in (0..40)] # G. C. Greubel, Sep 02 2019

G.f.: x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)).
E.g.f.: (exp(2*x) + 3*exp(x) - 7*exp(-x) + 3)/6.
From Paul Curtz, Dec 20 2020: (Start)
a(n) + (period 2 sequence: repeat [1, -2]) = A328284(n+2).
a(n+1) + (period 2 sequence: repeat [-2, 1]) = A001045(n).
a(n+1) + (period 2 sequence: repeat [-1, 0]) = A078008(n).
a(n+1) + (period 2 sequence : repeat [-3, 2]) = -(-1)^n*A084247(n).
a(n+4) = a(n+1) + 7*A001045(n).
a(n+4) + a(n+1) = A181565(n).
a(2*n+2) + a(2*n+3) = A087289(n) = 3*A007583(n).
a(2*n+1) = A163834(n), a(2*n+2) = A020988(n). (End)

First part of definition corrected by Klaus Brockhaus, Dec 05 2009

A133767 a(n) = (4n+3)(4n+5)(4*n+7).

Original entry on oeis.org

105, 693, 2145, 4845, 9177, 15525, 24273, 35805, 50505, 68757, 90945, 117453, 148665, 184965, 226737, 274365, 328233, 388725, 456225, 531117, 613785, 704613, 803985, 912285, 1029897, 1157205, 1294593, 1442445, 1601145, 1771077, 1952625, 2146173, 2352105, 2570805, 2802657

Offset: 0

Miklos Kristof, Jan 02 2008

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Maple
```
seq((4*n+3)*(4*n+5)*(4*n+7),n=0..40);
```

a[n_]:=(4*n+3)*(4*n+5)*(4*n+7); Array[a,35,0] (* Stefano Spezia, Aug 27 2025 *)

G.f.: 3*(35 + 91*x + x^2 + x^3)/(1-x)^4.
E.g.f: (105 + 588*x + 432*x^2 + 64*x^3)*exp(x).
Sum_{m>0} 4/a(m) = 5/6 - Pi/4.

a(31)-a(34) from Stefano Spezia, Aug 27 2025

cp's OEIS Frontend

User: Miklos Kristof

A190818 Expansion of e.g.f.: 1/(1-2*tanh(x)).

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A146301 a(n) = (8*n+3)*(8*n+7).

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A146302 a(n) = (8*n+5)*(8*n+9).

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A141759 a(n) = 16n^2 + 32n + 15.

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A135399 a(n) = (-1)^n + (-2)^n + 3^n (-1, -2 and 3 are the roots of the equation x^3 = 7*x + 6).

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A133818 a(n) = (8*n+3)*(8*n+5)*(8*n+7)*(8*n+9).

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A135400 a(n) = (4*n^4 - 4*n^3 - n^2 + 3*n)/2.

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A146301 a(n) = (8n+3)(8*n+7).

A146302 a(n) = (8n+5)(8*n+9).

A133818 a(n) = (8n+3)(8n+5)(8n+7)(8*n+9).

A135400 a(n) = (4n^4 - 4n^3 - n^2 + 3*n)/2.

A133766 a(n) = (4n+1)(4n+3)(4*n+5).

A135351 a(n) = (2^n + 3 - 7(-1)^n + 30^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

A133767 a(n) = (4n+3)(4n+5)(4*n+7).