A068377 -id:A068377 - cp's OEIS frontend

A118239 Engel expansion of cosh(1).

1, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8010, 8372

Offset: 1

Eric W. Weisstein, Apr 17 2006

Differs from A002939 only in first term.

This sequence is also the Pierce expansion of cos(1). - G. C. Greubel, Nov 14 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Engel Expansion.
Eric Weisstein's World of Mathematics, Pierce Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784, A002939, A068377.

Join[{1}, Table[(2 n - 2) (2 n - 3), {n, 2, 50}]] (* Bruno Berselli, Aug 04 2015 *)
Join[{1}, LinearRecurrence[{3,-3,1},{2,12,30},25]] (* G. C. Greubel, Oct 27 2016 *)
PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Cos[1] , 7!], 50] (* G. C. Greubel, Nov 14 2016 *)

a(n)=max(4*n^2-10*n+6, 1) \\ Charles R Greathouse IV, Oct 22 2014

A118239 = lambda n: falling_factorial(n*2,2) if n>0 else 1
print([A118239(n) for n in (0..46)]) # Peter Luschny, Aug 04 2015

a(n) = A002939(n-1) = 2*(n-1)*(2*n-3) for n>1.
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(1 - x + 9*x^2 - x^3)/(1-x)^3. (End)
E.g.f.: -6 + x + 2*(3 - 3*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 27 2016
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = log(2) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - Pi/4 + log(2)/2. (End)

A068379 Engel expansion of sinh(1/2).

Original entry on oeis.org

2, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648, 25920, 27224, 28560

Offset: 1

Benoit Cloitre, Mar 03 2002

Cf. A006784 for Engel expansion definition.

The MathWorld link mentions the closed form of the Engel expansion of sinh(1) = A068377. - Georg Fischer, Nov 22 2020

			sinh(1/2) = 1/2 + 1/(2*24) + 1/(2*24*80) + 1/(2*24*80*168) + 1/(2*24*80*168*288) + ... = 0.52109530549374736162242562641... = A334367.

Vincenzo Librandi, Table of n, a(n) for n = 1..10001
Eric W. Weisstein's World of Mathematics, Engel Expansion.
Wikipedia, Engel Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002943, A006784, A033586, A068377, A068380, A334367.

[2] cat [8*(n*(2*n-3)+1): n in [2..50]]; // Vincenzo Librandi, Jan 31 2012

Table[If[n==1, 2, 8*(n*(2*n-3)+1)], {n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
LinearRecurrence[{3,-3,1},{2,24,80,168},50] (* Harvey P. Dale, Mar 21 2017 *)

a(n)=if(n<=1,2,8*(n*(2*n-3)+1)) \\ Charles R Greathouse IV, Jan 31 2012

a(n) = 8*(n*(2*n-3)+1) for n > 1, a(1)=2.
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: 2*x*(1+9*x+7*x^2-x^3)/(1-x)^3. (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = (3-log(2))/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/4 - Pi/16 - log(2)/8. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: 2*(4*exp(x)*(1 - x + 2*x^2) + (x - 4)).
a(n) = 2*A033586(n-1) for n >= 2.
a(n) = 4*A002943(n-1) for n >= 2. (End)

Edited, offset 1 and a(1)=2 in programs and b-file by Georg Fischer, Nov 22 2020

A068380 Engel expansion of sinh(1/3).

Original entry on oeis.org

3, 54, 180, 378, 648, 990, 1404, 1890, 2448, 3078, 3780, 4554, 5400, 6318, 7308, 8370, 9504, 10710, 11988, 13338, 14760, 16254, 17820, 19458, 21168, 22950, 24804, 26730, 28728, 30798, 32940, 35154, 37440, 39798, 42228, 44730, 47304, 49950, 52668, 55458, 58320, 61254

Offset: 1

Benoit Cloitre, Mar 03 2002

Cf. A006784 for the definition of the Engel expansion.

The MathWorld link mentions the closed form of the Engel expansion of sinh(1). - Georg Fischer, Nov 22 2020

			sinh(1/3) = 1/3 + 1/(3*54) + 1/(3*54*180) + 1/(3*54*180*378) + 1/(3*54*180*378*648) + ... = 0.33954055725615013910126061...

Eric Weisstein's World of Mathematics, Engel Expansion.
Wikipedia, Engel Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784, A002943, A068377, A068379.

LinearRecurrence[{3, -3, 1}, {3, 54, 180, 378}, 50]

PARI
```
a(n)=if(n<=1, 3, 18*(n*(2*n-3)+1));
```

my(x='x+O('x^43)); Vec(3*x*(1+15*x+9*x^2-x^3)/(1-x)^3) \\ Elmo R. Oliveira, May 29 2025

a(n) = 18*(n*(2*n-3)+1) for n > 1, a(1)=3. - Ralf Stephan, Sep 03 2003
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: 3*x*(x^3-9*x^2-15*x-1)/(x-1)^3. (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = (4-log(2))/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 - Pi/36 - log(2)/18. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: 3*(6*exp(x)*(1 - x + 2*x^2) + (x - 6)).
a(n) = 9*A002943(n-1) for n >= 2. (End)

Edited, offset 1 and a(1)=3 by Georg Fischer, Nov 23 2020

More terms from Elmo R. Oliveira, May 29 2025

A280187 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n), but 2 * (1^d + 2^d + 3^d + ... + d^d) is 0 (mod d) for each other d | n.

Original entry on oeis.org

6, 20, 110, 272, 506, 812, 2162, 2756, 3422, 4970, 6806, 7832, 11342, 12656, 17030, 18632, 22052, 27722, 29756, 31862, 36290, 38612, 51302, 54056, 56882, 62750, 65792, 68906, 72092, 85556, 96410, 100172, 120062, 124256, 128522

Offset: 1

Charles R Greathouse IV, Dec 28 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..4000

Primitive elements of A228870.

Subsequence of A002943. Also a subsequence of A028689, A036689, A053198, A068377, A079143, A128672, A220211 and other sequences ...- Paolo P. Lava, Jan 10 2017

has(n)=my(f=factor(n)[,1]); for(i=1,#f, if(n%(f[i]-1)==0 && f[i]>2, return(1))); 0
is(n)=if(n%2, return(0)); if(n%3==0, return(n==6)); if(n%20==0, return(n==20)); if(!has(n), return(0)); my(f=factor(n)[,1]); for(i=1,#f, if(has(n/f[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 28 2016

A098931 a(0) = 1, a(n) = 1 + 23 + 45 + 67 + ... + (2n)(2n+1) for n > 0.

Original entry on oeis.org

1, 7, 27, 69, 141, 251, 407, 617, 889, 1231, 1651, 2157, 2757, 3459, 4271, 5201, 6257, 7447, 8779, 10261, 11901, 13707, 15687, 17849, 20201, 22751, 25507, 28477, 31669, 35091, 38751, 42657, 46817, 51239, 55931, 60901, 66157, 71707, 77559, 83721

Offset: 0

Miklos Kristof, Oct 20 2004

If a(n) = a0, a1, a2, a3, ... then Sum(a(n))= a0, a0+a1, a0+a1+a2, a0+a1+a2+a3, ...

			a(0) = 1;
a(1) = 1 + 2*3 = 7;
a(2) = 1 + 2*3 + 4*5 = 27, etc.

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A068377.

[1+3*n^2+n*(5+4*n^2)/3: n in [0..40]]; // Vincenzo Librandi, Jul 28 2015

seq((4/3)*n^3+3*n^2+(5/3)*n+1, n=0..100); # Robert Israel, Jul 28 2015

Table[1 + 3 n^2 + n (5 + 4 n^2)/3, {n, 0, 40}] (* Robert G. Wilson v, Oct 23 2004 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 7, 27, 69}, 40] (* Vincenzo Librandi, Jul 28 2015 *)

a(n)=n*(4*n^2+9*n+5)/3+1 \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 1 + 3*n^2 + n*(5 + 4*n^2)/3.
G.f.: (1 + 3*x + 5*x^2 - x^3)/(1-x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Vincenzo Librandi, Jul 28 2015
From Robert Israel, Jul 28 2015: (Start)
E.g.f.: (1+6*x+7*x^2+(4/3)*x^3)*exp(x).
a(n) = 1 + Sum(A068377(i),i=1..n+1). (End)

Edited and extended by Robert G. Wilson v, Oct 23 2004

A226940 a(0)=0; if a(n-1) is odd, a(n) = n + a(n-1), otherwise a(n) = n - a(n-1).

Original entry on oeis.org

0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42, -30, 43, 57, 72, -56, 73, 91, 110, -90, 111, 133, 156, -132, 157, 183, 210, -182, 211, 241, 272, -240, 273, 307, 342, -306, 343, 381, 420, -380, 421, 463, 506, -462, 507, 553, 600, -552, 601, 651, 702, -650, 703, 757

Offset: 0

Enrico Santilli, Jun 23 2013

Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A081348 (second bisection); A002939, A054554, A054569, A068377.

[IsZero(n) select 0 else IsOdd(Self(n)) select n+Self(n) else n-Self(n): n in [0..60]]; // Bruno Berselli, Jul 01 2013

LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42}, 60] (* Bruno Berselli, Jul 01 2013 *)

makelist(coeff(taylor(x*(1+3*x+6*x^2-2*x^3+4*x^4+4*x^5+2*x^6-6*x^7+3*x^8+x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3), x, 0, n), x, n), n, 0, 60); /* Bruno Berselli, Jul 01 2013 */

G.f.: x*(1 +3*x +6*x^2 -2*x^3 +4*x^4 +4*x^5 +2*x^6 -6*x^7 +3*x^8 +x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3). [Bruno Berselli, Jul 01 2013]
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12). [Bruno Berselli, Jul 01 2013]
a(4n) = -A002939(n), a(4n+1) = A054569(n+1), a(4n+2) = A054554(n+2), a(4n+3) = A068377(n+2). [Bruno Berselli, Jul 02 2013]

More terms from Bruno Berselli, Jul 01 2013

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A118239 Engel expansion of cosh(1).

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A068379 Engel expansion of sinh(1/2).

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A068380 Engel expansion of sinh(1/3).

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A280187 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n), but 2 * (1^d + 2^d + 3^d + ... + d^d) is 0 (mod d) for each other d | n.

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A098931 a(0) = 1, a(n) = 1 + 2*3 + 4*5 + 6*7 + ... + (2n)*(2n+1) for n > 0.

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A226940 a(0)=0; if a(n-1) is odd, a(n) = n + a(n-1), otherwise a(n) = n - a(n-1).

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A098931 a(0) = 1, a(n) = 1 + 23 + 45 + 67 + ... + (2n)(2n+1) for n > 0.