A001539 -id:A001539 - cp's OEIS frontend

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

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

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.

Original entry on oeis.org

0, 3, 1, 15, 3, 35, 6, 63, 10, 99, 15, 143, 21, 195, 28, 255, 36, 323, 45, 399, 55, 483, 66, 575, 78, 675, 91, 783, 105, 899, 120, 1023, 136, 1155, 153, 1295, 171, 1443, 190, 1599, 210, 1763, 231, 1935, 253, 2115, 276, 2303, 300, 2499, 325

Offset: 0

Paul Curtz, Oct 21 2011

See, in A181318(n), A060819(n)*A060819(n+p): A060819(n)^2, A064038(n), a(n), A160050(n), A061037(n), A178242(n). The second differences a(n+2)-2*a(n+1)+a(n) = -5, 16, -26, 44, -61, 86, -110, 142, -173, 212, -250, 296, -341, 394, -446, 506, taken modulo 9 are periodic with the palindromic period 4, 7, 1, 8, 2, 5, 7, 7, 7, 5, 2, 8, 1, 7, 4.

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

Cf. quadrisections A014105, A001539, A000384, A141759.

Cf. A060819, A000217, A000466, A142705, A000034, A005563, A010689, A028552, A050534.

List([0..60], n -> n*(n+2)*(9-7*(-1)^n)/16); # G. C. Greubel, Feb 21 2019

[n*(n+2)*(9-7*(-1)^n)/16: n in [0..60]]; // Vincenzo Librandi, Nov 25 2011

A198148:=n->n*(n+2)*(9-7*(-1)^n)/16; seq(A198148(k), k=0..100); # Wesley Ivan Hurt, Oct 16 2013

LinearRecurrence[{0,3,0,-3,0,1},{0,3,1,15,3,35},60] (* Vincenzo Librandi, Nov 25 2011 *)

a(n)=n*(n+2)*(9-7*(-1)^n)/16 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n+2)*(9-7*(-1)^n)/16 for n in (0..60)] # G. C. Greubel, Feb 21 2019

a(n) = A060819(n)*A060819(n+2).
a(2n) = n*(n+1)/2 = A000217(n).
a(2n+1) = (2*n+1)*(2*n+3) = A000466(n+1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6), n>5.
a(n+1) - a(n) = (7*(-1)^n *(2*n^2+6*n+3) +18*n +27)/16.
a(n) = A142705(n) / A000034(n+1).
a(n) = A005563(n) / A010689(n+1). - Franklin T. Adams-Watters, Oct 21 2011
G.f. x*(3 +x +6*x^2 -x^4)/(1-x^2)^3. - R. J. Mathar, Oct 25 2011
a(n)*a(n+1) = a(A028552(n)) = A050534(n+2). - Bruno Berselli, Oct 26 2011
a(n) = numerator( binomial((n+2)/2,2) ). - Wesley Ivan Hurt, Oct 16 2013
E.g.f.: x*((24+x)*cosh(x) + (3+8*x)*sinh(x))/8. - G. C. Greubel, Sep 20 2018
Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Aug 12 2022

A001533 a(n) = (8n+1)(8*n+7).

Original entry on oeis.org

7, 135, 391, 775, 1287, 1927, 2695, 3591, 4615, 5767, 7047, 8455, 9991, 11655, 13447, 15367, 17415, 19591, 21895, 24327, 26887, 29575, 32391, 35335, 38407, 41607, 44935, 48391, 51975, 55687, 59527, 63495, 67591, 71815, 76167, 80647, 85255, 89991, 94855, 99847, 104967

Offset: 0

N. J. A. Sloane

From Klaus Purath, Aug 18 2022: (Start)
This is A028560(8*n+1), and thus a(n) + 9 is a square. (See formulas)
7 is the only prime number of this sequence in which all odd prime factors occur.
Each prime factor p appears exactly twice in any interval of p consecutive terms. If a(m) and a(n) are within such an interval containing p, then m + n == -1 (mod p). (End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001539, A004771, A017077, A017113, A028560, A250129.

a[n_] := (8 n + 1)*(8 n + 7); Array[a, 40, 0] (* Amiram Eldar, Sep 08 2022 *)
LinearRecurrence[{3,-3,1},{7,135,391},40] (* Harvey P. Dale, Jan 07 2023 *)

a(n)=(8*n+1)*(8*n+7) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*A001539(n) - 5.
a(n) = 128*n + a(n-1) with a(0)=7. - Vincenzo Librandi, Nov 12 2010
Sum_{n>=0} 1/a(n) = (Psi(7/8)-Psi(1/8))/48 = 0.1580099..., see A250129. - R. J. Mathar, May 30 2022 [ = (sqrt(2)+1)*Pi/48. - Amiram Eldar, Sep 08 2022]
From Klaus Purath, Aug 18 2022: (Start)
a(n) = A028560(8*n+1).
a(n) + 9 = ((a(n+1) - a(n-1))/32)^2 = A017113(n)^2.
a(2*n) = (a(n+1) - a(n-1))*n + 7. (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017077(n)*A004771(n).
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(cot(Pi/16)) + sin(Pi/8) * log(cot(3*Pi/16)))/12.
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/8)*cos(sqrt(5/2)*Pi/4).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/8)*cos(sqrt(2)*Pi/4). (End)
G.f.: -(7+114*x+7*x^2)/(x-1)^3. - R. J. Mathar, Apr 23 2024
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(7 + 64*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A154633 a(n) = (4n+1)(4n+3)(4n+5)(4*n+7).

Original entry on oeis.org

105, 3465, 19305, 62985, 156009, 326025, 606825, 1038345, 1666665, 2544009, 3728745, 5285385, 7284585, 9803145, 12924009, 16736265, 21335145, 26822025, 33304425, 40896009, 49716585, 59892105, 71554665, 84842505, 99900009, 116877705, 135932265, 157226505, 180929385

Offset: 0

Jaume Oliver Lafont, Jan 13 2009

3 divides a(n).

For n=5k, 5k+1, 5k+2 and 5k+3, a(n) is a multiple of 5. For n=5k+4, a(n)-9 is a multiple of 100. - Michel Marcus, Aug 21 2013

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

Cf. A133766, A001539, A157142.

a[n_] := (4*n + 1)*(4*n + 3)*(4*n + 5)*(4*n + 7); Array[a, 40, 0] (* Amiram Eldar, Feb 27 2022 *)

a(n) = (4*n+1)*(4*n+3)*(4*n+5)*(4*n+7); \\ Michel Marcus, Aug 21 2013

Sum_{n>=0} 1/a(n) = (3*Pi - 8)/144.
G.f.: 3*(35 + 980*x + 1010*x^2 + 20*x^3 + 3*x^4)/(1-x)^5.
a(n) = (4*n+1)*(4*n+3)*(4*n+5)*(4*n+7).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
Sum_{n>=0} (-1)^n/a(n) = 1/18 - Pi/(48*sqrt(2)). - Amiram Eldar, Feb 27 2022

A239545 Decimal expansion of Sum_{k>=0} (-1)^k/((2k+1)(2k+3)(2k+5)).

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Mar 21 2014

			0.0593657483653908214744970895766045271913128415885548942885347407051...

L. B. W. Jolley, Summation of Series, Dover, 1961, p. 44 (series n. 240)

Cf. A000796, A001539, A005408 (odd numbers) A021016 (Sum_{k>=0} 1/((2k+1)*(2k+3)*(2k+5))), A061550.

Mathematica
```
RealDigits[Pi/8 - 1/3, 10, 90, -1][[1]]
```

Pi/8-1/3 \\ Charles R Greathouse IV, May 05 2014

Equals Pi/8 - 1/3 = A019675 - A010701.

Equals Sum_{k>=1} 1/((4*k+1)*(4*k+3)) = Sum_{k>=1} 1/A001539(k). - Amiram Eldar, Jul 04 2020

cp's OEIS Frontend

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

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A198148 a(n) = n*(n+2)*(9 - 7*(-1)^n)/16.

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A001533 a(n) = (8*n+1)*(8*n+7).

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A154633 a(n) = (4*n+1)*(4*n+3)*(4*n+5)*(4*n+7).

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A239545 Decimal expansion of Sum_{k>=0} (-1)^k/((2k+1)*(2k+3)*(2k+5)).

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Mathematica

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Formula

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

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.

A001533 a(n) = (8n+1)(8*n+7).

A154633 a(n) = (4n+1)(4n+3)(4n+5)(4*n+7).

A239545 Decimal expansion of Sum_{k>=0} (-1)^k/((2k+1)(2k+3)(2k+5)).