A159741 -id:A159741 - cp's OEIS frontend

A159742 If an array is made of columns of -nacci sequences (Fibonacci, tribonacci, etc.), all starting with 1,1,2,..., the NW-to-SE diagonals can be extended by computation. This sequence is diagonal 6. See A159741 for details.

13, 44, 108, 236, 492, 1004, 2028, 4076, 8172, 16364, 32748, 65516, 131052, 262124, 524268, 1048556, 2097132, 4194284, 8388588, 16777196, 33554412, 67108844, 134217708, 268435436, 536870892, 1073741804, 2147483628, 4294967276, 8589934572, 17179869164

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

[13] cat [4*(2^(n+2) - 5): n in [2..30]]; // G. C. Greubel, May 22 2018

T := proc(n,m) option remember ; if n < 0 then 0; elif n <= 1 then 1; elif n = 2 then 2; else add(procname(n-i,m),i=1..m) ; fi: end: A159742 := proc(n) T(n+5,n+1) ; end: seq(A159742(n),n=1..40) ; # R. J. Mathar, Apr 22 2009

CoefficientList[Series[(2*z^2 + 5*z + 13)/(2*z^2 - 3*z + 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
Join[{13}, Table[4*(2^(n + 2) - 5), {n, 2, 50}]] (* G. C. Greubel, May 22 2018 *)
LinearRecurrence[{3,-2},{13,44,108},30] (* Harvey P. Dale, Jul 10 2018 *)

for(n=1, 30, print1(if(n==1, 13, 4*(2^(n+2) - 5)), ", ")) \\ G. C. Greubel, May 22 2018

From R. J. Mathar, Apr 22 2009: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), n>3.
a(n) = 16*2^n - 20, n>1. (End)

More terms from R. J. Mathar, Apr 22 2009

A159746 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 9. See A159741 for details.

Original entry on oeis.org

55, 274, 773, 1793, 3840, 7936, 16128, 32512, 65280, 130816, 261888, 524032, 1048320, 2096896, 4194048, 8388352, 16776960, 33554176, 67108608, 134217472, 268435200, 536870656, 1073741568, 2147483392, 4294967040, 8589934336, 17179868928, 34359738112

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

LinearRecurrence[{3,-2},{55,274,773,1793,3840,7936},30] (* Harvey P. Dale, Aug 18 2022 *)

Interpolate a(n)=(4*n^3-30*n^2+92*n+657)/3 offset 0 & b(n)=(-4*n^3+24*n^2-65*n+840)/3 offset 0. Add 55 as a leader. Binomially transform. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 08 2009]

a(n) = 2^(7+n)-256 for n>=5. - R. J. Mathar, Feb 04 2021

a(7) and beyond by R. J. Mathar, Feb 04 2021

A159748 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 11. See A159741 for details.

Original entry on oeis.org

144, 927, 2872, 6930, 15109, 31489, 64256, 129792, 260864, 523008, 1047296, 2095872, 4193024, 8387328, 16775936, 33553152, 67107584, 134216448, 268434176, 536869632, 1073740544, 2147482368, 4294966016, 8589933312, 17179867904

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

nonn

Cf. A159741 .

T := proc(n,m) option remember ; if n < 0 then 0; elif n <= 1 then 1; elif n = 2 then 2; else add(procname(n-i,m),i=1..m) ; fi: end: A159748 := proc(n) T(n+10,n+1) ; end: seq(A159748(n),n=1..40) ; # R. J. Mathar, Apr 22 2009

a(n) = 3*a(n-1) - 2*a(n-2), n > 8. - R. J. Mathar, Apr 22 2009

Interpolate a(n) = (4*n^5 - 60*n^4 + 415*n^3 - 1665*n^2 + 3826*n + 11745)/15 offset 0 and b(n) = -4*n^5 + 50*n^4 - 305*n^3 + 1127.5*n^2 - 2443.5 + 17430)/15 offset 0. Add the leading 144. Binomially transform. - Al Hakanson (hawkuu(AT)gmail.com), Jun 08 2009

More terms from R. J. Mathar, Apr 22 2009

A159743 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 7. See A159741 for details.

Original entry on oeis.org

21, 81, 208, 464, 976, 2000, 4048

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

Interpolate a(n)=2*n+60 offset 0 & b(n)=-2*n+67 offset 0. Add 21 as a leader. Binomially transform. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 08 2009]

A159744 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 8. See A159741 for details.

Original entry on oeis.org

34, 149, 401, 912, 1936, 3984, 8080

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

Interpolate a(n)=-2*n^2+9*n+115 offset 0 & b(n)=2*n^2-7*n+137 offset 0. Add 34 as a leader. Binomially transform. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 08 2009]

A159747 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 10. See A159741 for details.

Original entry on oeis.org

89, 504, 1490, 3525, 7617, 15808, 32192, 64960, 130496, 261568, 523712, 1048000, 2096576, 4193728, 8388032, 16776640, 33553856, 67108288, 134217152, 268434880, 536870336, 1073741248, 2147483072, 4294966720, 8589934016, 17179868608, 34359737792, 68719476160, 137438952896

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009

Interpolate a(n)=(-2*n^4+22*n^3-107.5*n^2+276.5*n+1245)/3 offset 0 & b(n)=(2*n^4-18*n^3+77.5*n^2-184.5*n+1713)/3 offset 0. Add a leading 89. Binomially transform. [From Al Hakanson (hawkuu(AT)gmail.com), Jun 08 2009]

a(n) = T(n+9,n+1) with T defined in A159741. a(n) = 2^(n+8)-576 for n>=6. - R. J. Mathar, Feb 04 2021

a(8) and beyond by R. J. Mathar, Feb 04 2021

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Reinhard Zumkeller, Feb 28 2010

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

A175161 a(n) = 8*(2^n + 1).

Original entry on oeis.org

16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832, 2147483656, 4294967304, 8589934600, 17179869192

Offset: 0

Reinhard Zumkeller, Feb 28 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n + 1): A000051 (m=1), A052548 (m=2), A140504 (m=4), A153973 (m=6), A231643 (m=5), this sequence (m=8), A175162 (m=16), A175163 (m=32).

Cf. A159741, A173786, A175166.

I:=[16,24]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

8*(2^Range[0, 40] + 1) (* G. C. Greubel, Jul 08 2021 *)
LinearRecurrence[{3,-2},{16,24},40] (* Harvey P. Dale, Feb 10 2022 *)

[8*(2^n +1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = A173786(n+3, 3).
a(n) = A175166(2*n)/A159741(n) for n > 0.
a(n) = 3*a(n-1) -2*a(n-2) with a(0)=16, a(1)=24. - Vincenzo Librandi, Dec 28 2010
G.f.: 8*(2 - 3*x)/((1-x)*(1-2*x)). - Chai Wah Wu, Jun 20 2020
a(n) = 8 * A000051(n). - Alois P. Heinz, Jun 20 2020
E.g.f.: 8*(exp(2*x) + exp(x)). - G. C. Greubel, Jul 08 2021

A175164 a(n) = 16*(2^n - 1).

Original entry on oeis.org

0, 16, 48, 112, 240, 496, 1008, 2032, 4080, 8176, 16368, 32752, 65520, 131056, 262128, 524272, 1048560, 2097136, 4194288, 8388592, 16777200, 33554416, 67108848, 134217712, 268435440

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), this sequence (m=16), A175165 (m=32), A175166 (m=64).

Cf. A140504, A173787.

I:=[0,16]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

16*(2^Range[0,40] - 1) (* G. C. Greubel, Jul 08 2021 *)

def A175164(n): return (1<Chai Wah Wu, Jun 27 2023

[16*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+4) - 16.
a(n) = A173787(n+4, 4).
a(2*n) = A140504(n+2)*A028399(n).
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=16. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 16*x/((1-x)*(1-2*x)).
E.g.f.: 16*(exp(2*x) - exp(x)). (End)

A175166 a(n) = 64*(2^n - 1).

Original entry on oeis.org

0, 64, 192, 448, 960, 1984, 4032, 8128, 16320, 32704, 65472, 131008, 262080, 524224, 1048512, 2097088, 4194240, 8388544, 16777152, 33554368, 67108800, 134217664, 268435392, 536870848, 1073741760

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), A175165 (m=32), this sequence (m=64).

Cf. A173787, A175161.

I:=[0,64]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

LinearRecurrence[{3,-2},{0,64},30] (* Harvey P. Dale, Apr 08 2015 *)

def A175166(n): return (1<Chai Wah Wu, Jun 27 2023

[64*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+6) - 64.
a(n) = A173787(n+6, 6).
a(2*n) = A175161(n)*A159741(n) for n > 0.
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=64. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 64*x/((1-x)*(1-2*x)).
E.g.f.: 64*(exp(2*x) - exp(x)). (End)

cp's OEIS Frontend

A159742 If an array is made of columns of -nacci sequences (Fibonacci, tribonacci, etc.), all starting with 1,1,2,..., the NW-to-SE diagonals can be extended by computation. This sequence is diagonal 6. See A159741 for details.

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A159746 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 9. See A159741 for details.

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A159748 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 11. See A159741 for details.

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A159743 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 7. See A159741 for details.

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A159744 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 8. See A159741 for details.

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A159747 If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the NW to SE diagonals can be extended by computation. The above is diagonal 10. See A159741 for details.

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A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

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Sage

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A175161 a(n) = 8*(2^n + 1).

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A175164 a(n) = 16*(2^n - 1).

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