A010688 -id:A010688 - cp's OEIS frontend

A146541 Binomial transform of A010688.

1, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Philippe Deléham, Oct 31 2008

Hankel transform is := 1,-48,0,0,0,0,0,0,0,...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000007, A011782, A000079, A003945, A046055, A146523, A082505, A135092.

Essentially the same as A132479, A077552, A020707.

Join[{1},2^Range[3,40]] (* Harvey P. Dale, Feb 28 2016 *)

Vec((1+6*x)/(1-2*x) + O(x^50)) \\ Colin Barker, Mar 17 2016

a(0)=1, a(n) = 2^(n+2) for n>0.
a(n) = Sum_{k, 0..n} A109466(n,k)*A146534(k).
a(n) = A132479(n), n>1. - R. J. Mathar, Nov 02 2008
G.f.: (1+6*x) / (1-2*x). - Colin Barker, Mar 17 2016

Corrected and extended by Harvey P. Dale, Feb 28 2016

A036543 a(n) = T(3,n), array T given by A048471.

Original entry on oeis.org

1, 9, 33, 105, 321, 969, 2913, 8745, 26241, 78729, 236193, 708585, 2125761, 6377289, 19131873, 57395625, 172186881, 516560649, 1549681953, 4649045865, 13947137601, 41841412809, 125524238433, 376572715305, 1129718145921

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+2) for n=1, 2, 3, ...

Cf. A146541 (inv. bin. transf.)

[4*3^n-3: n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

4*3^Range[0,25]-3 (* or *) LinearRecurrence[{4,-3},{1,9},25] (* Harvey P. Dale, Aug 16 2011 *)

vector(30, n, n--; 4*3^n-3) \\ G. C. Greubel, Nov 23 2018

[4*3^n-3 for n in range(30)] # G. C. Greubel, Nov 23 2018

Binomial transform of A084242. Second binomial transform of periodic sequence A010688. - Paul Barry, May 23 2003
From Paul Barry, May 23 2003: (Start)
a(n) = 4*3^n - 3;
G.f.: (1+5*x)/((1-x)*(1-3*x));
E.g.f.: 4*exp(3*x) - 3*exp(x). (End)
a(n) = 4*a(n-1) - 3*a(n-2); a(0)=1, a(1)=9. - Harvey P. Dale, Aug 16 2011
a(n) = 3*a(n-1) + 6. - Vincenzo Librandi, Nov 11 2011
a(n) = A171498(n) - 2. - Philippe Deléham, Apr 13 2013

A176415 Periodic sequence: repeat 7,1.

Original entry on oeis.org

7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7

Offset: 0

Klaus Brockhaus, Apr 17 2010

Interleaving of A010727 and A000012.
Also continued fraction expansion of (7+sqrt(77))/2.
Also decimal expansion of 71/99.
Essentially first differences of A047521.
Binomial transform of A176414.
Inverse binomial transform of 2*A020707 preceded by 7.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 4*x^3 + 10*x^4 + 10*x^5 + ... is the o.g.f. for A058187. - Peter Bala, Mar 13 2015

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

Cf. A010727 (all 7's sequence), A000012 (all 1's sequence), A092290 (decimal expansion of (7+sqrt(77))/2), A010688 (repeat 1, 7), A047521 (congruent to 0 or 7 mod 8), A176414 (expansion of (7+8*x)/(1+2*x)), A020707 (2^(n+2)), A058187.

&cat[ [7, 1]: n in [0..52] ];
[ 4+3*(-1)^n: n in [0..104] ];

PadRight[{},120,{7,1}] (* Harvey P. Dale, Dec 30 2018 *)

a(n)=7-n%2*6 \\ Charles R Greathouse IV, Oct 28 2011

a(n) = 4+3*(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 7, a(1) = 1.
a(n) = -a(n-1)+8 for n > 0; a(0) = 7.
a(n) = 7*((n+1) mod 2)+(n mod 2).
a(n) = A010688(n+1).
G.f.: (7+x)/(1-x^2).
Dirichglet g.f.: (1+6*2^(-s))*zeta(s). - R. J. Mathar, Apr 06 2011
Multiplicative with a(2^e) = 7, and a(p^e) = 1 for p >= 3. - Amiram Eldar, Jan 01 2023

A132728 Triangle T(n, k) = 4 - 3*(-1)^k, read by rows.

Original entry on oeis.org

1, 1, 7, 1, 7, 1, 1, 7, 1, 7, 1, 7, 1, 7, 1, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1

Offset: 0

Roger L. Bagula, Nov 17 2007

			Triangle begins as:
  1;
  1, 7;
  1, 7, 1;
  1, 7, 1, 7;
  1, 7, 1, 7, 1;
  1, 7, 1, 7, 1, 7;
  1, 7, 1, 7, 1, 7, 1;
  1, 7, 1, 7, 1, 7, 1, 7;
  1, 7, 1, 7, 1, 7, 1, 7, 1;
  1, 7, 1, 7, 1, 7, 1, 7, 1, 7;
  1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1;

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A010688, A047393, A132742.

[4 -3*(-1)^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 14 2021

Table[PadRight[{},n,{1,7}],{n,20}]//Flatten (* Harvey P. Dale, Aug 02 2019 *)
Table[4 -3*(-1)^k, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)

flatten([[4 -3*(-1)^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021

From G. C. Greubel, Feb 14 2021: (Start)
T(n, k) = 4 - 3*(-1)^k.
Sum_{k=0..n} T(n, k) = (8*n + 5 - 3*(-1)^n)/2 = A047393(n+2). (End)
Bivariate g.f.: (1 + 7*x*y)/((1 - x)*(1 - x*y)*(1 + x*y)). - J. Douglas Morrison, Jul 19 2021

Edited and corrected by Joerg Arndt, Dec 26 2018

Offset and title changed by G. C. Greubel, Feb 14 2021

A146534 a(n) = 4C(2n,n) - 30^n.

Original entry on oeis.org

1, 8, 24, 80, 280, 1008, 3696, 13728, 51480, 194480, 739024, 2821728, 10816624, 41602400, 160466400, 620470080, 2404321560, 9334424880, 36300541200, 141381055200, 551386115280, 2153031497760, 8416395854880, 32933722910400, 128990414732400, 505642425751008, 1983674131792416

Offset: 0

Philippe Deléham, Oct 31 2008

nonn

Cf. A000984, A010688, A029609, A029651, A039599, A088218, A100320, A144706, A146533, A240530.

a[n_]:=4*Binomial[2n,n]-3*KroneckerDelta[n,0]; Array[a,27,0] (* Stefano Spezia, Feb 14 2025 *)

a(n) = Sum_{k=0..n} A039599(n,k)*A010688(k).
From Stefano Spezia, Feb 14 2025: (Start)
G.f.: 4/sqrt(1 - 4*x) - 3.
E.g.f.: 4*exp(2*x)*BesselI(0, 2*x) - 3. (End)

a(22)-a(26) from Stefano Spezia, Feb 14 2025

A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

Original entry on oeis.org

1, 7, 10, 25, 46, 97, 190, 385, 766, 1537, 3070, 6145, 12286, 24577, 49150, 98305, 196606, 393217, 786430, 1572865, 3145726, 6291457, 12582910, 25165825, 50331646, 100663297, 201326590, 402653185, 805306366, 1610612737, 3221225470, 6442450945, 12884901886

Offset: 0

Paul Curtz, Dec 28 2016

a(n) mod 9 = period 2: repeat [1, 7].
The last digit from 7 is of period 4: repeat [7, 0, 5, 6].
The bisection A096045 = 1, 10, 46, ... is based on Bernoulli numbers.
a(n) is a companion to A051049(n).
With an initial 0, A051049(n) is an autosequence of the first kind.
With an initial 2, this sequence is an autosequence of the second kind.
See the reference.
Difference table:
1,   7, 10, 25, 46,  97, ... = this sequence.
6,   3, 15, 21, 51,  93, ... = 3*A014551(n)
-3, 12,  6, 30, 42, 102, ... = -3 followed by 6*A014551(n).
The main diagonal of the difference table gives A003945: 1, 3, 6, 12, 24, ...

			a(0) = 1, a(1) = 2*1 + 5 = 7, a(2) = 2*7 - 4 = 10, a(3) = 2*10 + 5 = 25.

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

Cf. A003945, A005010, A010688, A010710, A014551, A051049, A096045, A199116.

seq(3*2^n-(-1)^n*(1+irem(n+1,2)),n=0..32); # Peter Luschny, Dec 29 2016

LinearRecurrence[{2,1,-2},{1,7,10},50] (* Paolo Xausa, Nov 13 2023 *)

Vec((1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 28 2016

a(2n) = 3*4^n - 2, a(2n+1) = 6*4^n + 1.
a(n+2) = a(n) + 9*2^n, a(0) = 1, a(1) = 7.
a(n) = 2*A051049(n+1) - A051049(n).
From Colin Barker, Dec 28 2016: (Start)
a(n) = 3*2^n - 2 for n even.
a(n) = 3*2^n + 1 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2.
G.f.: (1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)

A133190 a(n) = 2a(n-1) - a(n-2) + 2a(n-3).

Original entry on oeis.org

1, 3, 3, 5, 13, 27, 51, 101, 205, 411, 819, 1637, 3277, 6555, 13107, 26213, 52429, 104859, 209715, 419429, 838861, 1677723, 3355443, 6710885, 13421773, 26843547, 53687091, 107374181, 214748365, 429496731, 858993459, 1717986917, 3435973837

Offset: 0

Paul Curtz, Dec 17 2007

Index entries for linear recurrences with constant coefficients, signature (2,-1,2).

A010688 := proc(n) if n mod 2 = 0 then 1; else 7; fi ; end:
A133190 := proc(n) (4*2^n+(-1)^floor(n/2)*A010688(n))/5 ; end:
seq(A133190(n),n=0..30) ; # R. J. Mathar, Jan 13 2008

LinearRecurrence[{2,-1,2},{1,3,3},40] (* Harvey P. Dale, Jun 22 2022 *)

From R. J. Mathar, Jan 13 2008: (Start)
O.g.f.: (2*x+1)*(x-1)/((2*x-1)*(x^2+1)).
a(n) = (4*2^n + (-1)^floor(n/2)*A010688(n))/5. (End)

A138122 Cousin primes, the lower of which is 7 (mod 10).

Original entry on oeis.org

7, 11, 37, 41, 67, 71, 97, 101, 127, 131, 277, 281, 307, 311, 397, 401, 457, 461, 487, 491, 757, 761, 877, 881, 907, 911, 937, 941, 967, 971, 1087, 1091, 1297, 1301, 1447, 1451, 1567, 1571, 1597, 1601, 1867, 1871, 2137, 2141, 2347, 2351, 2377, 2381, 2437

Offset: 1

Roger L. Bagula, May 04 2008

nonn

Start from the intersection of A023200 and A030432, then add the associated members of A046132. The last digits are obviously periodic as A010688. - R. J. Mathar, Nov 26 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A132243, A132242, A076746.

a[0] = 7; a[n_] := a[n] = a[n - 1] + 10; Flatten[Table[If[PrimeQ[a[n]] && PrimeQ[a[n] + 4], {a[n],a[n] + 4}, {}], {n, 0, 1000}]]

Replaced Mathematica definition by humanly readable phrase. - R. J. Mathar, Nov 26 2008

A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 5, 1, 3, 1, 1, 6, 1, 4, 1, 2, 1, 7, 1, 5, 1, 3, 1, 1, 8, 1, 6, 1, 4, 1, 2, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2

Offset: 2

Paul Curtz, Feb 14 2010

One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:
  -1, -1, 0, 1,  4,  9, 20,  41,  84, 169,  340,  681, 1364 ...: A084639;
  -1, -1, 1, 3,  9, 19, 41,  83, 169, 339,  681, 1363, 2729;
  -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
  -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
B(n,k) = (n-1)*A001045(k) - T(n,k).
First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).

			The array T(n,k) starts in row n=2 with columns k>=0 as:
  1,  2, 1,  2, 1,  2, 1,  2, 1,  2, 1,  2 ... A000034;
  1,  3, 1,  3, 1,  3, 1,  3, 1,  3, 1,  3 ... A010684;
  1,  4, 1,  4, 1,  4, 1,  4, 1,  4, 1,  4 ... A010685;
  1,  5, 1,  5, 1,  5, 1,  5, 1,  5, 1,  5 ... A010686;
  1,  6, 1,  6, 1,  6, 1,  6, 1,  6, 1,  6 ... A010687;
  1,  7, 1,  7, 1,  7, 1,  7, 1,  7, 1,  7 ... A010688;
  1,  8, 1,  8, 1,  8, 1,  8, 1,  8, 1,  8 ... A010689;
  1,  9, 1,  9, 1,  9, 1,  9, 1,  9, 1,  9 ... A010690;
  1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
Antidiagonal triangle begins as:
  1;
  1,  2;
  1,  3,  1;
  1,  4,  1,  2;
  1,  5,  1,  3,  1;
  1,  6,  1,  4,  1,  2;
  1,  7,  1,  5,  1,  3,  1;
  1,  8,  1,  6,  1,  4,  1,  2;
  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 13,  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 14,  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;

G. C. Greubel, Antidiagonals n = 0..50 of the array, flattened

Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.

Cf. A001045, A024206, A084639, A093178, A173114.

T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* G. C. Greubel, Dec 03 2021 *)

flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # G. C. Greubel, Dec 03 2021

From G. C. Greubel, Dec 03 2021: (Start)
T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
Sum_{k=0..n} T(n-k, k) = A024206(n).
Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).
T(2*n-2, n) = A093178(n). (End)

A267317 a(n) = final digit of 2^n-1.

Original entry on oeis.org

0, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

Decimal expansion of 25/1818.

Period 4: repeat [1, 3, 7, 5] for n > 0.

Eric Weisstein's World of Mathematics, Mersenne Number
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000225, A000689, A010688, A010703, A010879, A080172.

[0] cat &cat[[1, 3, 7, 5]^^25]; // Bruno Berselli, Jan 13 2016

A267317:=n->(2^n-1) mod 10: seq(A267317(n), n=0..150); # Wesley Ivan Hurt, Jun 15 2016

Mathematica
```
Table[Mod[2^n - 1, 10], {n, 0, 120}]
```

a(n) = if(n==0, 0, if(n%4==0, 5, if(n%4==1, 1, if(n%4==2, 3, if(n%4==3, 7))))) \\ Felix Fröhlich, Jan 19 2016

a(n) = lift(Mod(2^n-1, 10)) \\ Felix Fröhlich, Jan 19 2016

G.f.: x*(1 + 2*x + 5*x^2)/(1 - x + x^2 - x^3).
a(n) = A010879(A000225(n)).
a(n) = A000689(n) - 1.
a(n) = (1+(-1)^n)*(-1)^(n*(n-1)/2)/2 + 3*(1-(-1)^n)*(-1)^(n*(n+1)/2)/2 + 4 for n > 0, a(0) = 0. [Bruno Berselli, Jan 13 2016]
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = a(n-4) for n>4.
a(2k+2) = A010703(k), a(2k+1) = A010688(k). (End)
From Wesley Ivan Hurt, Jul 06 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 3.
a(n) = 4 + cos(n*Pi/2) - 3*sin(n*Pi/2) for n > 0. (End)
E.g.f.: -5 + cos(x) - 3*sin(x) + 4*exp(x). - Ilya Gutkovskiy, Jul 06 2016

cp's OEIS Frontend

A146541 Binomial transform of A010688.

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A036543 a(n) = T(3,n), array T given by A048471.

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A176415 Periodic sequence: repeat 7,1.

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A132728 Triangle T(n, k) = 4 - 3*(-1)^k, read by rows.

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A146534 a(n) = 4*C(2n,n) - 3*0^n.

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A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

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A133190 a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).

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A146534 a(n) = 4C(2n,n) - 30^n.

A133190 a(n) = 2a(n-1) - a(n-2) + 2a(n-3).