A051049 -id:A051049 - cp's OEIS frontend

A166692 Triangle T(n,k) read by rows: T(n,k) = 2^(k-1), k>0, T(n,0) = (n+1) mod 2.

1, 0, 1, 1, 1, 2, 0, 1, 2, 4, 1, 1, 2, 4, 8, 0, 1, 2, 4, 8, 16, 1, 1, 2, 4, 8, 16, 32, 0, 1, 2, 4, 8, 16, 32, 64, 1, 1, 2, 4, 8, 16, 32, 64, 128, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

Offset: 0

Paul Curtz, Oct 18 2009

Variant of A166918.

			Triangle begins as:
  1;
  0, 1;
  1, 1, 2;
  0, 1, 2, 4;
  1, 1, 2, 4, 8;
  0, 1, 2, 4, 8, 16;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A005578, A011782, A051049, A131577, A106624, A166494, A166918.

A166692:= func< n,k | k eq 0 select ((n+1) mod 2) else 2^(k-1) >;
[A166692(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 24 2023

Join[{1,0},Flatten[Riffle[Table[2^Range[0,n],{n,0,10}],{1,0}]]] (* Harvey P. Dale, Jan 18 2015 *)

def A166692(n,k): return ((n+1)%2) if (k==0) else 2^(k-1)
flatten([[A166692(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 24 2023

T(2n, k) = A011782(k).
T(2n+1, k) = A131577(k).
Sum_{k=0..n} T(n,k) = A051049(n).
From G. C. Greubel, Apr 24 2023: (Start)
T(2*n, n) = A011782(n).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A005578(n).
Sum_{k=0..n} T(n-k, k) = A106624(n). (End)

More terms from Harvey P. Dale, Jan 18 2015

A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 4, 2, 1, 2, 3, 7, 3, 1, 0, 6, 9, 11, 4, 1, 2, 5, 16, 19, 16, 5, 1, 0, 8, 20, 36, 34, 22, 6, 1, 2, 7, 29, 55, 71, 55, 29, 7, 1, 0, 10, 35, 85, 125, 127, 83, 37, 8, 1, 2, 9, 46, 119, 211, 251, 211, 119, 46, 9, 1, 0, 12, 54, 166, 329, 463, 461, 331, 164, 56, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

row sums = A051049: (1, 1, 4, 7, 16, 31, 64, ...).

			First few rows of the triangle:
  1;
  0, 1;
  2, 1,  1;
  0, 4,  2,  1;
  2, 3,  7,  3,  1;
  0, 6,  9, 11,  4,  1;
  2, 5, 16, 19, 16,  5,  1;
  0, 8, 20, 36, 34, 22,  6, 1;
  2, 7, 29, 55, 71, 55, 29, 7, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A051049.

T:= function(n,k)
    if k=n then return 1;
    else return Binomial(n,k) + (-1)^(n-k);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

T:= func< n,k | k eq n select 1 else Binomial(n,k) +(-1)^(n-k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

seq(seq( `if`(k=n, 1, binomial(n,k) + (-1)^(n-k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, Binomial[n, k] + (-1)^(n-k)] ;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==n, 1, binomial(n,k) + (-1)^(n-k)); \\ G. C. Greubel, Nov 20 2019

def T(n, k):
    if (k==n): return 1
    else: return binomial(n,k) + (-1)^(n-k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A007318 + A000012(signed) - Identity matrix, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = (-1)^(n-k) + binomial(n,k), with T(n,n)=1. - G. C. Greubel, Nov 20 2019

More terms added by G. C. Greubel, Nov 20 2019

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)

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

Original entry on oeis.org

3, 7, 12, 25, 48, 97, 192, 385, 768, 1537, 3072, 6145, 12288, 24577, 49152, 98305, 196608, 393217, 786432, 1572865, 3145728, 6291457, 12582912, 25165825, 50331648, 100663297, 201326592, 402653185, 805306368, 1610612737, 3221225472, 6442450945, 12884901888

Offset: 0

Paul Curtz, Jan 01 2017

a(n) mod 9 is a periodic sequence of length 2: repeat [3, 7].
From 7, the last digit is of period 4: repeat [7, 2, 5, 8].
(Main sequence for the signature (2,1,-2): 0, 0, 1, 2, 5, 10, 21, 42, ... = 0 followed by A000975(n) = b(n), which first differences are A001045(n) (Paul Barry, Oct 08 2005). Then, 0 followed by b(n) is an autosequence of the first kind. The corresponding autosequence of the second kind is 0, 0, 2, 3, 8, 15, 32, 63, ... . See A277078(n).)
Difference table of a(n):
3,   7, 12, 25, 48,  97, 192, ...
4,   5, 13, 23, 49,  95, 193, ...  = -(-1)^n* A140683(n)
1,   8, 10, 26, 46,  98, 190, ...  = A259713(n)
7,   2, 16, 20, 52,  92, 196, ...
-5, 14,  4, 32, 40, 104, 184, ...
... .

			a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 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. A005010, A051049, A140657, A140683, A164346, A199116, A259713.

a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)
CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* Michael De Vlieger, Jan 01 2017 *)

Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Jan 01 2017

a(2n) = 3*4^n, a(2n+1) = 6*4^n + 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
a(n+2) = a(n) + 9*2^n.
a(n) = 2^(n+2) - A051049(n).
From Colin Barker, Jan 01 2017: (Start)
a(n) = 3*2^n for n even.
a(n) = 3*2^n + 1 for n odd.
G.f.: (3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)
Binomial transform of 3, followed by (-1)^n* A140657(n).

More terms from Colin Barker, Jan 01 2017

A321643 a(n) = 5*2^n - (-1)^n.

Original entry on oeis.org

4, 11, 19, 41, 79, 161, 319, 641, 1279, 2561, 5119, 10241, 20479, 40961, 81919, 163841, 327679, 655361, 1310719, 2621441, 5242879, 10485761, 20971519, 41943041, 83886079, 167772161, 335544319, 671088641, 1342177279, 2684354561, 5368709119, 10737418241, 21474836479

Offset: 0

Paul Curtz, Dec 03 2018

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

Cf. A010690, A010701, A020714, A051049, A070366, A110286.

Cf. A001045, A081808, A321483.

List([0..30],n->5*2^n-(-1)^n); # Muniru A Asiru, Dec 05 2018

[5*2^n-(-1)^n$n=0..30]; # Muniru A Asiru, Dec 05 2018

a[n_] := 5*2^n - (-1)^n; Array[a, 30, 0] (* Amiram Eldar, Dec 03 2018 *)

Vec((4 + 7*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 04 2018

for n in range(0,30): print(5*2**n - (-1)**n) # Stefano Spezia, Dec 05 2018

a(n+2) - a(n) = a(n+1) + a(n) = 15*2^n, n >= 0.
a(n) - 2*a(n-1) = period 2: repeat [3, -3], n > 0, a(0)=4, a(1)=11.
a(n+1) = 10*A051049(n) + period 2: repeat [1, 9].
a(n) = 12*2^n - A321483(n), n >= 0.
a(n) = 2^(n+2) + 3*A001045(n), n >= 0.
a(n) == A070366(n+4) (mod 9).
From Colin Barker, Dec 04 2018: (Start)
G.f.: (4 + 7*x) / ((1 + x)*(1 - 2*x)).
a(n) = a(n-1) + 2*a(n-2) for n > 1. (End)
E.g.f.: exp(-x)*(5*exp(3*x) - 1). - Elmo R. Oliveira, Aug 17 2024

A131086 Triangle read by rows: T(n,k) = 2*binomial(n,k) - (-1)^(n-k) (0 <= k <= n).

Original entry on oeis.org

1, 3, 1, 1, 5, 1, 3, 5, 7, 1, 1, 9, 11, 9, 1, 3, 9, 21, 19, 11, 1, 1, 13, 29, 41, 29, 13, 1, 3, 13, 43, 69, 71, 41, 15, 1, 1, 17, 55, 113, 139, 113, 55, 17, 1, 3, 17, 73, 167, 253, 251, 169, 71, 19, 1, 1, 21, 89, 241, 419, 505, 419, 241, 89, 21, 1, 3, 21, 111, 329

Offset: 0

Gary W. Adamson, Jun 14 2007

Row sums = A051049 starting (1, 4, 7, 16, 31, 64, ...).

			First few rows of the triangle are
  1;
  3,  1;
  1,  5,  1;
  3,  5,  7,  1;
  1,  9, 11,  9,  1;
  3,  9, 21, 19, 11,  1;
  1, 13, 29, 41, 29, 13,  1;
  ...

Cf. A000012, A051049.

T := proc (n, k) if k <= n then 2*binomial(n, k)-(-1)^(n-k) else 0 end if end proc: for n from 0 to 11 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007

G.f. = G(t,z) = (1 + 3z - tz - 2tz^2)/((1+z)(1-tz)(1-z-tz)). - Emeric Deutsch, Jun 21 2007

More terms from Emeric Deutsch, Jun 21 2007

Sequence corrected by N. J. A. Sloane, Sep 30 2007

A155998 Triangle read by rows: T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 - (-1)^k)/2.

Original entry on oeis.org

0, 1, 1, 0, 4, 0, 1, 3, 3, 1, 0, 8, 0, 8, 0, 1, 5, 10, 10, 5, 1, 0, 12, 0, 40, 0, 12, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 16, 0, 112, 0, 112, 0, 16, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 20, 0, 240, 0, 504, 0, 240, 0, 20, 0

Offset: 0

Roger L. Bagula, Feb 01 2009

Row sums are: A155559(n) = {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...}.

			Triangle begins as:
  0;
  1,  1;
  0,  4,  0;
  1,  3,  3,   1;
  0,  8,  0,   8,   0;
  1,  5, 10,  10,   5,   1;
  0, 12,  0,  40,   0,  12,  0;
  1,  7, 21,  35,  35,  21,  7,   1;
  0, 16,  0, 112,   0, 112,  0,  16, 0;
  1,  9, 36,  84, 126, 126, 84,  36, 9,  1;
  0, 20,  0, 240,   0, 504,  0, 240, 0, 20, 0;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*(2 - (-1)^k*(1 + (-1)^n))/2 ))); # G. C. Greubel, Dec 01 2019

[Binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2: k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 01 2019

seq(seq( binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2, k=0..n), n=0..12); # G. C. Greubel, Dec 01 2019

f[n_, k_]:= Binomial[n, k]*(1 - (-1)^k)/2; Table[f[n,k]+f[n,n-k], {n, 0, 10}, {k, 0, n}]//Flatten
Table[Binomial[n, k]*(2-(-1)^k*(1+(-1)^n))/2, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 01 2019 *)

T(n,k) = binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2; \\ G. C. Greubel, Dec 01 2019

[[binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Dec 01 2019

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 - (-1)^k)/2.
From G. C. Greubel, Dec 01 2019: (Start)
T(n, k) = binomial(n, k)*(2 - (-1)^k*(1 + (-1)^n))/2.
Sum_{k=0..n} T(n,k) = 2^n = A155559(n) for n >= 1.
Sum_{k=0..n-1} T(n,k) = (2^(n+1) - (1-(-1)^n))/2 = A051049(n), n >= 1. (End)

A174182 Numerator of the first column, n-th row of the table of the Akiyama-Tanigawa transform starting from a top row of Bernoulli numbers.

Original entry on oeis.org

1, 3, 17, 13, 481, 69, 1595, 53, 64561, 19333, -24278897, -4223787, 425750784331, 2082755237, -759610365139, -1935668618507, 91825384919760257, 3104887811555781, -333936446105117072383, -8039608511659164907, 496858217433153687034811, 31900258438443561908965, -1108179772136293880993162549, -186044136772398390757763787, 167280081459577193334628789960171

Offset: 0

Paul Curtz, Mar 11 2010

Starting with a top row of Bernoulli numbers, the Akiyama-Tanigawa transform generates further rows as follows:
  1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66,...
  3/2, -4/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33,..
  17/6, -11/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110,...
  13/2, -143/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165,...
  481/30, -277/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22,..
  69/2, -73/21, -129/14, -38566/105, -20995/42, 211515/77,...
The numerators of the leftmost column define the current sequence.
The denominators appear to be the same as A141056.

D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A174129, A141056, A051049.

a[0, k_] := BernoulliB[k]; a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); Table[a[n, 0], {n, 0, 24}] // Numerator (* Jean-François Alcover, Sep 18 2012 *)

(a(n)-A174129(n))/A141056(n) = A000225(n).

A318143 Coefficients of the polynomials generated by the e.g.f. cosh(xz)(x-1)/(x-exp(z*(x-1))), triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 4, 4, 0, 1, 11, 17, 7, 1, 1, 26, 76, 66, 16, 0, 1, 57, 317, 467, 237, 31, 1, 1, 120, 1212, 2962, 2612, 806, 64, 0, 1, 247, 4321, 17215, 24145, 13519, 2641, 127, 1, 1, 502, 14644, 92554, 199192, 178486, 65884, 8434, 256, 0

Offset: 0

Peter Luschny, Aug 19 2018

			[n\k][0,   1,    2,     3,     4,     5,    6,   7,  8]
[0]   1;
[1]   1,   0;
[2]   1,   1,    1;
[3]   1,   4,    4,     0;
[4]   1,  11,   17,     7,     1;
[5]   1,  26,   76,    66,    16,     0;
[6]   1,  57,  317,   467,   237,    31,    1;
[7]   1, 120, 1212,  2962,  2612,   806,   64,   0;
[8]   1, 247, 4321, 17215, 24145, 13519, 2641, 127, 1;

Row sums are (-1)^n*A009179(n).
Alternating row sums are 1.
Polynomials evaluated at x = 0 are 1.
T(n, n-1) = A051049(n-1) for n >= 1.
T(n, 1) = A000295(n) for n >= 0.
Cf. A046802, A173018.

gf := cosh(x*z)*(x-1)/(x-exp(z*(x-1))):
ser := series(gf, z, 12): p := n -> normal(n!*coeff(ser, z, n)):
seq(seq(coeff(p(n),x,k), k=0..n), n=0..10);

cp's OEIS Frontend

A166692 Triangle T(n,k) read by rows: T(n,k) = 2^(k-1), k>0, T(n,0) = (n+1) mod 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A321643 a(n) = 5*2^n - (-1)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Python

Formula

A131086 Triangle read by rows: T(n,k) = 2*binomial(n,k) - (-1)^(n-k) (0 <= k <= n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A318143 Coefficients of the polynomials generated by the e.g.f. cosh(xz)(x-1)/(x-exp(z*(x-1))), triangle read by rows, T(n,k) for 0 <= k <= n.