A091042 -id:A091042 - cp's OEIS frontend

A212592 a(n) is the difference between multiples of 7 with even and odd digit sum in base 6 in interval [0,6^n).

1, 6, 11, 106, 201, 2022, 3843, 38794, 73745, 744646, 1415547, 14293930, 27172313, 274381478, 521590643, 5266936010, 10012281377, 101102361990, 192192442603, 1940727511786, 3689262580969, 37253563629926, 70817864678883, 715107089849866

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 22 2012

In general for all z, given a sequence of the form: a(n) is the difference between multiples of 2z+1 with even and odd digit sum in base 2z in interval [0,(2z)^n); then a(n) = (a(n+1) + a(n-1))/2 when n is even. The equation applies here where z=3. - Bob Selcoe, Jun 10 2014

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (0,21,0,-35,0,7).

Cf. A038754, A212500, A091042.

LinearRecurrence[{0, 21, 0, -35, 0, 7}, {1, 6, 11, 106, 201, 2022}, 24] (* Bruno Berselli, May 22 2012 *)

For n>=7, a(n) = 21*a(n-2)-35*a(n-4)+7*a(n-6).
G.f.: x*(1+6*x-10*x^2-20*x^3+5*x^4+6*x^5)/(1-21*x^2+35*x^4-7*x^6). [Bruno Berselli, May 22 2012]
a(n) = 2a(n-1) - a(n-2) when n is odd; a(n) = (a(n+1) + a(n-1))/2 when n is even. - Bob Selcoe, Jun 10 2014

A082762 Trinomial transform of Lucas numbers (A000032).

Original entry on oeis.org

1, 8, 44, 232, 1216, 6368, 33344, 174592, 914176, 4786688, 25063424, 131233792, 687149056, 3597959168, 18839158784, 98643116032, 516502061056, 2704439902208, 14160631169024, 74146027405312, 388233639755776, 2032817728913408, 10643971814457344

Offset: 0

Emanuele Munarini, May 21 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Yuhan Jiang, The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model, arXiv:2312.09427 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A000032, A027907, A091042, A292277.

I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1)-4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

a[n_]:=(MatrixPower[{{2,2},{2,4}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(3 + s)^n + (1 - s)(3 - s)^n)/2]]; Array[f, 21, 0] (* Robert G. Wilson v, Mar 07 2011 *)
LinearRecurrence[{6,-4}, {1, 8}, 30] (* G. C. Greubel, Dec 21 2017 *)

x='x+O('x^30); Vec((1 + 2*x)/(1 - 6*x + 4*x^2)) \\ G. C. Greubel, Dec 21 2017

a(n) = Sum_{k=0..2*n} Trinomial(n,k)*Lucas(k+1), where Trinomial(n,k) = trinomial coefficients (A027907).
a(n) = 2^n*Lucas(2*n+1), where Lucas = A000032.
From Philippe Deléham, Mar 01 2004: (Start)
a(n) = 2^n*A002878(n) = 2^(-n)*Sum_{k>=0} C(2*n+1,2*k)*5^k; see A091042.
a(0) = 1, a(1) = 8, a(n+1) = 6*a(n) - 4*a(n-1). (End)
From Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009: (Start)
a(n) = ((1+sqrt(5))*(3+sqrt(5))^n + (1-sqrt(5))*(3-sqrt(5))^n)/2.
Third binomial transform of 1, 5, 5, 25, 25, 125. (End)
G.f.: (1 + 2*x)/(1 - 6*x + 4*x^2). - Colin Barker, Mar 23 2012

A212668 a(n) = (16/3)(n+1)n(n-1) + 8n^2 + 1.

Original entry on oeis.org

9, 65, 201, 449, 841, 1409, 2185, 3201, 4489, 6081, 8009, 10305, 13001, 16129, 19721, 23809, 28425, 33601, 39369, 45761, 52809, 60545, 69001, 78209, 88201, 99009, 110665, 123201, 136649, 151041, 166409, 182785, 200201, 218689, 238281, 259009, 280905, 304001

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 32*n^5).

Colin Barker, Table of n, a(n) for n = 1..1000
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A038754, A084990, A091042, A212500, A212592.

[(16/3)*(n+1)*n*(n-1)+8*n^2+1: n in [1..40]]; // Vincenzo Librandi, Dec 01 2015

LinearRecurrence[{4, -6, 4, -1}, {9, 65, 201, 449}, 40] (* Vincenzo Librandi, Dec 01 2015 *)
CoefficientList[Series[x (9+29x-5x^2-x^3)/(1-x)^4,{x,0,40}],x] (* Harvey P. Dale, Mar 29 2023 *)

a(n)=16*(n+1)*n*(n-1)/3+8*n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

Vec(x*(9+29*x-5*x^2-x^3)/(1-x)^4 + O(x^100)) \\ Colin Barker, Nov 30 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^5(Pi*i/(2*n+1)) * sin(2*Pi*i/(2*n+1)).

G.f.: x*(9+29*x-5*x^2-x^3) / (1-x)^4. - Colin Barker, Nov 30 2015

A091044 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.

Original entry on oeis.org

1, 2, 2, 3, 10, 3, 4, 28, 28, 4, 5, 60, 126, 60, 5, 6, 110, 396, 396, 110, 6, 7, 182, 1001, 1716, 1001, 182, 7, 8, 280, 2184, 5720, 5720, 2184, 280, 8, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10, 11

Offset: 1

Wolfdieter Lang, Jan 23 2004

The odd-numbered columns of this triangle can be reduced: see triangle A091043.
The odd-numbered rows coincide with the ones of the reduced triangle A091043.
binomial(2*n,2*m+1) is even for n >= m + 1 >= 1, hence every T(n,m) is a positive integer.
The GCD (greatest common divisor) of the entries of each odd-numbered row n=2*k+1, k>=0, is 1.
The GCD of the entries of the even-numbered row n=2*k, k>=1, is A006519(n) (highest power of 2 in n=2*k).

			Triangle begins:
  [1];
  [2,2];
  [3,10,3];
  [4,28,28,4];
  [5,60,126,60,5];
  [6,110,396,396,110,6];
  ...
n = 6 = 2*3: gcd(6,110,396) = 2 = A006519(6);
n = 5: gcd(5,60,126) = 1 = A006519(5).

Indranil Ghosh, Rows 1..125, flattened
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 4.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.
Hernan de Alba, W. Carballosa, J. Leaños, and L. M. Rivera, Independence and matching numbers of some token graphs, arXiv preprint arXiv:1606.06370 [math.CO], 2016.
Wolfdieter Lang, First 9 rows.

Cf. A000302 (row sums), A001109, A006519, A007318, A053124, A091042, A091043, A111910.

Flatten[Table[Binomial[2n,2m+1]/2,{n,1,11},{m,0,n-1}]] (* Indranil Ghosh, Feb 22 2017 *)

{A(i, j) = binomial(2*i + 2*j - 2, 2*i - 1) / 2}; /* Michael Somos, Oct 15 2017 */

T(n, m)= binomial(2*n, 2*m+1)/2, n >= m + 1 >= 1, else 0.
  Put a(n) = n!*(n+1/2)!/(1/2)!. T(n+1,k) = (n+1)*a(n)/(a(k)*a(n-k)).
  T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1). Cf. A111910. - Peter Bala, Oct 13 2011
From Peter Bala, Jul 29 2013: (Start)
O.g.f.: 1/(1 - 2*t*(x + 1) + t^2*(x - 1)^2)= 1 + (2 + 2*x)*t + (3 + 10*x + 3*x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(4*sqrt(x))*( (1 + sqrt(x))^(2*n) - (sqrt(x) - 1)^(2*n) ) and has n-1 real zeros given by the formula -cot^2(k*Pi/(2*n)) for k = 1,2,...,n-1. Cf A091042.
The row polynomial R(n,x) satisfies (x - 1)^n*R(n,x/(x - 1)) = U(n,2*x - 1), the n-th row polynomial of A053124.
Row sums A000302. Sum {k = 0..n-1} 2^k*T(n,k) = A001109(n). (End)
From Werner Schulte, Jan 13 2017: (Start)
(1) T(n,m) = T(n-1,m) + T(n-1,m-1)*(2*n-1-m)/m for 0 < m < n-1 with T(n,0) = n and T(n,n) = 0;
(2) T(n,m) = 2*T(n-1,m) + 2*T(n-1,m-1) - T(n-2,m) + 2*T(n-2,m-1) - T(n-2,m-2) for 0 < m < n-1 with T(n,0) = T(n,n-1) = n and T(n,m) = 0 if m < 0 or m >= n;
(3) The row polynomials p(n,x) = Sum_{m=0..n-1} T(n,m)*x^m satisfy the recurrence equation p(n+2,x) = (2+2*x)*p(n+1,x) - (x-1)^2*p(n,x) for n >= 1 with initial values p(1,x) = 1 and p(2,x) = 2+2*x.
(End)
G.f.: x*y /(1 - 2*(x+y) + (x-y)^2) with the entries regarded as an infinite square array A(i, j) read by antidiagonals. - Michael Somos, Oct 15 2017

A212669 a(n) = 2/15 * (32n^5 + 80n^4 + 40n^3 - 20n^2 + 3*n).

Original entry on oeis.org

18, 340, 2022, 7400, 20602, 48060, 99022, 186064, 325602, 538404, 850102, 1291704, 1900106, 2718604, 3797406, 5194144, 6974386, 9212148, 11990406, 15401608, 19548186, 24543068, 30510190, 37585008, 45915010, 55660228, 66993750, 80102232, 95186410, 112461612

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 64*n^6).

Colin Barker, Table of n, a(n) for n = 1..1000
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212592, A212592, A212668.

Vec(2*x*(9+116*x+126*x^2+4*x^3+x^4)/(1-x)^6 + O(x^50)) \\ Colin Barker, Dec 01 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^6(Pi*i/(2*n+1)).

G.f.: 2*x*(9+116*x+126*x^2+4*x^3+x^4) / (1-x)^6. - Colin Barker, Dec 01 2015

A212593 a(n) is the difference between multiples of 9 with even and odd digit sum in base 8 in interval [0,8^n).

Original entry on oeis.org

1, 8, 15, 232, 449, 7400, 14351, 237832, 461313, 7648968, 14836623, 246015528, 477194433, 7912700328, 15348206223, 254499628104, 493651049985, 8185582834056, 15877514618127, 263276481572712, 510675448527297, 8467876653984360

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (0,36,0,-126,0,84,0,-9).

Cf. A038754, A212500, A212592, A091042.

LinearRecurrence[{0, 36, 0, -126, 0, 84, 0, -9}, {1, 8, 15, 232, 449, 7400, 14351, 237832}, 22] (* Bruno Berselli, May 22 2012 *)

For n>=9, a(n) = 36*a(n-2)-126*a(n-4)+84*a(n-6)-9*a(n-8).

G.f.: x*(1+8*x-21*x^2-56*x^3+35*x^4+56*x^5-7*x^6-8*x^7)/((1-3*x^2)*(1-33*x^2+27*x^4-3*x^6)). [Bruno Berselli, May 22 2012]

A212670 a(n) = 1/15(128n^5 + 320n^4 + 80n^3 - 200n^2 + 92n - 15).

Original entry on oeis.org

27, 615, 3843, 14351, 40363, 94711, 195859, 368927, 646715, 1070727, 1692195, 2573103, 3787211, 5421079, 7575091, 10364479, 13920347, 18390695, 23941443, 30757455, 39043563, 49025591, 60951379, 75091807, 91741819, 111221447, 133876835, 160081263, 190236171

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 128*n^7).

Colin Barker, Table of n, a(n) for n = 1..1000
Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177v2 [math.NT], 2007
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212593, A212594, A212668, A212669.

Table[(1/15) (8 n^2 - 4 n + 1) (16 n^3 + 48 n^2 + 32 n - 15), {n, 29}] (* Bruno Berselli, May 24 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{27,615,3843,14351,40363,94711},30] (* Harvey P. Dale, Apr 30 2018 *)

Vec(x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6 + O(x^50)) \\ Colin Barker, Dec 01 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^7(Pi*i/(2*n+1))*sin(2*Pi*i/(2*n+1)).

G.f.: x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6. [Bruno Berselli, May 24 2012]

A212594 a(n) is the difference between multiples of 11 with even and odd decimal digit sum in interval [0,10^n).

Original entry on oeis.org

1, 10, 19, 430, 841, 20602, 40363, 995710, 1951057, 48162410, 94373763, 2329795534, 4565217305, 112701782490, 220838347675, 5451852478622, 10682866609569, 263728727794378, 516774588979187, 12757653047779310, 24998531506579433, 617140623134480698

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (0,55,0,-330,0,462,0,-165,0,11).

Cf. A038754, A212500, A212592, A212593, A091042.

m:=23; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+10*x-36*x^2-120*x^3+126*x^4+252*x^5-84*x^6-120*x^7+9*x^8+10*x^9)/(1-55*x^2+330*x^4-462*x^6+165*x^8-11*x^10))); // Bruno Berselli, May 22 2012

LinearRecurrence[{0, 55, 0, -330, 0, 462, 0, -165, 0, 11}, {1, 10, 19, 430, 841, 20602, 40363, 995710, 1951057, 48162410}, 22] (* Bruno Berselli, May 22 2012 *)

For n>=11, a(n) = 55*a(n-2)-330*a(n-4)+462*a(n-6)-165*a(n-8)+11*a(n-10).

G.f.: x*(1+10*x-36*x^2-120*x^3+126*x^4+252*x^5-84*x^6-120*x^7+9*x^8+10*x^9)/(1-55*x^2+330*x^4-462*x^6+165*x^8-11*x^10). [Bruno Berselli, May 22 2012]

A212705 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^8).

Original entry on oeis.org

54, 3220, 38794, 237832, 995710, 3256540, 8954258, 21645200, 47366982, 95758500, 181475866, 325939096, 559444366, 923676652, 1474657570, 2286163232, 3453646934, 5098701492, 7374096042, 10469422120, 14617383838

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 24 2012

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177v2 [math.NT], 2007
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212593, A212594, A212668, A212669, A212670.

a(n) = 2/(2*n+1)*sum{i=1..n}tan^8(pi*i/(2*n+1)).
a(n) = 2/315*n*(1088*n^6+3808*n^5+3920*n^4+280*n^3-868*n^2+322n-45).
G.f.: 2*x*(27+1394*x+7273*x^2+7308*x^3+1373*x^4+34*x^5-x^6)/(1-x)^8. [Bruno Berselli, May 24 2012]

A212706 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^9).

Original entry on oeis.org

81, 5825, 73745, 461313, 1951057, 6418369, 17712657, 42921473, 94087249, 190446273, 361259537, 649305089, 1115101521, 1841932225, 2941740049, 4561961985, 6893373521, 10179012289, 14724250641, 20908086785, 29195724113, 40152508353, 54459292177, 72929296897

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 24 2012

Colin Barker, Table of n, a(n) for n = 1..1000
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212592, A212592, A212668, A212669, A212670, A212705.

[1+n/315*(4352*n^6+15232*n^5+12992*n^4-5600*n^3- 5152*n^2+5488*n-2112): n in [1..25]]; // Vincenzo Librandi, Dec 02 2015

Table[1 + n/315 (4352 n^6 + 15232 n^5 + 12992 n^4 - 5600 n^3 - 5152 n^2 + 5488 n - 2112), {n, 30}] (* Vincenzo Librandi, Dec 02 2015 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{81,5825,73745,461313,1951057,6418369,17712657,42921473},30] (* Harvey P. Dale, Aug 05 2025 *)

Vec(x*(81+5177*x+29413*x^2+29917*x^3+4883*x^4+171*x^5-9*x^6-x^7)/(1-x)^8 + O(x^40)) \\ Colin Barker, Dec 01 2015

a(n) = 2/(2*n+1) * Sum_{i=1..n} tan^9(Pi*i/(2*n+1)) * sin(2*Pi*i/(2*n+1)).
a(n) = 1+n/315*(4352*n^6 + 15232*n^5 + 12992*n^4 - 5600*n^3 - 5152*n^2 + 5488*n - 2112).
G.f.: x*(81+5177*x+29413*x^2+29917*x^3+4883*x^4+171*x^5-9*x^6-x^7) / (1-x)^8. - Colin Barker, Dec 01 2015

Typo in data fixed by Colin Barker, Dec 01 2015

cp's OEIS Frontend

A212592 a(n) is the difference between multiples of 7 with even and odd digit sum in base 6 in interval [0,6^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A082762 Trinomial transform of Lucas numbers (A000032).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A212668 a(n) = (16/3)*(n+1)*n*(n-1) + 8*n^2 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A091044 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A212669 a(n) = 2/15 * (32*n^5 + 80*n^4 + 40*n^3 - 20*n^2 + 3*n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A212593 a(n) is the difference between multiples of 9 with even and odd digit sum in base 8 in interval [0,8^n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A212670 a(n) = 1/15*(128*n^5 + 320*n^4 + 80*n^3 - 200*n^2 + 92*n - 15).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A212594 a(n) is the difference between multiples of 11 with even and odd decimal digit sum in interval [0,10^n).

Views

Author

A212668 a(n) = (16/3)(n+1)n(n-1) + 8n^2 + 1.

A212669 a(n) = 2/15 * (32n^5 + 80n^4 + 40n^3 - 20n^2 + 3*n).

A212670 a(n) = 1/15(128n^5 + 320n^4 + 80n^3 - 200n^2 + 92n - 15).

A212705 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^8).

A212706 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^9).