A101945 -id:A101945 - cp's OEIS frontend

A157268 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1, read by rows.

1, 1, 1, 1, 6, 1, 1, 17, 17, 1, 1, 40, 126, 40, 1, 1, 87, 606, 606, 87, 1, 1, 182, 2413, 5856, 2413, 182, 1, 1, 373, 8679, 40337, 40337, 8679, 373, 1, 1, 756, 29376, 232726, 497066, 232726, 29376, 756, 1, 1, 1523, 95668, 1205968, 4527078, 4527078, 1205968, 95668, 1523, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   17,    17,       1;
  1,   40,   126,      40,       1;
  1,   87,   606,     606,      87,       1;
  1,  182,  2413,    5856,    2413,     182,       1;
  1,  373,  8679,   40337,   40337,    8679,     373,     1;
  1,  756, 29376,  232726,  497066,  232726,   29376,   756,    1;
  1, 1523, 95668, 1205968, 4527078, 4527078, 1205968, 95668, 1523, 1;

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

Cf. A007318 (m=0), this sequence (m=1).
Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157272, A157273, A157274, A157275, A157277, A157278.
Cf. A101945.

f[n_,k_]:= If[k<=Floor[n/2], 2^k, 2^(n-k)];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 04 2022 *)

def f(n,k): return 2^k if (k <= n//2) else 2^(n-k)
@CachedFunction
def T(n,k,m):  # A157207
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 04 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A101945(n-1), n >= 1. - G. C. Greubel, Feb 04 2022

Edited by G. C. Greubel, Feb 04 2022

A157275 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k if k <= floor(n/2) otherwise 2(n-k), and m = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 17, 17, 1, 1, 40, 126, 40, 1, 1, 87, 606, 606, 87, 1, 1, 182, 2413, 5604, 2413, 182, 1, 1, 373, 8679, 38117, 38117, 8679, 373, 1, 1, 756, 29376, 219020, 426002, 219020, 29376, 756, 1, 1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 95668, 1523, 1

Offset: 0

Roger L. Bagula, Feb 26 2009

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   17,    17,       1;
  1,   40,   126,      40,       1;
  1,   87,   606,     606,      87,       1;
  1,  182,  2413,    5604,    2413,     182,       1;
  1,  373,  8679,   38117,   38117,    8679,     373,     1;
  1,  756, 29376,  219020,  426002,  219020,   29376,   756,    1;
  1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 95668, 1523, 1;

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

Cf. A007318 (m=0), this sequence (m=1), A157277 (m=2), A157278 (m=3).
Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.
Cf. A101945.

f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 05 2022 *)

def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n,k,m):  # A157275
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A101945(n-1), for n >= 1. - G. C. Greubel, Feb 05 2022

Edited by G. C. Greubel, Feb 05 2022

A101946 a(n) = 62^n - 3n - 5.

Original entry on oeis.org

1, 4, 13, 34, 79, 172, 361, 742, 1507, 3040, 6109, 12250, 24535, 49108, 98257, 196558, 393163, 786376, 1572805, 3145666, 6291391, 12582844, 25165753, 50331574, 100663219, 201326512, 402653101, 805306282, 1610612647, 3221225380, 6442450849, 12884901790

Offset: 0

Gary W. Adamson, Dec 22 2004

Sequence generated from a 3 X 3 matrix, companion to A101945.
Characteristic polynomial of M = x^3 - 4x^2 + 5x - 2.
Sequence can also be generated by the same method as A061777 with slightly different rules. Refer to A061777, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones, i.e., "vertex to side" expansion version. Let us assign the label "1" to the triangle at the origin; at n-th generation add a triangle at each expandable vertex, i.e., each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping triangles will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The triangles count is A005448. See illustration. - Kival Ngaokrajang, Sep 26 2014
The number of ways to select 0 or more nodes of a 2 X n rectangular grid such that the selected nodes are connected, but do not fill any 2 X 2 square. This question arises in logic puzzles such as Nurikabe. - Hugo van der Sanden, Feb 22 2024

			a(4) = 79 = 4*34 - 5*13 + 2*4 = 4*a(3) - 5*a(2) + 2*a(1).
a(4) = right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 46 a(4)], where 46 = A033484(4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1-2, 4.
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See pp. 1, 3, 5, 7, 17, 38, 40.
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A033484, A101945.

Cf. A000225, A005448, A061777.

[6*2^n -3*n-5: n in [0..40]]; // G. C. Greubel, Feb 06 2022

a[0]=1; a[1]=4; a[2]=13; a[n_]:= a[n]= 4a[n-1] -5a[n-2] +2a[n-3]; Table[ a[n], {n, 0, 30}] (* Or *)
a[n_] := (MatrixPower[{{1, 0, 0}, {2, 2, 0}, {1, 2, 1}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 12 2005 *)
Table[6*2^n-3n-5,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{1,4,13},40] (* Harvey P. Dale, Jun 03 2017 *)

a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))\\ Kival Ngaokrajang, Sep 26 2014

Vec(-(2*x^2+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

[3*(2^(n+1) -n-2) +1 for n in (0..40)] # G. C. Greubel, Feb 06 2022

a(0)=1, a(1)=4, a(2)=13 and for n>2, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
a(n) = right term in M^n * [1 1 1], where M = the 3X3 matrix [1 0 0 / 2 2 0 / 1 2 1]. M^n * [1 1 1] = [1 A033484(n) a(n)].
a(0) = 1, for n >= 1, a(n) = 3*A000225(n) + a(n-1). - Kival Ngaokrajang, Sep 26 2014
G.f.: (1+2*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Sep 26 2014
E.g.f.: 6*exp(2*x) - (5+3*x)*exp(x). - G. C. Greubel, Feb 06 2022

New definition from Ralf Stephan, May 17 2007

A135855 A007318 * a tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column.

Original entry on oeis.org

1, 5, 1, 10, 6, 1, 16, 16, 7, 1, 23, 32, 23, 8, 1, 31, 55, 55, 31, 9, 1, 40, 86, 110, 86, 40, 10, 1, 50, 126, 196, 196, 126, 50, 11, 1, 61, 176, 322, 392, 322, 176, 61, 12, 1, 73, 237, 498, 714, 714, 498, 237, 73, 13, 1

Offset: 0

Gary W. Adamson, Dec 01 2007

			First few rows of the triangle:
   1;
   5,  1;
  10,  6,   1;
  16, 16,   7,  1;
  23, 32,  23,  8,  1;
  31, 55,  55, 31,  9,  1;
  40, 86, 110, 86, 40, 10, 1;
  ...

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

Cf. A007318, A022815, A052905, A101945, A134465.

A135855:= func< n,k | Binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)) >;
[A135855(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 06 2022

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, (n^2+7*n+2)/2, If[k==n, 1, T[n-1, k-1] + T[n-1, k]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 06 2022 *)

@CachedFunction
def T(n,k): # A135855
    if (k==0): return (n^2+7*n+2)/2
    elif (k==n): return 1
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 06 2022

Binomial transform of an infinite tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column; i.e., (1, 1, 1, ...) in the main diagonal, (4, 4, 4, 0, 0, 0, ...) in the subdiagonal and (1, 1, 1, ...) in the subsubdiagonal.
T(n, 0) = A052905(n).
Sum_{k=0..n} T(n, k) = A101945(n).
From G. C. Greubel, Feb 06 2022: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k), with T(n, n) = 1, T(n, 0) = A052905(n).
T(n, k) = binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)).
T(n, 1) = A134465(n).
T(n, 2) = A022815(n-1).
T(n, n-1) = n+3.
T(n, n-2) = A052905(n+2). (End)

A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

3, 2, 6, 1, 5, 12, 0, 4, 11, 24, -1, 3, 10, 23, 48, -2, 2, 9, 22, 47, 96, -3, 1, 8, 21, 46, 95, 192, -4, 0, 7, 20, 45, 94, 191, 384, -5, -1, 6, 19, 44, 93, 190, 383, 768, -6, -2, 5, 18, 43, 92, 189, 382, 767, 1536, -7, -3, 4, 17, 42, 91, 188, 381, 766, 1535, 3072

Offset: 0

Paul Curtz, Jan 07 2024

Similar to A367559.

			Table begins:
   3  6 12 24 48 96 ...
   2  5 11 23 47 95 ...
   1  4 10 22 46 94 ...
   0  3  9 21 45 93 ...
  -1  2  8 20 44 92 ...
  -2  1  7 19 43 91 ...
  ...

Cf. A000225, A000918, A007283, A033484, A048488, A083329, A165751, A367559.

Cf. diagonals: A123720, A079583, A095151, A101945 (after -1).

T[n_, k_] := 3*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 15 2024 *)

T(0,k) = 3*2^k     =   A007283(k).
T(1,k) = 3*2^k - 1 =   A083329(k+1).
T(2,k) = 3*2^k - 2 =   A033484(k).
T(3,k) = 3*2^k - 3 = 3*A000225(k).
T(4,k) = 3*2^k - 4 =  -A165751(k).
T(5,k) = 3*2^k - 5 =   A048488(k-1)
T(6,k) = 3*2^k - 6 = 3*A000918(k).

cp's OEIS Frontend

A157268 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1, read by rows.

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A157275 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 1, read by rows.

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A101946 a(n) = 6*2^n - 3*n - 5.

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A135855 A007318 * a tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column.

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Magma

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A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

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Mathematica

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A157268 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1, read by rows.

A157275 Triangle T(n, k, m) = (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mf(n,k)T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2k if k <= floor(n/2) otherwise 2(n-k), and m = 1, read by rows.

A101946 a(n) = 62^n - 3n - 5.