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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A178568 Triangle read by rows, antidiagonals of an array (row r >= 1, column n >= 1) generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 1, 4, 4, 4, 3, 1, 5, 5, 9, 5, 2, 1, 6, 6, 16, 7, 6, 3, 1, 7, 7, 25, 9, 12, 7, 1, 1, 8, 8, 36, 11, 20, 13, 8, 4, 1, 9, 9, 49, 13, 30, 21, 27, 9, 3, 1, 10, 10, 64, 15, 42, 31, 64, 16, 10, 5, 1, 11, 11, 81, 17, 56, 43, 125, 25, 21, 11, 2
Offset: 1

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Author

Gary W. Adamson, May 29 2010

Keywords

Comments

Companion to A178239 (the latter generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1)).
Row sums of the triangle = A169826: (1, 2, 5, 8, 16, 27, 45, 69, 109, ...).
Polcoeff row r of the array as f(x) satisfies f(x)/f(x^2) = (1 + r*x + x^2).
Let q(x) = (1 + r*x + x^2). Then polcoeff row r = q(x)*q(x^2)*q(x^4)*q(x^8)*...
Right border of the triangle = A002487: (1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, ...).
Terms in r-th row mod r (for r > 1): (1, 0, 1, 0, 1, 0, ...).

Examples

			First few rows of the array:
  1,   1,   2,   1,   3,   2,   3,   1,   4,   3,   5,   2,   5,   3, ..
  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14, ..
  1,   3,   4,   9,   7,  12,  13,  27,  16,  21,  19,  36,  25,  39, ..
  1,   4,   5,  16,   9,  20,  21,  64,  25,  36,  29,  80,  41,  84, ..
  1,   5,   6,  25,  11,  30,  31, 125,  36,  55,  41, 150,  61, 155, ..
  1,   6,   7,  36,  13,  42,  43, 216,  49,  78,  55, 252,  85, 258, ..
  1,   7,   8,  49,  15,  56,  57, 343,  64, 105,  71, 392, 113, 399, ..
  1,   8,   9,  64,  17,  72,  73, 512,  81, 136,  89, 576, 145, 584, ..
  ...
First few rows of the triangle:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  3,   1;
  1,  4,  4,   4,  3;
  1,  5,  5,   9,  5,   2;
  1,  6,  6,  16,  7,   6,  3;
  1,  7,  7,  25,  9,  12,  7,   1;
  1,  8,  8,  36, 11,  20, 13,   8,  4;
  1,  9,  9,  49, 13,  30, 21,  27,  9,  3;
  1, 10, 10,  64, 15,  42, 31,  64, 16, 10,  5;
  1, 11, 11,  81, 17,  56, 43, 125, 25, 21, 11,  2;
  1, 12, 12, 100, 19,  72, 57, 216, 36, 36, 19, 12,  5;
  1, 13, 13, 121, 21,  90, 73, 343, 49, 55, 29, 36, 13,  3;
  1, 14, 14, 144, 23, 110, 91, 512, 64, 78, 41, 80, 25, 14, 4;
  ...

Crossrefs

Array rows r=1 to r=10: A002487, A000027, A178590, A244643, A342610, A237711, A342611, A342614, A342615, A178569.
Cf. A178239.

Programs

  • PARI
    A(r,n) = my(x=0,y=1); forstep(i=if(n,logint(n,2)),0,-1, if(bittest(n,i), x+=y;y*=r, y+=x;x*=r)); x;
T(r,n) = A(r-n+1,n); \\ Kevin Ryde, Mar 18 2021

Formula

a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1).
Given (1, r, 1, 0, 0, 0, ...) in each column of an infinite lower triangular matrix M; shifted down twice from the previous column. r-th row of the array = lim_{n->inf} M^n.
For the r-th row, a(2^k+n) = r*a(n) + a(2^k-n). - Andrey Zabolotskiy, Oct 21 2021

A342615 a(0) = 0, a(1) = 1; a(2*n) = 9*a(n), a(2*n+1) = a(n) + a(n+1).

Original entry on oeis.org

0, 1, 9, 10, 81, 19, 90, 91, 729, 100, 171, 109, 810, 181, 819, 820, 6561, 829, 900, 271, 1539, 280, 981, 919, 7290, 991, 1629, 1000, 7371, 1639, 7380, 7381, 59049, 7390, 7461, 1729, 8100, 1171, 2439, 1810, 13851, 1819, 2520, 1261, 8829, 1900, 8271, 8209, 65610, 8281, 8919, 2620
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 16 2021

Keywords

Links

Crossrefs

Cf. A002487, A178569, A178590, A237711, A244643, A342610, A342611, A342614.

Programs

  • Maple
    N:= 100: # for a(0) to a(N)
g:= x*mul(1+9*x^(2^k)+x^(2^(k+1)),k=0..ilog2(N)):
S:= series(g,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, May 04 2023
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], 9 a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 51}]
nmax = 51; CoefficientList[Series[x Product[(1 + 9 x^(2^k) + x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + 9*x^(2^k) + x^(2^(k+1))).

A342611 a(0) = 0, a(1) = 1; a(2*n) = 7*a(n), a(2*n+1) = a(n) + a(n+1).

Original entry on oeis.org

0, 1, 7, 8, 49, 15, 56, 57, 343, 64, 105, 71, 392, 113, 399, 400, 2401, 407, 448, 169, 735, 176, 497, 463, 2744, 505, 791, 512, 2793, 799, 2800, 2801, 16807, 2808, 2849, 855, 3136, 617, 1183, 904, 5145, 911, 1232, 673, 3479, 960, 3241, 3207, 19208, 3249, 3535, 1296, 5537
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 16 2021

Keywords

Crossrefs

Cf. A002487, A178569, A178590, A237711, A244643, A342610, A342614, A342615.

Programs

  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], 7 a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 52}]
nmax = 52; CoefficientList[Series[x Product[(1 + 7 x^(2^k) + x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + 7*x^(2^k) + x^(2^(k+1))).

A342614 a(0) = 0, a(1) = 1; a(2*n) = 8*a(n), a(2*n+1) = a(n) + a(n+1).

Original entry on oeis.org

0, 1, 8, 9, 64, 17, 72, 73, 512, 81, 136, 89, 576, 145, 584, 585, 4096, 593, 648, 217, 1088, 225, 712, 665, 4608, 721, 1160, 729, 4672, 1169, 4680, 4681, 32768, 4689, 4744, 1241, 5184, 865, 1736, 1305, 8704, 1313, 1800, 937, 5696, 1377, 5320, 5273, 36864, 5329, 5768, 1881, 9280
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 16 2021

Keywords

Crossrefs

Cf. A002487, A178569, A178590, A237711, A244643, A342610, A342611, A342615.

Programs

  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], 8 a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 52}]
nmax = 52; CoefficientList[Series[x Product[(1 + 8 x^(2^k) + x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

G.f.: x * Product_{k>=0} (1 + 8*x^(2^k) + x^(2^(k+1))).
Showing 1-4 of 4 results.

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