cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 8, 15, 12, 6, 16, 31, 27, 14, 9, 32, 63, 58, 30, 21, 10, 64, 127, 121, 62, 48, 24, 11, 128, 255, 248, 126, 106, 54, 26, 13, 256, 511, 503, 254, 227, 116, 57, 29, 17, 512, 1023, 1014, 510, 475, 242, 120, 61, 38, 18, 1024, 2047, 2037, 1022, 978, 496, 247, 125, 86, 42, 19, 2048, 4095, 4084, 2046, 1992, 1006, 502, 253, 192, 96, 45, 20
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2015

Keywords

Comments

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).
Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

Examples

			The top left corner of the array:
   1,  2,   4,   8,  16,   32,   64,  128,  256,   512,  1024, ...
   3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095, ...
   5, 12,  27,  58, 121,  248,  503, 1014, 2037,  4084,  8179, ...
   6, 14,  30,  62, 126,  254,  510, 1022, 2046,  4094,  8190, ...
   9, 21,  48, 106, 227,  475,  978, 1992, 4029,  8113, 16292, ...
  10, 24,  54, 116, 242,  496, 1006, 2028, 4074,  8168, 16358, ...
  11, 26,  57, 120, 247,  502, 1013, 2036, 4083,  8178, 16369, ...
  13, 29,  61, 125, 253,  509, 1021, 2045, 4093,  8189, 16381, ...
  17, 38,  86, 192, 419,  894, 1872, 3864, 7893, 16006, 32298, ...
  18, 42,  96, 212, 454,  950, 1956, 3984, 8058, 16226, 32584, ...
  19, 45, 102, 222, 469,  971, 1984, 4020, 8103, 16281, 32650, ...
  20, 47, 105, 226, 474,  977, 1991, 4028, 8112, 16291, 32661, ...
  22, 51, 112, 237, 490,  999, 2020, 4065, 8158, 16347, 32728, ...
  23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, ...
  25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, ...
  28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, ...
  ...

Links

Crossrefs

Inverse permutation: A267102.
Transpose: A265903.
Cf. A265900 (main diagonal).
Cf. A162598 (row index of n in array), A265332 (column index of n in array).
Cf. A004001, A051135, A088359, A087686.
Column 1: A188163.
Column 2: A266109.
Row 1: A000079 (2^n).
Row 2: A000225 (2^n - 1, from 3 onward).
Row 3: A000325 (2^n - n, from 5 onward).
Row 4: A000918 (2^n - 2, from 6 onward).
Row 5: A084634 (?, from 9 onward).
Row 6: A132732 (2^n - 2n + 2, from 10 onward).
Row 7: A000295 (2^n - n - 1, from 11 onward).
Row 8: A036563 (2^n - 3).
Row 9: A084635 (?, from 17 onward).
Row 12: A048492 (?, from 20 onward).
Row 13: A249453 (?, from 22 onward).
Row 14: A183155 (2^n - 2n + 1, from 23 onward. Cf. also A244331).
Row 15: A000247 (2^n - n - 2, from 25 onward).
Row 16: A028399 (2^n - 4).
Cf. also permutations A267111, A267112.

Programs

Formula

For the first column k=1, A(n,1) = A188163(n), for columns k > 1, A(n,k) = A087686(1+A(n,k-1)).

A084636 Binomial transform of (1,0,1,0,1,0,2,0,2,0,2,0,...).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 33, 71, 157, 349, 768, 1662, 3534, 7398, 15291, 31297, 63595, 128555, 258930, 520240, 1043540, 2090956, 4186757, 8379499, 16766313, 33541481, 67093588, 134199826, 268414602, 536846754, 1073713983, 2147451717, 4294930839, 8589893143
Offset: 0

Views

Author

Paul Barry, Jun 06 2003

Keywords

Comments

Partial sums are A084637 (without leading 1).
The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,2,2,2,...) with b(n) = Sum_{k=0..4} C(n,k) + 2*Sum_{k=5..n} C(n,k) = 2^(n+1) - (n^4 -2*n^3 + 11*n^2 + 14*n + 24)/24. This gives the partial sums of A084635.

Links

Crossrefs

Cf. A000225, A000325, A084634, A084635, A084637.

Programs

  • Magma
    [(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18) +0^n: n in [0..50]]; // G. C. Greubel, Mar 19 2023
  • Mathematica
    Table[Boole[n==0] +(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18), {n,0,50}] (* G. C. Greubel, Mar 19 2023 *)
  • SageMath
    [(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18) +0^n for n in range(51)] # G. C. Greubel, Mar 19 2023

Formula

a(n) = Sum_{k=0..2} C(n, 2*k) + 2*Sum_{k=3..floor(n/2)} C(n, 2*k).
a(n) = (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24 + 2*Sum_{k=3..floor(n/2)} C(n, 2*k).
O.g.f.: (1-2*x+2*x^2)*(1-4*x+5*x^2-2*x^3+x^4)/((1-x)^5*(1-2*x)). - R. J. Mathar, Apr 07 2008
a(n) = A000225(n) - (1/24)*n*(n-1)*(n^2 - 5*n + 18) + [n=0]. - G. C. Greubel, Mar 19 2023

A084637 Binomial transform of (1,0,1,0,1,0,1,1,1,1,1,...).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 65, 136, 293, 642, 1410, 3072, 6606, 14004, 29295, 60592, 124187, 252742, 511672, 1031912, 2075452, 4166408, 8353165, 16732664, 33498977, 67040458, 134134046, 268333872, 536748474, 1073595228, 2147309211, 4294760928, 8589691767
Offset: 0

Views

Author

Paul Barry, Jun 06 2003

Keywords

Comments

The sequence starting 1,2,4,... is the binomial transform of (1, 1, 1, 1, 1, 1, 2, 2, 2, ...) with A035038(n) = Sum_{k=0..5} C(n,k) + 2*Sum_{k=6..n} C(n,k) = 2^n - (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. This gives the partial sums of A084636.

Links

Crossrefs

Cf. A000225, A000325, A035038, A084634, A084635, A084636.

Programs

  • Magma
    [2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120: n in [0..50]]; // G. C. Greubel, Mar 19 2023
  • Mathematica
    Table[2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120, {n,0,50}] (* G. C. Greubel, Mar 19 2023 *)
  • PARI
    Vec((1-7*x+21*x^2-35*x^3+35*x^4-21*x^5+7*x^6)/((1-x)^6*(1-2*x)) + O(x^50)) \\ Colin Barker, Mar 17 2016
  • SageMath
    [2^n -n*(n^4-10*n^3+55*n^2-110*n+184)/120 for n in range(51)] # G. C. Greubel, Mar 19 2023

Formula

a(n) = Sum_{k=0..2} C(n, 2*k) + Sum_{k=6..n} C(n, k).
a(n) = 2^n - n*(n^4 - 10*n^3 + 55*n^2 - 110*n + 184)/120.
G.f.: (1-7*x+21*x^2-35*x^3+35*x^4-21*x^5+7*x^6) / ((1-x)^6*(1-2*x)). - Colin Barker, Mar 17 2016

A084638 Binomial transform of (1,0,1,0,1,0,1,0,2,0,2,0,2,....).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 129, 265, 558, 1200, 2610, 5682, 12288, 26292, 55587, 116179, 240366, 493108, 1004780, 2036692, 4112144, 8278552, 16631717, 33364381, 66863358, 133903816, 268037862, 536371734, 1073120208, 2146715436, 4294024647, 8588785575
Offset: 0

Views

Author

Paul Barry, Jun 06 2003

Keywords

Comments

The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,1,1,2,2...) with a(n) = Sum_{k=0..6} C(n,k) + 2*Sum_{k=7..n} C(n,k) = 2^(n+1) - A008859(n). This gives the partial sums of A084637.

Links

Crossrefs

Cf. A000225, A000325, A008859, A084634, A084635, A084636, A084637.

Programs

  • Magma
    [2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n: n in [0..50]]; // G. C. Greubel, Mar 20 2023
  • Mathematica
    Table[2^n -4 -(1/6!)*(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160) + Boole[n==0], {n,0,50}] (* G. C. Greubel, Mar 20 2023 *)
  • PARI
    Vec((1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8)/((1-x)^7*(1-2*x)) + O(x^50)) \\ Colin Barker, Mar 17 2016
  • SageMath
    [2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n for n in range(51)] # G. C. Greubel, Mar 20 2023

Formula

a(n) = Sum_{k=0..3, C(n, 2*k)} + 2*Sum_{k=4..floor(n/2), C(n, 2*k)}.
a(n) = (n^6-15*n^5+115*n^4-405*n^3+964*n^2-660*n+720)/720 + 2*Sum_{k=4..floor(n/2), C(n, 2k)}.
G.f.: (1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8) / ((1-x)^7*(1-2*x)). - Colin Barker, Mar 17 2016
Showing 1-4 of 4 results.

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