A177274 -id:A177274 - cp's OEIS frontend

A373169 Square array read by ascending antidiagonals: T(n,k) = noz(T(n,k-1) + (k-1)*(n-2) + 1), with T(n,1) = 1, n >= 2, k >= 1, where noz(n) = A004719(n).

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 9, 1, 5, 1, 6, 12, 16, 6, 6, 1, 7, 15, 22, 25, 12, 7, 1, 8, 18, 28, 35, 36, 19, 8, 1, 9, 21, 34, 45, 51, 49, 27, 9, 1, 1, 24, 4, 55, 66, 7, 64, 36, 1, 1, 11, 18, 46, 29, 81, 91, 29, 81, 46, 2, 1, 12, 3, 43, 75, 6, 112, 12, 54, 1, 57, 3

Offset: 2

Paolo Xausa, May 27 2024

Row n is the zeroless analog of the positive n-gonal numbers.

			The array begins:
  n\k|  1  2   3   4   5    6    7    8    9   10  ...
  ----------------------------------------------------
   2 |  1, 2,  3,  4,  5,   6,   7,   8,   9,   1, ... = A177274
   3 |  1, 3,  6,  1,  6,  12,  19,  27,  36,  46, ... = A243658 (from n = 1)
   4 |  1, 4,  9, 16, 25,  36,  49,  64,  81,   1, ... = A370812
   5 |  1, 5, 12, 22, 35,  51,   7,  29,  54,  82, ... = A373171
   6 |  1, 6, 15, 28, 45,  66,  91,  12,  45,  82, ... = A373172
   7 |  1, 7, 18, 34, 55,  81, 112, 148, 189, 235, ...
   8 |  1, 8, 21,  4, 29,   6,  43,  86, 135,  19, ...
   9 |  1, 9, 24, 46, 75, 111, 154,  24,  81, 145, ...
  10 |  1, 1, 18, 43, 76, 117, 166, 223, 288, 361, ...
  ...      |                                     \______ A373170 (main diagonal)
        A004719 (from n = 2)

Paolo Xausa, Table of n, a(n) for n = 2..11326 (first 150 antidiagonals, flattened).
Wikipedia, Polygonal number.

Cf. rows 2..6: A177274, A243658, A370812, A373171, A373172.
Cf. A373170 (main diagonal).
Cf. A004719, A057145.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A373169[n_, k_] := A373169[n, k] = If[k == 1, 1, noz[A373169[n, k-1] + (k-1)*(n-2) + 1]];
Table[A373169[n - k + 1, k], {n, 2, 15}, {k, n - 1}]

noz(n) = fromdigits(select(sign, digits(n)));
T(n,k) = if (k==1, 1, noz(T(n,k-1) + (k-1)*(n-2) + 1));
matrix(7,7,n,k,T(n+1,k)) \\ Michel Marcus, May 30 2024

A177270 Decimal expansion of (684125+sqrt(635918528029))/1033802.

Original entry on oeis.org

1, 4, 3, 3, 1, 2, 7, 4, 2, 6, 7, 2, 4, 3, 7, 0, 8, 2, 7, 2, 5, 3, 9, 5, 7, 3, 1, 9, 2, 7, 1, 0, 2, 8, 9, 4, 7, 4, 3, 0, 8, 2, 6, 3, 5, 5, 0, 7, 0, 5, 3, 0, 1, 5, 7, 1, 1, 3, 0, 0, 5, 3, 9, 2, 6, 4, 1, 6, 9, 7, 9, 3, 1, 4, 5, 8, 0, 3, 4, 1, 6, 9, 9, 3, 5, 7, 6, 9, 6, 4, 7, 6, 2, 6, 1, 6, 9, 8, 0, 5, 9, 5, 8, 2, 9

Offset: 1

Klaus Brockhaus, May 07 2010

Continued fraction expansion of (684125+sqrt(635918528029))/1033802 is A177274.

Agrees with A060997 for n < 13.

			(684125+sqrt(635918528029))/1033802 = 1.43312742672437082725...

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A060997, A177271 (decimal expansion of sqrt(635918528029)), A177274 (repeat 1, 2, 3, 4, 5, 6, 7, 8, 9).

First[RealDigits[(684125+Sqrt[635918528029])/1033802,10,120]] (* Paolo Xausa, Jan 09 2024 *)

A177271 Decimal expansion of sqrt(635918528029).

Original entry on oeis.org

7, 9, 7, 4, 4, 5, 0, 0, 0, 0, 0, 2, 5, 0, 8, 0, 0, 9, 9, 5, 6, 7, 9, 5, 5, 8, 4, 5, 7, 7, 0, 2, 8, 2, 6, 7, 9, 1, 1, 8, 8, 3, 1, 4, 7, 5, 2, 4, 6, 2, 4, 2, 1, 7, 4, 8, 3, 7, 3, 9, 2, 0, 0, 9, 2, 3, 7, 7, 2, 6, 0, 4, 9, 3, 7, 1, 7, 8, 6, 4, 0, 9, 4, 7, 9, 3, 8, 5, 3, 3, 2, 5, 5, 2, 2, 9, 5, 9, 7, 7, 3, 9, 3, 0, 0

Offset: 6

Klaus Brockhaus, May 07 2010

Continued fraction expansion of sqrt(635918528029) is 797445 followed by (repeat 398722, 1, 1, 398722, 1594890).

sqrt(635918528029) = sqrt(17)*sqrt(53)*sqrt(193)*sqrt(3656953).

			sqrt(635918528029) = 797445.00000250800995679558...

Daniel Starodubtsev, Table of n, a(n) for n = 6..10000

Cf. A010473 (decimal expansion of sqrt(17)), A010506 (decimal expansion of sqrt(53)), A177272 (decimal expansion of sqrt(193)), A177273 (decimal expansion of sqrt(3656953)), A177274 (continued fraction expansion of (684125+sqrt(635918528029))/1033802), A177270 (decimal expansion of (684125+sqrt(635918528029))/1033802).

First[RealDigits[Sqrt[635918528029],10,120]] (* Paolo Xausa, Jan 09 2024 *)

A262734 Period 16: repeat (1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 0

Ilya Gutkovskiy, Sep 29 2015

Decimal expansion of 111111112/900000009.

For n which lies in the interval [16*(k-1), 8*(2*k-1)], where k>0 -> pattern {1, 2, 3, 4, 5, 6, 7, 8, 9}; for n which lies in the interval [16*k - 7, 16*k - 1], where k>0 -> pattern {8, 7, 6, 5, 4, 3, 2}.

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

Cf. A158289, A177274, A010889, A138531, A181975, A199264, A068073, A028356.

&cat[[1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2]: n in [0..10]]; // Vincenzo Librandi, Sep 29 2015

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, -1, 1}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, 120] (* Vincenzo Librandi, Sep 29 2015 *)

Vec(-(2*x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)*(x^8+1)) + O(x^100)) \\ Colin Barker, Sep 29 2015

111111112/900000009. \\ Altug Alkan, Sep 29 2015

vector(200, n, default(realprecision, n+2); floor(111111112/900000009*10^n)%10) \\ Altug Alkan, Nov 12 2015

-1 + a(16*(k - 1)) = -2 + a(8*k + 3*(-1)^k - 4) = -3 + a(2*(4*k + (-1)^k - 2)) = -4 + a(8*k + (-1)^k - 4) = -5 + a(4*(2*k - 1)) = -6 + a(8*k - (-1)^k - 4) = -7 + a(-2*(-4*k + (-1)^k + 2)) = -8 + a(8*k - 3*(-1)^k - 4) = -9 + a(8*(2*k - 11)) = 0, for k>0.
a(0) = 1, a(n) = a(n+1) - 1, for 16*(k - 1) <= n < 8*(2*k - 1), and a(n) = a(n + 1) + 1, for 8*(2*k - 1) <= n < 16*k, where k>0.
From Colin Barker, Sep 29 2015: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n>8.
G.f.: -(2*x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1) / ((x-1)*(x^8+1)). (End)

A306309 The "zeroless Pascal triangle" read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 1, 1, 5, 1, 1, 6, 6, 2, 6, 6, 1, 1, 7, 12, 8, 8, 12, 7, 1, 1, 8, 19, 2, 16, 2, 19, 8, 1, 1, 9, 27, 21, 18, 18, 21, 27, 9, 1, 1, 1, 36, 48, 39, 36, 39, 48, 36, 1, 1, 1, 2, 37, 84, 87, 75, 75, 87, 84, 37, 2, 1

Offset: 0

Rémy Sigrist, Feb 06 2019

Left and right edges are all 1's, interior entries are obtained by removing zeros from the sum of the two numbers above them.
For any k >= 0 and n >= 0, let d_k(n) = T(n+k, k).
For any k >= 0, d_k is eventually periodic: by induction:
- for k = 0: for any n >= 0, d_0(n) = 1, hence d_0 is 1-periodic,
- suppose that the property is true for some k >= 0,
- d_k is eventually p_k-periodic, and so d_k is bounded, say by m_k,
- d_{k+1}(n+1) - d_{k+1}(n) = d_k(n+1) <= m_k,
- so the first difference of d_{k+1} is bounded by m_k,
- A004719 has arbitrary large gaps,
  and we can choose a range of m_k+1 terms that do not belong to A004719,
  say x_k..x_k+m_k (with x_k > 1),
- d_{k+1}(0) = 1 < x_k,
  and if d_{k+1}(n) < x_k, then d_{k+1)(n+1) < x_k,
  so d_{k+1} is bounded by x_k,
- let D_{k+1}(n) = d_{k+1}(n*p_k},
- D_{k+1} is bounded,
  so D_{k+1}(n + q_k) = D_{k+1}(n) for some n >= 0 and q_k > 0,
- we can assume that n*p_k is beyond the transient part of d_k,
- d_{k+1}(n*p_k + q_k*p_k + 1) = d_{k+1}(n*p_k+q_k*p_k) + d_k(n*p_k+q_k*p_k + 1)
                               = d_{k+1}(n*p_k)         + d_k(n*p_k + 1)
                               = d_{k+1}(n*p_k + 1),
- we can generalize: for any m >= n*p_k, d_{k+1}(m + q_k*p_k) = d_{k+1)(m),
- and d_{k+1} is (at least q_k*p_k-)periodic, QED.

			Triangle begins:
                    1
                  1   1
                1   2   1
              1   3   3   1
            1   4   6   4   1
          1   5   1   1   5   1
        1   6   6   2   6   6   1
      1   7  12   8   8  12   7   1
    1   8  19   2  16   2  19   8   1
  1   9  27  21  18  18  21  27   9   1
...

Rémy Sigrist, Colored representation of the first 1000 rows (where the hue is function of T(n, k))
Rémy Sigrist, PARI program for A306309

Cf. A004719, A007318, A177274.

PARI
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See Links section.
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T(n, 0) = T(n, n) = 1 for n >= 0.
T(n, k) = A004719(T(n-1, k-1) + T(n-1, k)) for n >= 0 and k = 1..n-1.
T(n, 1) = A177274(n-1) for any n > 0.

cp's OEIS Frontend

A373169 Square array read by ascending antidiagonals: T(n,k) = noz(T(n,k-1) + (k-1)*(n-2) + 1), with T(n,1) = 1, n >= 2, k >= 1, where noz(n) = A004719(n).

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A177270 Decimal expansion of (684125+sqrt(635918528029))/1033802.

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A177271 Decimal expansion of sqrt(635918528029).

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A262734 Period 16: repeat (1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2).

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A306309 The "zeroless Pascal triangle" read by rows.

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