A001129 -id:A001129 - cp's OEIS frontend

A102122 Iccanobirt numbers (12 of 15): a(n) = R(R(a(n-1)) + a(n-2) + a(n-3)), where R is the digit reversal function A004086.

0, 0, 1, 1, 2, 4, 7, 31, 42, 26, 531, 302, 67, 909, 8721, 4522, 48811, 72152, 6487, 908821, 844702, 6572211, 9726782, 29139201, 58129562, 86185456, 139627251, 949140792, 656458225, 9962261161, 6171227123, 20114953831, 68392496992

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Cf. A102111-A102125.

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[R[a[n-1]]+a[n-2]+a[n-3]];Table[a[n], {n, 0, 40}]
nxt[{a_,b_,c_}]:={b,c,IntegerReverse[IntegerReverse[c]+b+a]}; NestList[nxt,{0,0,1},40][[;;,1]] (* Harvey P. Dale, Sep 10 2024 *)

a(n) = A004086(A102114(n)).

A102123 Iccanobirt numbers (13 of 15): a(n) = R(R(a(n-1)) + a(n-2) + R(a(n-3))), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 31, 42, 26, 711, 761, 49, 279, 8811, 1651, 44311, 38141, 55006, 45901, 34108, 990681, 161132, 5891031, 6129461, 8041777, 45820251, 74839842, 60558487, 202825861, 635089352, 309192535, 7549098331, 8252802091

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Cf. A102111-A102125.

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[R[a[n-1]]+a[n-2]+R[a[n-3]]];Table[a[n], {n, 0, 40}]

a(n) = A004086(A102115(n)).

A102118 Iccanobirt numbers (8 of 15): a(n) = R(a(n-1) + a(n-2) + a(n-3)), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 31, 24, 26, 18, 86, 31, 531, 846, 8041, 8149, 63071, 16297, 71578, 649051, 629637, 6620531, 9129987, 55108361, 97885807, 551421261, 924514407, 5741283751, 9149127127, 58252941851, 92725334137, 511304721061

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A102111, A102112, A102113, A102114, A102115, A102116, A102117, A102119, A102120, A102121, A102122, A102123, A102124, A102125.

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[a[n-1]+a[n-2]+a[n-3]];Table[a[n], {n, 0, 40}]
Transpose[NestList[{#[[2]],#[[3]],FromDigits[Reverse[IntegerDigits[Total[ #]]]]}&,{0,0,1},40]][[1]] (* Harvey P. Dale, Dec 04 2012 *)

Incorrect formula removed by Georg Fischer, Dec 18 2020

A101764 Iccanobif semiprime indices: Indices of semiprime numbers in A014258.

Original entry on oeis.org

8, 10, 13, 17, 23, 26, 28, 29, 31, 39, 42, 53, 55, 56, 73, 83, 94, 98, 101, 113, 114, 115, 121, 167, 217, 255, 266, 326, 327, 333, 367, 389, 397, 404, 409, 423, 425, 467, 497, 570, 631, 639, 749, 761

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 15 2004

This sequence also includes 815, 862, 943, 1013, 1106, 1204, 1319, 1398, 1419, 1554, 1669, 1729, 1762, 1801, 1847, 1874, 1930, 1977, and 2123. It might or might not include 791, 927, 1022, 1027, 1110, 1129, 1307, 1558, 1662, 1694, 1723, 1747, 1850, 1934, 1954, 1978, 2014, 2069, and 2077, but the required factoring proved rather difficult. There are no further terms below 2123. - Lucas A. Brown, Nov 12 2022

Lucas A. Brown, A101764.py.

Cf. A001358, A001129, A014258, A014259, A014260, A101763, A101765, A101766.

nxt[{a_,b_}]:={b,IntegerReverse[a+b]}; Flatten[Position[NestList[nxt,{0,1},500][[All,1]],?(PrimeOmega[#]==2&)]]-1 (* _Harvey P. Dale, Jun 05 2022 *)

Missing 367 inserted and new terms 570-761 added by Lucas A. Brown, Nov 12 2022

A101765 Iccanobif semiprime indices: Indices of semiprime numbers in A014259.

Original entry on oeis.org

8, 9, 15, 16, 18, 22, 32, 37, 46, 53, 61, 62, 64, 79, 82, 106, 121, 129, 149, 153, 229, 241, 266, 301, 381, 411, 502

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 15 2004

This sequence also includes 742, 987, 1147, 1246, 1337, 1373, 1454, 1493, 1537, 1835, 1967, and 2265. It might or might not include 622, 630, 647, 817, 1247, 1402, 1422, 1477, 1649, 1781, 1818, 1867, 1874, and 2115, but the required factoring proved rather difficult. There are no further terms below 2265. - Lucas A. Brown, Nov 12 2022

Lucas A. Brown, A101765.py.

Cf. A001358, A001129, A014258, A014259, A014260, A101763, A101764, A101766.

a(27) from Lucas A. Brown, Nov 12 2022

A101766 Iccanobif semiprime indices: Indices of semiprime numbers in A014260.

Original entry on oeis.org

8, 16, 18, 21, 26, 38, 42, 44, 49, 54, 55, 57, 61, 67, 77, 78, 115, 123, 134, 145, 151, 154, 202, 218, 249, 286, 349, 403, 498, 539, 647

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 15 2004

This sequence also includes 790, 1161, 1347, 1418, 1595, 1761, and 2018. It might or might not include 769, 1394, 1795, 1983, 2093, 2178, but the required factoring proved rather difficult. There are no further terms below 2178. - Lucas A. Brown, Nov 12 2022

Lucas A. Brown, A101766.py.

Cf. A001358, A001129, A014258, A014259, A014260, A101763, A101764, A101765.

nxt[{a_,b_}]:={b,IntegerReverse[a]+b}; Flatten[Position[NestList[nxt,{0,1},600][[All,1]],?(PrimeOmega[#]==2&)]]-1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 04 2018 *)

Offset changed to 1 by Jinyuan Wang, Aug 01 2021

a(31) from Lucas A. Brown, Nov 12 2022

A237568 Fibonacci-like sequence of numbers with nondecreasing positive digits. Let a^+ denote the number that is obtained from a if its positive digits are written in nondecreasing order, while zeros remain in their places. Let a<+>b = (a + b)^+. a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 12, 25, 37, 26, 36, 26, 26, 25, 15, 40, 55, 59, 114, 137, 125, 226, 135, 136, 127, 236, 336, 257, 359, 166, 255, 124, 379, 305, 468, 377, 458, 358, 168, 256, 244, 500, 447, 479, 269, 478, 477, 559, 1036, 1559, 2559, 1148, 3707, 4558, 2568, 1267, 3358, 2456, 1458, 1349, 2708, 4057, 5667, 2479, 1468, 3479, 4479, 5789, 10268, 15067, 23355, 22348

Offset: 0

Vladimir Shevelev, Feb 09 2014

Note that operation n^+ differs from the one in A004185. If a term of the sequence has k digits, then it is followed by terms with >=k digits. The sequence has 7 terms with 1 digit, 13 terms with 2 digits, 30 terms with 3 digits, etc. The corresponding maximal terms are 8, 59, 559, etc.

The sequence is eventually periodic with period of length 144 and the first position of period 237. - Peter J. C. Moses, Feb 09 2014

Peter J. C. Moses, Table of n, a(n) for n = 0..952

Cf. A000045, A001129, A004185, A069638.

a[0]:=0;a[1]:=1;a[n_]:=a[n]=FromDigits[Insert[DeleteCases[Sort[#],0],0,1+#-Range[Length[#]]&[Position[#,0]]]&[IntegerDigits[a[n-1]+a[n-2]]]]; Map[a,Range[0,99]] (* Peter J. C. Moses, Feb 09 2014 *)

A237575 Fibonacci-like numbers with nonincreasing positive digits. Let a denote the number that is obtained from a if its digits are written in nonincreasing order. Let a<+>b = (a + b). a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 31, 93, 421, 541, 962, 5310, 7622, 93221, 843100, 963321, 8642110, 9654310, 98642210, 986522100, 8654311100, 9864332000, 88654311100, 98865431100, 987754221100, 9866652211000, 86544432110000, 98644321110000, 888755322110000

Offset: 0

Vladimir Shevelev, Feb 09 2014

Peter J. C. Moses, Table of n, a(n) for n = 0..499

Cf. A000045, A001129, A004185, A069638, A237568.

a:= proc(n) option remember; `if`(n<2, n, parse(cat(
      sort(convert(a(n-1)+a(n-2), base, 10), `>`)[])))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 31 2022

a[0]:=0;a[1]:=1;a[n_]:=a[n]=FromDigits[Reverse[Sort[IntegerDigits[a[n-1]+a[n-2]]]]];Map[a,Range[0,20]] (* Peter J. C. Moses, Feb 09 2014 *)

Correction and extension by Peter J. C. Moses

A101762 Iccanobif prime indices: Indices of prime numbers in A014260.

Original entry on oeis.org

3, 4, 5, 7, 11, 13, 19, 22, 25, 30, 39, 71, 81, 98, 1041, 2942, 4377, 10410

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 15 2004

No more terms through 11000.

Cf. A000040, A001129, A014258, A014259, A014260, A101759, A101760, A101761.

Offset changed to 1 by Jinyuan Wang, Aug 02 2021

A210791 Triangle of coefficients of polynomials u(n,x) jointly generated with A210792; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 3, 1, 4, 17, 14, 5, 1, 5, 36, 42, 30, 8, 1, 6, 72, 104, 111, 58, 13, 1, 7, 141, 233, 329, 251, 111, 21, 1, 8, 275, 494, 862, 848, 553, 206, 34, 1, 9, 538, 1016, 2097, 2479, 2112, 1158, 377, 55, 1, 10, 1058, 2056, 4870, 6608, 6875

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
Column 2: 1,2,3,4,5,6,7,8,...
Row sums: A007051.
Alternating row sums: A000129.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012

			First five rows:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  7,  3;
  1,  4, 17, 14,  5;
First three polynomials u(n,x):
  1
  1 + x
  1 + 2x + 2x^2.
From _Philippe Deléham_, Mar 29 2012: (Start)
(1, 0, 0, 2, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   1,   0;
  1,   2,   2,   0;
  1,   3,   7,   3,   0;
  1,   4,  17,  14,   5,   0;
  1,   5,  36,  42,  30,   8,   0;
  1,   6,  72, 104, 111,  58,  13,   0; (End)

Cf. A210792, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 2; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210791 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210792 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A007051 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A001129 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A001333 *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 29 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*x - y*x + 2*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

cp's OEIS Frontend

A102122 Iccanobirt numbers (12 of 15): a(n) = R(R(a(n-1)) + a(n-2) + a(n-3)), where R is the digit reversal function A004086.

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A102123 Iccanobirt numbers (13 of 15): a(n) = R(R(a(n-1)) + a(n-2) + R(a(n-3))), where R is the digit reversal function A004086.

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A102118 Iccanobirt numbers (8 of 15): a(n) = R(a(n-1) + a(n-2) + a(n-3)), where R is the digit reversal function A004086.

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A101764 Iccanobif semiprime indices: Indices of semiprime numbers in A014258.

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A101765 Iccanobif semiprime indices: Indices of semiprime numbers in A014259.

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A101766 Iccanobif semiprime indices: Indices of semiprime numbers in A014260.

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A237568 Fibonacci-like sequence of numbers with nondecreasing positive digits. Let a^+ denote the number that is obtained from a if its positive digits are written in nondecreasing order, while zeros remain in their places. Let a<+>b = (a + b)^+. a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).

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A237575 Fibonacci-like numbers with nonincreasing positive digits. Let a** denote the number that is obtained from a if its digits are written in nonincreasing order. Let a<+>b = (a + b)**. a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).

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A101762 Iccanobif prime indices: Indices of prime numbers in A014260.

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A237575 Fibonacci-like numbers with nonincreasing positive digits. Let a denote the number that is obtained from a if its digits are written in nonincreasing order. Let a<+>b = (a + b). a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).