A021113 -id:A021113 - cp's OEIS frontend

A162741 Fibonacci-Pascal triangle; same as Pascal triangle, but beginning another Pascal triangle to the right of each row starting at row 2.

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 7, 7, 5, 3, 2, 1, 1, 1, 5, 11, 14, 12, 8, 5, 3, 2, 1, 1, 1, 6, 16, 25, 26, 20, 13, 8, 5, 3, 2, 1, 1, 1, 7, 22, 41, 51, 46, 33, 21, 13, 8, 5, 3, 2, 1, 1, 1, 8, 29, 63, 92, 97, 79, 54, 34, 21, 13, 8, 5, 3, 2, 1, 1

Offset: 1

Mark Dols, Jul 12 2009, Jul 19 2009

Intertwined Pascal-triangles;

the first five rows seen as numbers in decimal representation: row(n) = 110*row(n-1) + 1. - corrected by Reinhard Zumkeller, Jul 16 2013

			.                                           1
.                                       1,  1, 1
.                                   1,  2,  2, 1, 1
.                               1,  3,  4,  3, 2, 1, 1
.                           1,  4,  7,  7,  5, 3, 2, 1, 1
.                       1,  5, 11, 14, 12,  8, 5, 3, 2, 1, 1
.                   1,  6, 16, 25, 26, 20, 13, 8, 5, 3, 2, 1,1
.               1,  7, 22, 41, 51, 46, 33, 21,13, 8, 5, 3, 2,1,1
.           1,  8, 29, 63, 92, 97, 79, 54, 34,21,13, 8, 5, 3,2,1,1
.       1,  9, 37, 92,155,189,176,133, 88, 55,34,21,13, 8, 5,3,2,1,1
.    1,10, 46,129,247,344,365,309,221,143, 89,55,34,21,13, 8,5,3,2,1,1
. 1,11,56,175,376,591,709,674,530,364,232,144,89,55,34,21,13,8,5,3,2,1,1 .

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Table 3.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A005408 (row length), A000225 (row sums), A000045 (central terms), A007318, A136431.
Cf. A021113. - Mark Dols, Jul 18 2009
Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A228074.

a162741 n k = a162741_tabf !! (n-1) !! (k-1)
a162741_row n = a162741_tabf !! (n-1)
a162741_tabf = iterate
   (\row -> zipWith (+) ([0] ++ row ++ [0]) (row ++ [0,1])) [1]
-- Reinhard Zumkeller, Jul 16 2013

T[, 1] = 1; T[n, k_] /; k == 2*n-2 || k == 2*n-1 = 1; T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k]; Table[T[n, k], {n, 1, 9}, {k, 1, 2*n-1}] // Flatten (* Jean-François Alcover, Oct 30 2017, after Reinhard Zumkeller *)

T(n,k) = T(n-1,k-1) + T(n-1,k), T(n,1)=1 and for n>1: T(n,2*n-2) = T(n,2*n-1)=1. - Reinhard Zumkeller, Jul 16 2013

A165154 a(n) = 100*a(n-1) + (-9)^(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 1, 91, 9181, 917371, 91743661, 9174307051, 917431236541, 91743118871131, 9174311930159821, 917431192628561611, 91743119266342945501, 9174311926602913490491, 917431192660573778585581, 91743119266054835992729771, 9174311926605506476065432061

Offset: 0

Mark Dols, Sep 05 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (91,900).

Cf. A021113, A164913, A165155, A172162, A172163.

[(1/109)*(100^n-(-9)^n): n in [0..20]]; // Vincenzo Librandi, Jun 10 2011

LinearRecurrence[{91,900}, {0,1}, 40] (* G. C. Greubel, Feb 09 2023 *)

Vec(x/((1+9*x)*(1-100*x)) + O(x^20)) \\ Colin Barker, Oct 02 2015

[(100^n-(-9)^n)/109 for n in range(41)] # G. C. Greubel, Feb 09 2023

From Colin Barker, Oct 02 2015: (Start)
a(n) = 91*a(n-1) + 900*a(n-2) for n>1, a(0)=0.
G.f.: x/((1+9*x)*(1-100*x)). (End)
E.g.f.: (1/109)*(exp(100*x) - exp(-9*x)). - G. C. Greubel, Feb 09 2023

a(0) prepended by Joerg Arndt, Oct 02 2015

A164913 Expansion of x/(1-9x-11x^2+10*x^3).

Original entry on oeis.org

1, 9, 92, 917, 9175, 91742, 917433, 9174309, 91743124, 917431185, 9174311939, 91743119246, 917431192693, 9174311926553, 91743119266140, 917431192660413, 9174311926605727, 91743119266054686, 917431192660551041, 9174311926605503645

Offset: 1

Mark Dols, Aug 31 2009

Index entries for linear recurrences with constant coefficients, signature (9,11,-10).

Cf. A000045, A021113.

a(n) = +9*a(n-1) +11*a(n-2) -10*a(n-3). - Joerg Arndt, May 14 2011

a(n) = 10*a(n-1)+F(n) for n=odd and a(n) = 10*a(n-1)-F(n) for n=even, with a(0) = 0 and F(n) is the n-th Fibonacci number.

A021222 Decimal expansion of 1/218.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A021113, A021440, A021876.

Join[{0,0},RealDigits[1/218,10,120][[1]]] (* Harvey P. Dale, Feb 28 2013 *)

A021440 Decimal expansion of 1/436.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.0022935779816513761467889908256880733944954128440...

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A021113, A021222, A021876.

First[RealDigits[1/436, 10, 100, -1]] (* Paolo Xausa, Feb 13 2025 *)

A021876 Decimal expansion of 1/872.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This constant, A021440, A021222 and A021113 all share, apart from initial digits, the same period of decimal digits of length A002322(109) = A002322(218) = A002322(436) = 108. - R. J. Mathar, Mar 19 2008

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A021113, A021222, A021440.

Join[{0,0},RealDigits[1/872,10,120][[1]]] (* Harvey P. Dale, Apr 08 2021 *)

A064737 Fibonacci sequence (A000045) mod 10 with carry.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 3, 2, 6, 8, 4, 3, 8, 1, 10, 1, 2, 4, 6, 0, 7, 7, 4, 2, 7, 9, 6, 6, 3, 10, 3, 4, 8, 2, 1, 4, 5, 9, 4, 4, 9, 3, 3, 7, 0, 8, 8, 6, 5, 2, 8, 0, 9, 9, 8, 8, 7, 6, 4, 1, 6, 7, 3, 1, 5, 6, 1, 8, 9, 7, 7, 5, 3, 9, 2, 2, 5, 7, 2, 10, 2, 3, 6, 9, 5, 5, 1, 7, 8, 5, 4, 10, 4, 5, 10, 5, 6, 2, 9, 1, 1

Offset: 0

Robert G. Wilson v, Oct 17 2001

Observation (from Kaprekar) this sequence can be the reverse of 1/109 (A021113) which is recurring and has a period of 108 digits:
0.[009174311926605504587155963302752293577981651376146788\
   990825688073394495412844036697247706422018348623853211]. - Eric Desbiaux, Oct 28 2008

Primes.utm.edu, Sum of 11^n*10^n [From _Eric Desbiaux_, Oct 28 2008]

Cf. A000045.

f[0] = 0; f[1] = f[2] = 1; f[n_] := f[n] = Block[{k}, If[f[n - 2] + f[n - 3] >= 10, k = 1, k = 0]; Mod[f[n - 1] + f[n - 2], 10] + k]; Table[ f[n], {n, 0, 100} ]

A021549 Decimal expansion of 1/545.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.00183486238532110091743119266055045871559633... .

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A021113.

Join[{0,0},RealDigits[1/545,10,120][[1]]] (* Harvey P. Dale, Jan 30 2019 *)

Equals A021113/5.

cp's OEIS Frontend

A162741 Fibonacci-Pascal triangle; same as Pascal triangle, but beginning another Pascal triangle to the right of each row starting at row 2.

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A165154 a(n) = 100*a(n-1) + (-9)^(n-1) for n>0, a(0)=0.

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A164913 Expansion of x/(1-9*x-11*x^2+10*x^3).

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A021222 Decimal expansion of 1/218.

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A021440 Decimal expansion of 1/436.

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A021876 Decimal expansion of 1/872.

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A064737 Fibonacci sequence (A000045) mod 10 with carry.

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A021549 Decimal expansion of 1/545.

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A164913 Expansion of x/(1-9x-11x^2+10*x^3).