A093645 -id:A093645 - cp's OEIS frontend

A022105 Fibonacci sequence beginning 1, 15.

1, 15, 16, 31, 47, 78, 125, 203, 328, 531, 859, 1390, 2249, 3639, 5888, 9527, 15415, 24942, 40357, 65299, 105656, 170955, 276611, 447566, 724177, 1171743, 1895920, 3067663, 4963583, 8031246, 12994829

Offset: 0

N. J. A. Sloane

a(n-1)=sum(P(15;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=14. These are the SW-NE diagonals in P(15;n,k), the (15,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1)

a(n) = A109754(14, n+1).

a(k) = A118654(4, k).

a0:=1; a1:=15; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

a={};b=1;c=15;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,12,1}];a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
LinearRecurrence[{1,1},{1,15},40] (* Harvey P. Dale, Oct 11 2015 *)

a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=15. a(-1):=14.
G.f.: (1+14*x)/(1-x-x^2).
a(n) = A101220(14,0,n+1). - Ross La Haye, May 02 2006

A022107 Fibonacci sequence beginning 1, 17.

Original entry on oeis.org

1, 17, 18, 35, 53, 88, 141, 229, 370, 599, 969, 1568, 2537, 4105, 6642, 10747, 17389, 28136, 45525, 73661, 119186, 192847, 312033, 504880, 816913, 1321793, 2138706, 3460499, 5599205, 9059704, 14658909

Offset: 0

N. J. A. Sloane

a(n-1)=sum(P(17;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=16. These are the SW-NE diagonals in P(17;n,k), the (17,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

a(n) = A109754(16, n+1) = A101220(16, 0, n+1).

a0:=1; a1:=17; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

a={};b=1;c=17;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,12,1}];a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
LinearRecurrence[{1,1},{1,17},40] (* Harvey P. Dale, Aug 04 2017 *)

a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=17. a(-1):=16.

G.f.: (1+16*x)/(1-x-x^2).

A022108 Fibonacci sequence beginning 1, 18.

Original entry on oeis.org

1, 18, 19, 37, 56, 93, 149, 242, 391, 633, 1024, 1657, 2681, 4338, 7019, 11357, 18376, 29733, 48109, 77842, 125951, 203793, 329744, 533537, 863281, 1396818, 2260099, 3656917, 5917016, 9573933, 15490949

Offset: 0

N. J. A. Sloane

a(n-1)=sum(P(18;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=17. These are the SW-NE diagonals in P(18;n,k), the (18,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

a(n) = A109754(17, n+1) = A101220(17, 0, n+1).

a0:=1; a1:=18; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

a={};b=1;c=18;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,12,1}];a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
LinearRecurrence[{1,1},{1,18},40] (* Harvey P. Dale, Apr 15 2018 *)

a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=18. a(-1):=17.

G.f.: (1+17*x)/(1-x-x^2).

A022391 Fibonacci sequence beginning 1, 21.

Original entry on oeis.org

1, 21, 22, 43, 65, 108, 173, 281, 454, 735, 1189, 1924, 3113, 5037, 8150, 13187, 21337, 34524, 55861, 90385, 146246, 236631, 382877, 619508, 1002385, 1621893, 2624278, 4246171, 6870449, 11116620, 17987069, 29103689, 47090758, 76194447, 123285205, 199479652, 322764857, 522244509

Offset: 0

N. J. A. Sloane

nonn

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(21;n-1-k,k), n>=1, with a(-1)=20. These are the SW-NE diagonals in P(21;n,k), the (21,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Kak, The Golden Mean and the Physics of Aesthetics, arXiv:physics/0411195 [physics.hist-ph], 2004.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

[Fibonacci(n+2) + 19*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Mar 02 2018

LinearRecurrence[{1, 1}, {1, 21}, 30] (* Jean-François Alcover, Feb 25 2018 *)
Table[Fibonacci[n + 2] + 19*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 02 2018 *)

for(n=0, 50, print1(fibonacci(n+2) + 19*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 02 2018

a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=21. a(-1):=20.

G.f.: (1+20*x)/(1-x-x^2).

Terms a(30) onward added by G. C. Greubel, Mar 02 2018

A056125 a(n) = (5n + 4)binomial(n+7,7)/4.

Original entry on oeis.org

1, 18, 126, 570, 1980, 5742, 14586, 33462, 70785, 140140, 262548, 469404, 806208, 1337220, 2151180, 3368244, 5148297, 7700814, 11296450, 16280550, 23088780, 32265090, 44482230, 60565050, 81516825, 108548856, 143113608, 186941656

Offset: 0

Barry E. Williams, Jul 07 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A052254.
Cf. A093645 ((10, 1) Pascal, column m=8).
Partial sums of A052254.

List([0..30], n-> (5*n+4)*Binomial(n+7,7)/4 ); # G. C. Greubel, Jan 19 2020

[(5*n+4)*Binomial(n+7,7)/4: n in [0..30]]; // G. C. Greubel, Jan 19 2020

seq( (5*n+4)*binomial(n+7,7)/4, n=0..30); # G. C. Greubel, Jan 19 2020

Table[((5n+4)Binomial[n+7,7])/4,{n,0,30}] (* or *) LinearRecurrence[{9,-36,84, -126,126,-84,36,-9,1},{1,18,126,570,1980,5742,14586,33462,70785},30] (* Harvey P. Dale, Jan 18 2013 *)

vector(31, n, (5*n-1)*binomial(n+6,7)/4 ) \\ G. C. Greubel, Jan 19 2020

[(5*n+4)*binomial(n+7,7)/4 for n in (0..30)] # G. C. Greubel, Jan 19 2020

G.f.: (1+9*x)/(1-x)^9.
a(0)=1, a(1)=18, a(2)=126, a(3)=570, a(4)=1980, a(5)=5742, a(6)=14586, a(7)=33462, a(8)=70785, a(n) = 9*a(n-1) -36*a(n-2) +84*a(n-3) -126*a(n-4) + 126*a(n-5) -84*a(n-6) +36*a(n-7) -9*a(n-8) +a(n-9). - Harvey P. Dale, Jan 18 2013
From G. C. Greubel, Jan 19 2020: (Start)
a(n) = 10*binomial(n+8,8) - 9*binomial(n+7,7).
E.g.f.: (20160 + 342720*x + 917280*x^2 + 823200*x^3 + 323400*x^4 + 62328*x^5 + 6076*x^6 + 284*x^7 + 5*x^8)*exp(x)/20160. (End)

A022392 Fibonacci sequence beginning 1, 22.

Original entry on oeis.org

1, 22, 23, 45, 68, 113, 181, 294, 475, 769, 1244, 2013, 3257, 5270, 8527, 13797, 22324, 36121, 58445, 94566, 153011, 247577, 400588, 648165, 1048753, 1696918, 2745671, 4442589, 7188260, 11630849, 18819109, 30449958, 49269067, 79719025, 128988092, 208707117, 337695209, 546402326

Offset: 0

N. J. A. Sloane

nonn

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(22;n-1-k,k), n>=1, with a(-1)=21. These are the SW-NE diagonals in P(22;n,k), the (22,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

[Fibonacci(n+2) + 20*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Mar 02 2018

Table[Fibonacci[n + 2] + 20*Fibonacci[n], {n, 0, 50}] (* or *) LinearRecurrence[{1,1}, {1,22}, 50] (* G. C. Greubel, Mar 02 2018 *)

for(n=0, 50, print1(fibonacci(n+2) + 20*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 02 2018

a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=22. a(-1):=21.

G.f.: (1+21*x)/(1-x-x^2).

Terms a(30) onward added by G. C. Greubel, Mar 02 2018

A022393 Fibonacci sequence beginning 1, 23.

Original entry on oeis.org

1, 23, 24, 47, 71, 118, 189, 307, 496, 803, 1299, 2102, 3401, 5503, 8904, 14407, 23311, 37718, 61029, 98747, 159776, 258523, 418299, 676822, 1095121, 1771943, 2867064, 4639007, 7506071, 12145078, 19651149, 31796227, 51447376, 83243603, 134690979, 217934582, 352625561, 570560143

Offset: 0

N. J. A. Sloane

nonn

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(23;n-1-k,k), n>=1, with a(-1)=22. These are the SW-NE diagonals in P(23;n,k), the (23,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

List([0..40],n->Fibonacci(n+2)+21*Fibonacci(n)); # Muniru A Asiru, Mar 03 2018

[Fibonacci(n+2) + 21*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Mar 02 2018

a[1]=1; a[2]=23; a[n_]:=a[n]=a[n - 1]+a[n - 2] (*  José María Grau Ribas, Feb 15 2010 *)
LinearRecurrence[{1,1},{1,23},30] (* Harvey P. Dale, Sep 30 2011 *)
Table[Fibonacci[n + 2] + 21*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 02 2018 *)

for(n=0, 50, print1(fibonacci(n+2) + 21*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 02 2018

a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=23. a(-1):=22.

G.f.: (1+22*x)/(1-x-x^2).

Terms a(30) onward added by G. C. Greubel, Mar 02 2018

A093646 Higher dimensional figurate numbers based on 12-gonal numbers A051624.

Original entry on oeis.org

1, 19, 145, 715, 2695, 8437, 23023, 56485, 127270, 267410, 529958, 999362, 1805570, 3142790, 5293970, 8662214, 13810511, 21511325, 32807775, 49088325, 72177105, 104442195, 148924425, 209489475, 291006300, 399555156, 542668764, 729610420

Offset: 0

Wolfdieter Lang, Apr 22 2004

Cf. A093645 ((10, 1) Pascal, column m=9).

a(n)= (10*n+9)*binomial(n+8, 8)/9. G.f.: (1+9*x)/(1-x)^10.

A180053 a(1)=1, a(2)=101, a(n) = 1001*a(n-1) for n > 2.

Original entry on oeis.org

1, 101, 101101, 101202101, 101303303101, 101404606404101, 101506011010505101, 101607517021515606101, 101709124538537121707101, 101810833663075658828808101, 101912644496738734487636909101, 102014557141235473222124546010101

Offset: 1

Mark Dols, Aug 08 2010

			In a column:
                1
              101
           101101
        101202101
     101303303101
  101404606404101

Index entries for linear recurrences with constant coefficients, signature (1001).

Cf. A093645.

LinearRecurrence[{1001}, {1, 101}, 15] (* Paolo Xausa, Jan 29 2024 *)
Join[{1},NestList[1001#&,101,10]] (* Harvey P. Dale, Jul 29 2024 *)

a(n)=max(101*1001^(n-2),1) \\ Charles R Greathouse IV, Oct 08 2011

G.f.: x(1-900*x)/(1-1001*x). - Philippe Deléham, Oct 08 2011

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A022105 Fibonacci sequence beginning 1, 15.

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A022107 Fibonacci sequence beginning 1, 17.

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A022108 Fibonacci sequence beginning 1, 18.

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A022391 Fibonacci sequence beginning 1, 21.

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A056125 a(n) = (5*n + 4)*binomial(n+7,7)/4.

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A022392 Fibonacci sequence beginning 1, 22.

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A022393 Fibonacci sequence beginning 1, 23.

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A056125 a(n) = (5n + 4)binomial(n+7,7)/4.