A093645 -id:A093645 - cp's OEIS frontend

A022101 Fibonacci sequence beginning 1, 11.

1, 11, 12, 23, 35, 58, 93, 151, 244, 395, 639, 1034, 1673, 2707, 4380, 7087, 11467, 18554, 30021, 48575, 78596, 127171, 205767, 332938, 538705, 871643, 1410348, 2281991, 3692339, 5974330, 9666669, 15640999, 25307668, 40948667, 66256335, 107205002, 173461337, 280666339

Offset: 0

N. J. A. Sloane

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(11;n-1-k,k) with n >= 1, a(-1)=10. These are the SW-NE diagonals in P(11;n,k), the (11,1) 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.
In general, for b Fibonacci sequence beginning with 1, h, we have:
b(n) = (2^(-1-n)*((1 - sqrt(5))^n*(1 + sqrt(5) - 2*h) + (1 + sqrt(5))^n*(-1 + sqrt(5) + 2*h)))/sqrt(5). - Herbert Kociemba, Dec 18 2011
Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (is this A001175?). - R. J. Mathar, Aug 10 2012

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

Cf. A000032, A000045.

a(n) = A109754(10, n+1) = A101220(10, 0, n+1).

a0:=1; a1:=11; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

LinearRecurrence[{1,1},{1,11},40] (* Harvey P. Dale, Aug 16 2015 *)

a(n) = 10*fibonacci(n)+fibonacci(n+1) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = a(n-1) + a(n-2), n >= 2, a(0)=1, a(1)=11. a(-1)=10.
G.f.: (1+10*x)/(1-x-x^2).
a(n-1) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) + 5*((1+sqrt(5))^(n-1) - (1-sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009
a(n) = 10*A000045(n) + A000045(n+1). - R. J. Mathar, Apr 07 2011
a(n) = 12*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017
a(n) = A000045(n+4) + A000032(n-4) for n > 0. - Bruno Berselli, Sep 27 2017

A051880 a(n) = binomial(n+4,4)(2n+1).

Original entry on oeis.org

1, 15, 75, 245, 630, 1386, 2730, 4950, 8415, 13585, 21021, 31395, 45500, 64260, 88740, 120156, 159885, 209475, 270655, 345345, 435666, 543950, 672750, 824850, 1003275, 1211301, 1452465, 1730575, 2049720, 2414280, 2828936, 3298680, 3828825, 4425015, 5093235

Offset: 0

Barry E. Williams, Dec 14 1999

Old name was: Partial sums of A051799.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A051799.
Cf. A093645 ((10, 1) Pascal, column m=5).
A diagonal of A280880.

Nest[Accumulate[#]&,Table[n(n+1)(10n-7)/6,{n,0,50}],2] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = C(n+4, 4)*(2n+1).
G.f.: (1+9*x)/(1-x)^6.
From Amiram Eldar, Sep 04 2025: (Start)
Sum_{n>=0} 1/a(n) = 128*log(2)/35 - 152/105.
Sum_{n>=0} (-1)^n/a(n) = 32*Pi/35 + 596/105 - 384*log(2)/35. (End)

Name changed by Alois P. Heinz, Jan 09 2017

A022102 Fibonacci sequence beginning 1, 12.

Original entry on oeis.org

1, 12, 13, 25, 38, 63, 101, 164, 265, 429, 694, 1123, 1817, 2940, 4757, 7697, 12454, 20151, 32605, 52756, 85361, 138117, 223478, 361595, 585073, 946668, 1531741, 2478409, 4010150, 6488559, 10498709

Offset: 0

N. J. A. Sloane

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(12;n-1-k,k) with n>=1, a(-1)=11. These are the SW-NE diagonals in P(12;n,k), the (12,1) 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.
In general, for b Fibonacci sequence beginning with 1, h, we have:
b(n) = (2^(-1-n)*((1 - sqrt(5))^n*(1 + sqrt(5) - 2*h) + (1 + sqrt(5))^n*(-1 + sqrt(5) + 2*h)))/sqrt(5). - Herbert Kociemba, Dec 18 2011
Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (is this A001175?). - R. J. Mathar, Aug 10 2012

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

Cf. A000045, A001175, A093645, A101220, A109754.

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

LinearRecurrence[{1,1},{1,12},40] (* Harvey P. Dale, Jan 23 2012 *)

a(n) = if(n==0, 1, if(n==1, 12, a(n-1)+a(n-2))) \\ Felix Fröhlich, Jun 09 2022

Vec((1+11*x)/(1-x-x^2) + O(x^20)) \\ Felix Fröhlich, Jun 09 2022

a(n) = a(n-1) + a(n-2), n >= 2, a(0)=1, a(1)=12. a(-1):=11.
G.f.: (1+11*x)/(1-x-x^2).
a(n) = A109754(11, n+1) = A101220(11, 0, n+1).
a(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) + (11/2)*((1+sqrt(5))^(n-1)-(1-sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)). Offset 1. a(3)=13. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009
a(n) = 11*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012
a(n) = 13*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017
a(n) = F(n+5) + F(n-1) - F(n-5) for F(n) the Fibonacci number A000045(n). - Greg Dresden and Aamen Muharram, Jun 09 2022

A022110 Fibonacci sequence beginning 1, 20.

Original entry on oeis.org

1, 20, 21, 41, 62, 103, 165, 268, 433, 701, 1134, 1835, 2969, 4804, 7773, 12577, 20350, 32927, 53277, 86204, 139481, 225685, 365166, 590851, 956017, 1546868, 2502885, 4049753, 6552638, 10602391, 17155029, 27757420, 44912449, 72669869, 117582318, 190252187

Offset: 0

N. J. A. Sloane

a(n-1) = Sum(P(20;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1) = 19. These are the SW-NE diagonals in P(20;n,k), the (20,1) 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.

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

a(n) = A109754(19, n+1) = A101220(19, 0, n+1).

a0:=1; a1:=20; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

a={};b=1;c=20;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, 20}, 35] (* Paolo Xausa, Feb 22 2024 *)

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

G.f.: (1+19*x)/(1-x-x^2).

A051799 Partial sums of A007587.

Original entry on oeis.org

1, 14, 60, 170, 385, 756, 1344, 2220, 3465, 5170, 7436, 10374, 14105, 18760, 24480, 31416, 39729, 49590, 61180, 74690, 90321, 108284, 128800, 152100, 178425, 208026, 241164, 278110, 319145, 364560, 414656, 469744, 530145, 596190

Offset: 0

Barry E. Williams, Dec 11 1999

4-dimensional pyramidal number, composed of consecutive 3-dimensional slices; each of which is a 3-dimensional 12-gonal (or dodecagonal) pyramidal number; which in turn is composed of consecutive 2-dimensional slices 12-gonal numbers. - Jonathan Vos Post, Mar 17 2006

Convolution of A000027 with A051624 (excluding 0). - Bruno Berselli, Dec 07 2012

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A007587, A001622, A051624.
Cf. A093645 ((10, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

/* A000027 convolved with A051624 (excluding 0): */ A051624:=func; [&+[(n-i+1)*A051624(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Accumulate[Table[n(n+1)(10n-7)/6,{n,0,50}]] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.
G.f.: (1+9*x)/(1-x)^5.
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = (125*log(5) + 10*sqrt(5*(5-2*sqrt(5)))*Pi - 50*sqrt(5)*log(phi) - 84)/104, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = (50*sqrt(5)*log(phi) + 5*sqrt(50-10*sqrt(5))*Pi - 256*log(2) + 90)/52. (End)

A022104 Fibonacci sequence beginning 1, 14.

Original entry on oeis.org

1, 14, 15, 29, 44, 73, 117, 190, 307, 497, 804, 1301, 2105, 3406, 5511, 8917, 14428, 23345, 37773, 61118, 98891, 160009, 258900, 418909, 677809, 1096718, 1774527, 2871245, 4645772, 7517017, 12162789

Offset: 0

N. J. A. Sloane

a(n-1)=sum(P(14;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=13. These are the SW-NE diagonals in P(14;n,k), the (14,1) 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(13, n+1) = A101220(13, 0, n+1).

a0:=1; a1:=14; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

a={};b=1;c=14;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,9,1}];a (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
LinearRecurrence[{1,1},{1,14},40] (* Harvey P. Dale, Jun 12 2017 *)

a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=14. a(-1):=13.

G.f.: (1+13*x)/(1-x-x^2).

A022106 Fibonacci sequence beginning 1, 16.

Original entry on oeis.org

1, 16, 17, 33, 50, 83, 133, 216, 349, 565, 914, 1479, 2393, 3872, 6265, 10137, 16402, 26539, 42941, 69480, 112421, 181901, 294322, 476223, 770545, 1246768, 2017313, 3264081, 5281394, 8545475, 13826869

Offset: 0

N. J. A. Sloane

a(n-1)=sum(P(16;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=15. These are the SW-NE diagonals in P(16;n,k), the (16,1) 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.

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

a(n) = A109754(15, n+1) = A101220(15, 0, n+1).

a0:=1; a1:=16; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

a={};b=1;c=16;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},{1,16},40] (* Harvey P. Dale, Jun 22 2016 *)

a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=16. a(-1):=15.

G.f.: (1+15*x)/(1-x-x^2).

A022109 Fibonacci sequence beginning 1, 19.

Original entry on oeis.org

1, 19, 20, 39, 59, 98, 157, 255, 412, 667, 1079, 1746, 2825, 4571, 7396, 11967, 19363, 31330, 50693, 82023, 132716, 214739, 347455, 562194, 909649, 1471843, 2381492, 3853335, 6234827, 10088162, 16322989, 26411151, 42734140, 69145291, 111879431, 181024722

Offset: 0

N. J. A. Sloane

a(n-1) = Sum(P(19;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=18. These are the SW-NE diagonals in P(19;n,k), the (19,1) 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.

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

a(n) = A109754(18, n+1) = A101220(18, 0, n+1).

a0:=1; a1:=19; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

LinearRecurrence[{1, 1}, {1, 19}, 35] (* Paolo Xausa, Feb 22 2024 *)

a(n) = a(n-1)+a(n-2), n >= 2, a(0) = 1, a(1) = 19.

G.f.: (1+18*x)/(1-x-x^2).

A050406 Partial sums of A051880.

Original entry on oeis.org

1, 16, 91, 336, 966, 2352, 5082, 10032, 18447, 32032, 53053, 84448, 129948, 194208, 282948, 403104, 562989, 772464, 1043119, 1388464, 1824130, 2368080, 3040830, 3865680, 4868955, 6080256, 7532721, 9263296

Offset: 0

Barry E. Williams, Dec 21 1999

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A051880.

Cf. A093645 ((10, 1) Pascal, column m=6).

List([0..40], n-> Binomial(n+5,5)*(5*n+3)/3); # G. C. Greubel, Oct 30 2019

[Binomial(n+5,5)*(5*n+3)/3: n in [0..40]]; // G. C. Greubel, Oct 30 2019

seq(binomial(n+5,5)*(5*n+3)/3, n=0..40); # G. C. Greubel, Oct 30 2019

Nest[Accumulate[#]&,Table[n(n+1)(10n-7)/6,{n,0,50}],3] (* Harvey P. Dale, Nov 13 2013 *)

vector(41, n, binomial(n+4,5)*(5*n-2)/3) \\ G. C. Greubel, Oct 30 2019

[binomial(n+5,5)*(5*n+3)/3 for n in (0..40)] # G. C. Greubel, Oct 30 2019

a(n) = C(n+5, 5)*(5*n + 3)/3.
G.f.: (1+9*x)/(1-x)^7.
E.g.f.: (360 +5400*x +10800*x^2 +6600*x^3 +1575*x^4 +153*x^5 +5*x^6) *exp(x)/360. - G. C. Greubel, Oct 30 2019

Corrected by T. D. Noe, Nov 09 2006

A052254 Partial sums of A050406.

Original entry on oeis.org

1, 17, 108, 444, 1410, 3762, 8844, 18876, 37323, 69355, 122408, 206856, 336804, 531012, 813960, 1217064, 1780053, 2552517, 3595636, 4984100, 6808230, 9176310, 12217140, 16082820, 20951775

Offset: 0

Barry E. Williams, Feb 03 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1)

Cf. A050406.

Cf. A093645 ((10, 1) Pascal, column m=7).

List([0..30], n-> (10*n+7)*Binomial(n+6, 6)/7 ); # G. C. Greubel, Jan 19 2020

[(10*n+7)*Binomial(n+6, 6)/7: n in [0..30]]; // G. C. Greubel, Jan 19 2020

seq( (10*n+7)*binomial(n+6, 6)/7, n=0..30); # G. C. Greubel, Jan 19 2020

Table[10*Binomial[n+7,7] -9*Binomial[n+6,6], {n,0,30}] (* G. C. Greubel, Jan 19 2020 *)
Rest[Nest[Accumulate[#]&,Table[n(n+1)(10n-7)/6,{n,0,50}],4]] (* Harvey P. Dale, Aug 03 2020 *)

vector(31, n, (10*n-3)*binomial(n+5, 6)/7) \\ G. C. Greubel, Jan 19 2020

[(10*n+7)*binomial(n+6, 6)/7 for n in (0..30)] # G. C. Greubel, Jan 19 2020

a(n) = (10*n + 7)*binomial(n+6, 6)/7.
G.f.: (1+9*x)/(1-x)^8.
From G. C. Greubel, Jan 19 2020: (Start)
a(n) = 10*binomial(n+7, 7) - 9*binomial(n+6, 6).
E.g.f.: (7! + 80640*x + 189000*x^2 + 142800*x^3 + 45150*x^4 + 6552*x^5 + 427*x^6 + 10*x^7)*exp(x)/7!. (End)
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Nov 28 2021

cp's OEIS Frontend

A022101 Fibonacci sequence beginning 1, 11.

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A051880 a(n) = binomial(n+4,4)*(2*n+1).

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A022102 Fibonacci sequence beginning 1, 12.

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A022110 Fibonacci sequence beginning 1, 20.

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A051799 Partial sums of A007587.

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A022104 Fibonacci sequence beginning 1, 14.

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A022106 Fibonacci sequence beginning 1, 16.

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A051880 a(n) = binomial(n+4,4)(2n+1).