Edwin McCravy - cp's OEIS frontend

A377207 Number of n-digit numbers where every digit is either a 9 or adjacent to a 9.

1, 18, 99, 342, 2691, 13788, 65709, 407772, 2115981, 11108358, 63181719, 334551402, 1802963871, 9931645728, 53256984129, 288681869232, 1572458030361, 8484410567898, 46019764248939, 249748559819262, 1351163694059451, 7326501636596868, 39716608228492149, 215099382176679492

Offset: 1

Edwin McCravy, Oct 19 2024

The 9 in the definition can also be any other nonzero digit.

			The a(1) = 1 number is 9.
The a(2) = 18 numbers are 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.

Index entries for linear recurrences with constant coefficients, signature (1,9,81).

Vec((1 + 8*x)*(1 + 9*x)/(1 - x - 9*x^2 - 81*x^3) + O(x^25)) \\ Andrew Howroyd, Oct 20 2024

G.f.: x*(1 + 8*x)*(1 + 9*x)/(1 - x - 9*x^2 - 81*x^3). - Andrew Howroyd, Oct 20 2024

a(9) onwards from Andrew Howroyd, Oct 20 2024

A276764 1^2 + 3^2, 2^2 + 4^2, 5^2 + 7^2, 6^2 + 8^2, ...

Original entry on oeis.org

10, 20, 74, 100, 202, 244, 394, 452, 650, 724, 970, 1060, 1354, 1460, 1802, 1924, 2314, 2452, 2890, 3044, 3530, 3700, 4234, 4420, 5002, 5204, 5834, 6052, 6730, 6964, 7690, 7940, 8714, 8980, 9802, 10084, 10954, 11252, 12170, 12484, 13450, 13780, 14794, 15140

Offset: 1

Edwin McCravy, Nov 06 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Quora, What is the next number in this sequence: 10, 20, 74, 100?
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001844.

LinearRecurrence[{1,2,-2,-1,1},{10,20,74,100,202},50] (* Harvey P. Dale, Mar 02 2023 *)

Vec(2*x*(5+5*x+17*x^2+3*x^3+2*x^4) / ((1-x)^3*(1+x)^2) + O(x^60)) \\ Colin Barker, Nov 10 2016

a(n) = (2n - 1 - ((n+1) mod 2))^2 + (2n + (n mod 2))^2.
From Colin Barker, Nov 10 2016: (Start)
G.f.: 2*x*(5 + 5*x + 17*x^2 + 3*x^3 + 2*x^4) / ((1 - x)^3 * (1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = 8*n^2 - 8*n + 4 for n even.
a(n) = 8*n^2 + 2 for n odd.
(End)

A258340 a(n) = (7^n + 3^n - 2)/8.

Original entry on oeis.org

1, 7, 46, 310, 2131, 14797, 103216, 721420, 5046661, 35316787, 247187986, 1730227330, 12111325591, 84778481977, 593446982356, 4154121702040, 29078830390921, 203551748166367, 1424862043454326, 9974033723049550, 69818234317954651, 488727634995505957

Offset: 1

Edwin McCravy, Aug 05 2015

This sequence appeared on an test given to job interviewers.

Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Cf. A074608.

[(3^n+7^n-2)/8: n in [1..30]]; // Vincenzo Librandi, Aug 22 2015

Table[(7^n + 3^n - 2)/8, {n, 1, 30}] (* Bruno Berselli, Aug 24 2015 *)
LinearRecurrence[{11,-31,21},{1,7,46},30] (* Harvey P. Dale, May 01 2018 *)

first(m)=vector(m,i,(3^i+7^i-2)/8) \\ Anders Hellström, Aug 20 2015

[(7^n+3^n-2)/8 for n in (1..30)] # Bruno Berselli, Aug 24 2015

a(n) = (A074608(n) - 2)/8. - Michel Marcus, Aug 20 2015
G.f.: x*(1-4*x)/((1-x)*(1-3*x)*(1-7*x)). - Vincenzo Librandi, Aug 22 2015
a(n) = 11*a(n-1) - 31*a(n-2) + 21*a(n-3) with n>2, a(0)=0. - Bruno Berselli, Aug 24 2015
a(n) = Sum_{k=1..n} A027907(n,2k)*4^(k-1) . - J. Conrad, Aug 30 2016

A252729 Start with 1, add 1, subtract 2, multiply by 3, add 4, subtract 5, multiply by 6, add 7, etc.

Original entry on oeis.org

1, 2, 0, 0, 4, -1, -6, 1, -7, -63, -53, -64, -768, -755, -769, -11535, -11519, -11536, -207648, -207629, -207649, -4360629, -4360607, -4360630, -104655120, -104655095, -104655121, -2825688267, -2825688239, -2825688268, -84770648040, -84770648009

Offset: 1

Edwin McCravy, Dec 20 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A252729 := proc(n)
    option remember;
    if n =1 then
        1;
    elif modp(n,3) = 2 then
        procname(n-1)+n-1;
    elif modp(n,3) = 0 then
        procname(n-1)-(n-1);
    elif modp(n,3) = 1 then
        procname(n-1)*(n-1);
    fi ;
end proc:
seq(A252729(n),n=1..32) ; # R. J. Mathar, Mar 20 2015

nxt[{n_,a_}]:={n+1,Which[Mod[n+1,3]==1,a+n+1,Mod[n+1,3]==2,a-(n+1),True, a*(n+1)]}; Transpose[NestList[nxt,{0,1},35]][[2]] (* Harvey P. Dale, Mar 29 2015 *)

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User: Edwin McCravy

A377207 Number of n-digit numbers where every digit is either a 9 or adjacent to a 9.

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A276764 1^2 + 3^2, 2^2 + 4^2, 5^2 + 7^2, 6^2 + 8^2, ...

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A258340 a(n) = (7^n + 3^n - 2)/8.

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A252729 Start with 1, add 1, subtract 2, multiply by 3, add 4, subtract 5, multiply by 6, add 7, etc.

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